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so welcome everybody welcome back  uh let's get started with today's  lecture  we  um  where were we  on tuesday  we  covered  uh the main topic was the recursion  theorem  which allows  programs to self-reference  and we saw some applications of that  too so  we gave a new proof that atm is  undecidable  we looked at this language of minimal  turing machine descriptions  and we had a short digression into  mathematical logic where we looked at  how one shows that there are true but  unprovable  statements in any uh reasonable formal  system  okay so today we're i'm gonna shift  gears entirely um and we're moving into  the second half of the course uh  where we are beginning a study of  uh computational complexity theory  and uh we'll  say a little bit about that during the  course of the lecture  um of course but um you know the main  things that we're going to cover  in terms of content that you will need  um is defining the  complexity classes and the class p  um so and we'll prove a few theorems  along the way but that's the main  objective of today's lecture  computability theory which was the  subject of the first half of the course  and which is what going what the midterm  exam is going to cover  um  was a subject that was an active area of  of mathematical uh study  in the first part of the 20th century i  really got  it really dates back um into the late  19th century in fact where people were  trying to figure out how to formalize  mathematical reasoning um  [Music]  but that really really got going um in  the 1930s with the work of girdle and  church and turing um  who um  really formalized for the first time  what we mean by algorithm and that that  allowed the study of algorithms to  really get started um with and it had  its impact as i mentioned on the actual  uh design building and thinking about  real computers  um  [Music]  the main question if you kind of boil  the subject down to a a single question  is is some language decidable or not  um  in complexity theory which got started  kind of  when uh computability theory more or  less wrapped up as a subject largely  because they answered many of the  questions that they they answered pretty  much all of the questions that they were  asking so there really aren't uh  interesting unsolved questions left in  that field  and you really need mathematical  questions to keep a subject alive  unsolvable questions so complexity  theory got its start in the 1960s and it  continues on as an activator area of  research to the present day  um  and i guess if you could boil it down it  would be is is a language decidable  with some restriction on the resources  such as the amount of time or memory or  some other or some other kinds of  resources that you might provide you  know randomness and so on although those  are followed within the area of  computational complexity theory  so let's get ourselves started with an  example  here is a language that we've looked at  in the past  a to the k b to the k  and um let's look at it from now from  the perspective of complexity  all of the languages that we're going to  be studying in complexity are all going  to be decidable languages um so the  question of undecidability  undecidability in complexity theory is  not really of interest it's all  decidable languages but the question is  how decidable  you know what sort of resources do you  need to do the deciding  um  so  um for this language a how many steps  are needed  well we're going to spend a little time  just kind of  setting up  the definitions of the subject um and  and motivating them so in the in this  for this language a when i ask how many  steps are needed well it's going to  depend upon which input you have in mind  you know some inputs might require more  steps than others  so the way we're going to set the  subject up which is the the the standard  way that uh people in this field look at  it um and i think you know that applies  to you know  lots of  examples outside as well  um is that we're going to kind of group  all of the inputs of the same length  together  and look at the maximum  uh  uh cost the maximum number of steps you  need to solve any one of those inputs of  a given length and we'll do that for  each length  and the way we're going to frame it is  in terms of giving a maximum or what's  called an upper bound  on  the amount of time that you need  uh to solve all of those inputs of  length n  that's what's uh sometimes called worst  case complexity i'm sure many of you  seen this already but just to make sure  we're all together on this  um uh you might contrast that for  example with what's called average case  complexity where instead of looking at  the most difficult case among all  inputs of length n you take the average  of all inputs of length n and then you  have to then it's a little bit more  complicated because then you need to  have a probability distribution on those  inputs and so on we're not going to look  at it in this course from that  perspective we're only going to be  looking at what's called worst case  complexity  um  so let's um  begin then uh uh by taking uh looking at  this in more detail and uh  taking as our  starting point the theorem that says  that on a one tape turing machine  which is deciding this language a  you know a to the k b to the k  um you can do this on a one tape turing  machine um  m we're calling it  in at most  some constant times n squared steps  for any input of length n  where  the constant is going to be fixed  independent of it  okay  so  this is going to be  you know  having a constant in in in in  factor in the complexity is going to  come all come up often and so instead of  saying this over and over again we're  going to use the notation um  that  m uses order in squared steps um i'm  sure many of you seen that terminology  as well but just just for the purposes  of making sure we're all together on  that there is this big o and little o  notation i'm expecting i'm expecting you  to be familiar with that  big o is when you apply applied to  functions as it's done  you say uh f is big o of g as  for two functions f and g  it's basically if f is less than or  equal to g  if you're ignoring constant factors  and you say f is little o of g  if f is strictly less than g if you're  ignoring constant factors that's kind of  one sort of informal way of looking at  it the precise definition is given up  there on the slide and if you haven't  seen it before make sure you look at it  in the book where it's all carefully  described and defined  so that you're comfortable with these  notions because it's really we're going  to be using this without any further  discussion uh from here on  okay  so let's get to the proof then of this  theorem that you can do uh the language  a in order n squared steps  not super hard if i think if i ask you  to come up with an algorithm to solve a  this would be the algorithm that you  would find basically  first you would start off by scanning  the input  w to make sure it's of the right form  so a run of a's followed by a run of b's  of some lengths  um and if it's not of that form  uh then you're going to reject right  away  the next thing you'll do is then go  through a repeat loop  um in the turing machine  and you know if you imagine here is your  machine here is the input w you're going  to go through that repeating repeatedly  of course you can do this in a number of  different ways but here's the way i i'm  i have in mind for you uh on this slide  anyway  we're going to scan the entire tape  crossing off one  a single a and a single b  um on a  on a scan  and then you're going to keep doing that  until you've crossed off everything  um  unless you run out of a's or you run out  of b's  in that case you're going to reject  um  if you run out of a's or b's before you  run out of uh the other type  then you know that you started out with  an unequal number and so the machine is  going to reject  if you've managed to cross them all off  without running out of one before the  other then you'll accept  i know this is kind of obvious but i  think it's important to get us all  together on this at the beginning  so here's a little um animation of the  turing machine doing that  i'm not showing the motion of the head  but imagine the head scanning back and  forth crossing off these a's and b's  one a and one b on each pass  until they're all crossed off and then  it accepts  unless of course it runs out of a's or  b's before the other type then it  rejects  okay  now let's do a very quick informal  analysis of how much time this has taken  so the very first stage i'm calling each  of these things stages of the turing  machine to distinct  distinguish them from the steps of the  turing machine which the individual  transition function moves  so this is the entire stage of the  machine  the very first stage takes order n steps  because you have to make a scan across  the input  and then you know i'm not giving all the  full detail of course you're going to  scan and then you're going to return the  head back to its starting position i'm  not that's just going to be an extra  factor of n an extra an extra  an extra n  um so that's what we're talking about  being order n steps for the very first  stage um  and then as you do through goes with the  repeat loop each time you go through the  repeat loop you're gonna cross off  one a and one b  so  um that's gonna mean you're gonna have  to do this roughly order you're gonna  have to do this order n times in order  to cross off all the a's and b's  so there's gonna be edit order and  iterations of this repeat loop  each one of them is going to again  require a state a scan so that's order n  steps uh for each one of the iterations  um so adding that all up  um  the very top row gives us the order n  and then we have the order and  iterations times order n steps is order  n squared steps  and so the sum of these two is order n  square order n squared steps  due to the nature of arithmetic when you  have the big o notation  the dominant term  overrides all of the others  so  that's  in a nutshell  how we well this is our very first  example of analyzing the turing machine  algorithm  for a language okay so let me  ask you now whether there is some other  turing machine some other one tape  turing machine which can do better  than this turing machine um does in  terms of how much time it takes  so one idea that you might have  instead of crossing off a single a or in  the single b maybe you can cross off um  two a's and two b's or ten a's and ten  b's well that'll cut down the running  time by a factor of ten  but from the standpoint of this theorem  it's still going to be an order n  squared algorithm  um and so that's not gonna that really  doesn't change  um  the amount of time used from from the  perspective of um from our perspective  where we're going to be ignoring the  constant factors on the running time  um  so i ask you here can you do better than  just improving things by a constant  factor  there we go  okay  so here's a check-in on this on this uh  on this problem on the problem a solving  deciding a on a one tape turing machine  can we do better  than  uh order n squared as i just described  in that uh  in that  theorem and that algorithm for that  theorem or um you know can you get it  down to n log n  um or maybe you can get it down to order  n  um what do you think  uh obviously you know we're  we're just getting started but  just um just make your best guess  and let me just post that for you as a  poll  what's most important to me is that you  understand  the terminology that we're using and the  way we're discussing with the way we're  talking about it because that's going to  be setting setting us up for the  definitions that we're going to come a  little bit later in the lecture  okay  so um i'm going to close the poll  um uh we're kind of all over the place  on it but that's that's that that's good  um  uh  since we haven't really covered this  material yet material yet  in fact b is the correct answer we can  improve this algorithm down to order n  log n  but not all the way down to order n um  so let me get let me show you how you do  it in order n log n on the next slide  and  so here  is a one tape turing machine that can  decide a by using  only order and log n steps  instead of order and squared steps so  this is a significant improvement um  so  i'm just i'll describe the turing  machine here again is the picture of the  machine on an input um  and the very first thing i need to say  is that we're going to scan to make sure  the input is of the right form  and now we're going to again it's going  to be  making repeated passes over the input  but we're going to do it a little  differently now  instead of crossing off a single a and a  single b or some fixed number of a's and  a fixed number of b's we're going to  cross off every other a  and every other b  in in that way we're going to  essentially cut the number of a's and  b's in half  and that's why the number of iterations  is only going to be a log instead of um  linear  okay so we're going to cross off every  other a and every other b  um  and at the same time we're going to keep  track  of the parity the even-odd parity of the  number of a's  um that we've seen and the s and the  parity of the number of b's that we've  seen um that have not yet been crossed  off  um and we're going to compare those  parodies to make sure that they agree  if they ever disagree  we know we started off with different  numbers of a's and b's  um  and i'll i'll illustrate this in a  second just to make sure you where um  we understand this algorithm  okay so  i'm going to write down as a little  table of the parities that we've seen  and i'm going to illustrate the  algorithm with a little animation  okay so again we're now going to  scan across crossing off every other a  and then every other b  but now but before we get to the b's we  observe as we cross these off that we  had six a's now i'm not saying we count  them we just keep track of the even odd  parity that can be done in the finite  control of the machine  counting them would be more complicated  but karen but just keeping track of the  parity is not is is something that the  finite automaton could do  so the parity in this case because they  were six is going to be even  now we cross off the b's  same even parity now we're going to  return the head back to the beginning  i'm obviously not showing the head  moving here we return the head back to  the big beginning and now we scan across  again crossing off every other remaining  a and counting the parity of the  remaining a's  so here now  it's going to be this one and this one  are going to get crossed off  and there were three a's so that was odd  parity  and the same for the b's  three b's odd parity  and now we return our head back to the  beginning  cross again off uh every other a and  every other b  every a  so that was odd parry there was just one  crossing off the b  odd parity because it's just one they  all agree  um so the machine is going to accept  um because everything is now crossed off  and and the parity is agreed along the  way  let me just say for a second you know  obviously if you ever get disagreement  on the parities then you know that the  two uh  the number of a's and number b's had to  disagree  um  but how do we know that if the parities  always agree that we actually did start  out with the same number of a's as b's  and that you could see that in a number  of different ways but perhaps you know  uh a cute way to see it  is there is actually if you look at  these parodies here the sequence of  parodies actually in reverse  so if you let's say odd odd even um and  you look at the binary representation of  the number of a's which is six so the  binary representation would be one one  zero  the fact that you get odd odd even and  one one zero  is not a coincidence in fact the  sequence of parodies in reverse that you  get  is um  exactly  um the same as the binary representation  you'd have to  you know confirm that with a little  proof it's not hard uh but um that once  you've confirmed that you can see that  um if the  sequence of parities agree then the  numbers have to be the same because the  binary uh representations agreed  okay so now getting to the analysis here  again order n steps to do the check  login iterations  each scan takes order n steps  so the total uh running time here is  going to be order n log n  because it's going to be the log n times  n that's where the log n log n comes  from  now questions you might ask could i do  even better than that can i beat  n log n by more than any constant factor  um so can i do little o  of n log n  um and the answer is no  this is the best possible  a one tape turing machine for this  language um  uh so a one day turing machine cannot  decide a by using little o  of n log n steps  we're not going to prove that  i'm not going to ask you to be  responsible for the proof but in fact um  what you can show is that any language  that you can do on a one tape turing  machine in little o of n log n steps  turns out to be regular  okay so um uh you can prove a rather  strong theorem here not super difficult  to to prove but i don't want to spend a  lot of time on  um  uh proving bounds for turing machines  because you know really the whole  purpose of setting this up using turing  machines is to talk about algorithms in  general algorithms in general i'm not  going to be focusing on the nitty gritty  of turing machines  um  okay uh so um  what is  my um  okay yeah so i wanted to just stop and  make sure that we were  we are together here um so  a brief pause and feel free to send me  any questions  um  okay this is a good question if we can  keep track of parodies why can't we just  keep track of the number of a's or b's  um  well you could keep track of the parity  in the finite memory and so you can do  that with effectively no work  but the finite memory cannot is not  enough for doing an account up to you  know some arbitrarily large number  now you could store the count on the  tape  but that's going to cost you time to  maintain that counter  and so um  that's not going to be  so  simple  as keeping track of the parity  in the finite memory  um  okay  somebody's asking if a star b star is  regular yes a star b star is regular so  what  um but a to the k b to the k which is  our language is not regular uh right so  that's the language we're looking at  um  okay  so  getting questions about what happens  with multiple tapes we'll talk about  that in a second um  yes we could do an order end steps on a  regular computer sure but uh  you know  right for this slide anyway we're  looking at one tape turing machines um  okay getting questions about a big o and  little o um  for something to be little  little o means it's less than any  constant factor times the function um  so you have to look at the definition in  the book i think almost  enough people in the class probably know  this and have seen this already i don't  want to spend everybody's time on it but  uh please review it in the book  somebody's asking if you need to store  the parody on the tape no you can just  throw it in the finite memory um  uh  i mean storing the finite memory seems  to be the simplest thing  okay when we move on please feel free to  ask questions to the tas as well we have  two tas at least here attending uh with  me so  um  good  um  right um  all right so  we cannot do better than a one-tape  turing machine on a one-tape turing  machine and order n log n um and that's  something we can prove that we're not  going to do it here  however if you if you have if you change  the model  for example you use a two-tape turing  machine then yes as a lot of you are  suggesting in the chat you can do better  than that  so  if we now have a two-tape turing machine  or a multi-tape turing machine you can  do it in order n steps and that's  actually the point i i'm really trying  to make here um  so if you have here your um  two tape turing machine  um  then  two tips  same language  now what we're going to do is copy the  a's to the second tape  that we can do on a single pass  um and then once the a's have been  copied to the second tape we can  continue on reading the b's  and match them off with the a's that  appear on the second tape  so in order n steps you can do the  comparison instead of order n log n  steps  and of course if they match you're going  to accept otherwise reject  so let's just see here's a little  animation uh  uh uh  demonstrating that of course it's very  simple  um  so here is that here that if you could  see that it came maybe a little too fast  here let's just show it again here is  the heads moving across and the a is  coming down uh to the bottom  and now the head the upper head is going  to continue on reading the bees the  lower head is going to  go back to the beginning on the a's  and matching off the b's with the a's  and that's how we can verify or check  that they are the same number  okay so now in this case they were the  same number so the machine would accept  if they were the different number the  machine would not accept  um  and the analysis is very simple  um  uh each stage here is going to take a  linear number of steps order n steps  um because it just consists of a single  uh scan there are no uh there are no  loops in this machine  no no no repeat loops  um  okay  so  um  question on this  um  all right  why don't we move on then um  now observe  and the point i'm really trying to make  is that  uh  on a one type turing machine you can do  it in n log n but not any better but on  a two tape turning machine you can do it  in order n  um  so there's a difference  in how  much  time you need to spend how many steps  you need to spend depending upon the  model  and that's significant for us um  so the number of steps depends on the  model one tape turing machine was order  and login multitape was order n  we call them model dependence  if you contrast that with the situation  in the complexities in the computability  section of the course  we had model independence  the choice of the model didn't matter  and that was nice for us because  the theory of decidability didn't depend  upon whether you had a one tape turing  machine or a multi-tape turing machine  it was all the same  set of decidable and recognizable  languages  uh  so we didn't have to worry about which  model we're actually going to work with  we could work with any model even just  an informal model of algorithm  um was going to would be good enough  because we're going to end up  with the same  notion  in the end  now that's  goes away  in complexity theory  now we have  a difference depending upon the model  and from a mathematical standpoint  that's a little less nice  because  um you know which model do you work with  if you want to understand the complexity  of some problem you know that you have  at hand  um now you have to make a choice you're  going to work with the turing machine  around how many tapes or you're going to  look at some other model and you're  going to get different results so it's  somewhat less natural from a  mathematical standpoint just to talk  about the complexity of some problem  but we're going to kind of um  bring back something close enough to  model independence by observing that  even though we don't have uh model  independence as we did in computability  theory we can limit  how much dependence there is  so the amount of dependence is going to  be low  as we will see  provided you stick with a reasonable  class of deterministic models  so the dependence though it does exist  is not going to be um  that much it's going to be polynomial  dependence  and we'll say exactly what that means in  a second  um and from our standpoint that's going  to be uh um a small difference uh a  negligible difference that we're going  to ignore  um  so we're going to focus on questions  that do not depend on the model choice  among these reasonable deterministic  models  now you may say well that's not  interesting from a practical standpoint  because polynomial differences say the  difference between n squared and n cubed  certainly make a difference in practice  but  it really  uh depends  on the  um  the uh  um  you know what what kinds of questions  you're focusing in on so if you want to  look at something that's very uh  a very precise distinction say between  you know n squared and n cubed um then  you might want to focus in on  which model you want to be working with  and that's going to be more the domain  of an algorithms class  but from our standpoint we're going to  be looking at other still important  questions but there are questions that  um don't uh depend upon exactly which  polynomial you're going to have we're  going to be looking more at distinctions  between polynomial and exponential  and still there are important practical  questions that arise in in that  somewhat different um  setting  okay  so with that in mind  we're going to continue to use the one  tape turing machine as our basic model  of complexity um  since the the model um among the  reasonable deterministic models in the  end is not going to matter from the from  the perspective of the kinds of  questions we're going to be asking  um  okay  um  so  with that so  we are going to continue then it's  important to remember that from going  forward we're going to stick with the  one tape turing machine model maybe  that's something you would have expected  us to do anyway but i'm trying to  justify that you know through this  little discussion that we had so far  thus far  so now we're going to start defining  things  with the one tape model in mind  um  so so first of all if you have a turing  machine we're going to say it runs in a  certain amount of time  so if t is some sort of time bound  function like n squared or n log n  we'll say the machine runs in that  amount of time like n squared or n log n  if  that machine m  always halts within that number of steps  on all inputs of length n  so it always holds within t of n steps  on inputs of length n  then we'll say that the machine runs in  t of n time  so in other words if the machine runs in  n squared time  then the machine when you give it an  input of length 10 it's gotta it's got  to be guaranteed to hold within a  hundred steps  uh 10 10 squared 100 steps on every  input of length 10.  um that's what it means  um for the machine to be running in that  much time and it has to do that for  every every n for every input length  and with that we're going to come to the  following definition which is  highlighted in color because it's going  to be um  we're going to be using this definition  throughout the semester so it's  important to understand it  this is the definition of what's called  the time complexity classes  and what i'm going to do  is take some bound  t of n and again think of t of n like a  bound like n squared  so if you have time t of n or like time  n squared  that's going to be the collection of all  languages that you can decide  within time and square or within time t  of n  so in other words it's a collection of  all languages b  such that there's some  one tape turing machine here we're  focusing again on the one tape turning  machine there is some deterministic one  tape turing machine  that decides b  and that machine runs in that amount of  time  so this is a collection of languages  the time collect complexity class is a  set of languages i'm going to draw it  now as a diagram  um  so if you take the language again that  we've been  using as our standard example a to the  kb to the k  um  that's in time and log n as we observed  for uh uh  two slides two slides back or three  slides back  so on a one tape turning machine you can  do this language a in time n log n  so it's in the time complexity class n  log n this this  region here captures all of the  languages  that you can do  in order and log in time  for example  that also includes all of the regular  languages  why is that  well any regular language  can be done on a one tape turing machine  in time order n  because the turing machine only just  needs to scan across doesn't even need  to write just need to scan across from  left to right on the on the tape and in  n steps it has the answer  so all of the regular languages are  actually in time n certainly a subset of  time n log n and these all form a kind  of a hierarchy so if you increase the  bound you can imagine that the collect  class of languages grows  as you  allow the machine to have more and more  steps to do its computing  um so these are all the languages that  you can do in n squared  uh order n square time on a one tape  turing machine  and cube time on a one tape turing  machine and so on  to exponential time two to the end time  on a one tape turning machine these are  all collections of languages  um  getting larger and larger um as we  increase the bound um  so someone is asking kind of is  okay uh let's see let me let me get to  some of these questions here i'll try to  get them in order um  uh  okay um  so somebody says if you have a good  question if you have a regular computer  so an ordinary sort of random access  computer so we'll talk about that in a  second can do it in order and uh can you  do it on a multi-tape during machine  also in order and time actually i don't  know the answer teller offhand i suspect  the answer is no  that  ordinary computers have a random access  capability that turing machines do not  and so that there are going to be some  examples of problems that you can do  with a random with a regular computer  that you cannot do with a multi-tape  turing machine in order end time uh i'd  have to double check that though so  we can and so some question  what what we believe is true and what we  can prove to be true um as we'll see  there are a lot of things that we  believe to be true in this subject that  we don't know how to prove  um  somebody's asking kind of an interesting  sort of more advanced question is there  some function f  uh some some some some some function t  where um it's so big that it so that  time t of n gives you all of the  decidable problems  um  it would be a very big t but the the  answer actually to that question is yes  um but  that's a little bit exotic so let let's  not spend a lot of time on that right  here but i'm happy to talk about that  offline  um  it's a good question here somebody's  asking me does it mean that there are no  languages between order n and order n  log n because i pointed out that  anything below n log n is going to be  regular  and so as soon as you get below n log n  you can do it in order n and yes there  is what's called a gap  between order n and order n log n on a  one take turing machine this you don't  get anything new from order n until you  jump up so from order n to order n log  log n  nothing new shows up  um  so we'll talk about those kinds of  things a little a little bit down the  road when we look at actually the  relationship among these various uh  classes and what we call a hierarchy  theorem which shows you know what how  much bigger do you have to make the  bound in order in order to be sure  you'll get something new  um  all right  [Music]  okay somebody's asking is there a  model which has the same time complexity  as a normal computer  uh  well i mean there's the random access  model which is supposed to capture a  normal computer um  okay so  let me let me um  these are all great questions kind of  more  uh  riffing off of  this into in more into more advanced  directions  uh  let's move on um here's another check-in  um suppose we take this is a little bit  of a  check to see how well your how  comfortable you are with the notions  we've just presented and whether you can  think about some of the arguments that  we've made and apply them in into a new  language so take the language ww reverse  um  you know strings  followed by their  um  uh this followed by themselves backwards  this language via the even length  palindromes if you will  um what's the smallest  uh bound that you need  uh to be able to solve  uh  that language b  and i'll pose it as a  pose that is a question for you  so which which time complexity class  is  um  that language b in  is it time  order n or n log n n squared  so on what do you think  so we're about to come to the coffee  break so uh why don't we um  uh  i'll answer any questions that come up i  think we're  got everybody uh answered so i'm gonna  end the polling okay make sure you're in  if you want to be in  um  so  the correct answer is in fact uh order n  squared  um  you it would be hard for you know you  you know every  reasonable guess here would be order n  log n i mean you can come up with the  same procedure  that the as the one we showed at the  beginning uh the order n squared  procedure for eight to the kb to the k  works for ww reverse as well um you can  just cross off  sweep back and forth crossing off  um  the symbol from w  and going across to the other side  crossing off the symbol from w reverse  and that procedure will give you an n  squared an order n squared algorithm  um  you might imagine you can improve it to  order n log n but you cannot you can  prove that order n squared is the best  possible um  i  i'm a little unhappy that a lot of you  came up with order n frankly because i  already told you that order n is uh  these are just regular languages  anything that you can do in less than  little o of n log n is going to be  regular and we know this language is not  regular so this is a this was  uh  not a good answer  you know so  please pay attention  and um okay  so let us  stop sharing uh i will turn now to the  uh our our  our break  uh for five minutes  and i'm happy to you know try to take  questions along the way as we're um  uh we're waiting for the time to  to end so let's see let me  put this up here  okay  let me try to take some of your  questions  um  so someone has asked me about quantum  computers as reasonable models of  you know  you may say a quantum computer is a  reasonable model of computation  and that's fine i would not say it's a  reasonable model deterministic  computation um  uh at least from our standpoint though  let's not quibble about the words um  i'm not including quantum computers in  um the collection of  um machines that i have in mind right  now when i'm talking about the the the  reasonable models of deterministic  computation that we're going to be we're  going to be discussing  um  uh  let's see  okay a bunch of people apparently are  asking the tas  why all regular languages can be done in  order n  um  [Music]  so  if you  think about a dfa  which you know  processes  a  an input of length n  within steps  and the dfa is um  going to be a type of turing machine  that never writes on its tape  uh so if the dfa can do it in in in n  steps the turing machine can do it n  steps and so therefore every regular  language can be done in order and steps  on a turing machine um  not sure where the confusion is so  please  message me if you're not still not  getting it um  okay somebody's saying why are we using  one-take turing machines instead of  random access wouldn't it be better to  use the random access machines  um if you were using if you're trying to  do algorithms yes  that's a more reasonable model we're  trying to prove things about the  computation and from that standpoint we  want to use as simple as model as  possible  uh trying to prove things  um using random access computers it's  possible be very messy  so that's why we don't use uh random  access machines to prove the kinds of  things we're going to be proving  um about computation  you know that that are really the meat  and potatoes of this course  so uh  i mean there's  compelling reasons why you would want to  use a simple model like a turing machine  but not a  powerful model like a random access  computer  so somebody's asking me does  um  the class  the time uh order and log log n have any  elements yes it has all the regular  languages but nothing else order n log  log n is only read only the regular  languages you have to go all the way up  to or and log in before you get  something non-regular  uh somebody's asking me are we going to  talk about how the random access model  works no  you know there's uh beyond the scope of  this course  outside of what we're going to be doing  we're going to talk about turing  machines  not because we care so much about turing  machines but i'm trying to prove things  about computation and the turing  machines are a convenient vehicle for  doing that um  okay our candle has burned out uh why  don't we return then to um  the next uh slide so everybody come back  so this answers one of the questions i  got on the chat is what's what actually  is the dependency between multi-tape  turing machines and one tape turing  machines can we bound that in general  yes we can  we're going to show  that converting a multi-tape turing  machine to a one-tape turing machine can  at most blow up the amount of time  that's necessary by a squaring  um no i acknowledge it's a lot but it  still allows you um  uh but you know it's still small  compared with an exponential increase  and we're going to be focusing in this  course  on things like the difference between  polynomial and exponential not between  the different not not between the  difference uh of  n not between  not the difference between n squared and  n cubed that's going to be less of a  fact less of an issue from us  uh  okay  um  [Music]  so  the way i'm framing  uh  showing this theorem is that if you have  a multi-tape turing machine that can do  a language in a certain amount of time  then it's in the time complexity class  of that time bound squared  and the way i'm just saying that is  because this is the bound that's  utilizing the one tape model so another  way of saying that is converting  multi-tape to one tape  squares the amount of time you need at  most  so the way we're going to  prove that is  simply by going back and remembering the  conversion that we already presented  from multi-tape to one tape  and observe that if we analyze that  conversion it just uh ends up squaring  the amount of time that the multitape  used  okay so  why is that  so if you remember  let's just make sure we're all  altogether on this the way the single  tape machine s  simulates the multi-tape turing machine  m  is that it takes the contents of each of  m's tapes  you know up to the place where there's  infinitely many blanks obviously you  don't store the infinite part but the  the the the active portion of each of  them's tapes you're going to store them  consecutively in separate blocks on s's  tape on estes only tape  um and now every time m makes one move s  has to scan its entire tape to uh to uh  to  see what's under each of the heads and  to do all the updating  so to simulate one take step of m's  computation s is going to use order t of  n steps  where t of n is the total running time  that m is going to use so why is t of n  steps coming up here well that's because  you have to measure how s is going to  make a scan of course its tape  how big can its tape  be  well m  if it's trying to use as much tape as  possible  can use at most  t of n  um  tape on each of the t event cells on  each of its tapes so all together  there's just going to be some constant  number of times t of n cells on s's tape  do you see that  so each one of these is going to be at  most t of n long so this all together is  going to be order t of n log because you  know what can m do it could send its  head out  say the head on this tape here moving as  fast as possible to the right using as  much tape as it can but you can only use  t event cells in t of n time  so this is going to be ordered t of n  so one step of m's computation is going  to be t of n steps on s's computation  but m itself has t of n steps so it's  going to be t of n times t of n for the  total number of steps that s is going to  end up using  and that's where the squaring comes from  okay  similar results i'm not going to do lots  of simulations of one model by another i  think that you'll get the idea and you  can if you're interested you can uh  study those on your own but you can  convert multi-dimensional turing  machines to one one tape turing machines  um you know one tape ordinary uh  linear  one tape  one dimensional uh machines  and  the bottom line is that among all of the  reasonable models they're all what are  called polynomially related  uh if each can simulate the other with  at most the polynomial overhead  so if one of the machines can use this t  of n time the other machine that's  simulating it would use t to the k of n  time for some k  um that's what it means for the two  machines to be polynomially related and  all reasonable deterministic models are  polynomially related  so as we've already seen one tape in  multi-tape turing machines are  polynomial related because converting  multi-tape to one tape blows you up by  most squaring  so k equals two in this case  um multi-dimensional turing machines  again polynomial related  the random access machine which i'm not  going to define but it's the machine  that you might imagine you know you  would i'm sure they must define in some  form in the algorithms classes  polynomial related cellular automata  which are just arrays of a finite  automata that can communicate with each  other similar similarly  um  you know all the reasonable  deterministic models  um you know again classical models i'm  not talking about quantum computing  are polynomial polynomially related  so we are um  uh that kind of justifies our  uh choice in picking one of them as long  as we're going to ask questions which  don't depend upon the polynomial  so  let's then talk about the class p  so the class p this is an important  definition for us  this is um  the collection of all languages  that you can do  in time into the k  for some k  on a one take turing machine  or as i've written it over here i don't  know if this notation is  um unfamiliar to you but this is like  just a big sum  um but here it's a big union symbol  it's union over all values of k  of the time class n to the k  so this is  time n union to time n squared union  time n cubed union time into the fourth  and so on  um and we call these the polynomial time  decidable languages  okay so we're going to be spending um  you know uh a certain amount of effort  uh  exploring this class p and other similar  classes  somebody's asking why is it a union i'm  not sure how else you would write it so  if somebody hasn't if you have a  proposal for a different way to write it  that's fine but this is you know k this  is for all k  i don't know you know there's if you  only had a limit finite number of k's  you could just take the biggest one but  since it's for all k you need to write  it as a union  um  now i want to argue that the class p  um is an important class  and  you know why is it had so much impact on  the subject  um and in terms of applications as well  um  so one thing is that  the class p is invariant for all  reasonable deterministic models and what  do i mean by that  so we have defined the class p  in terms of these time classes here  which in turn are defined in terms of  the one tape model  so we have defined p by using one tape  turing machines  now if we had to find p  in terms of multi-tape turing machines  we get exactly the same class  because  one tape and multi-tape turing machines  are polynomially related to one another  and since we're  taking the union over all polynomials  that polynomial difference is going to  wash out  um  similarly we could define p  using any of the other reasonable  deterministic models we get exactly the  same class  so in a sense we get back what we um the  situation that we had in computability  theory when  the class of decidable languages  didn't depend on the choice of model  here the class p does not matter  depending upon the choice of reasonable  deterministic model  um  and we also kind of even in the case of  computability theory we have to stick  with kind of reasonable models that  cannot do an infinite amount of work in  one step  um  so  uh  you know when i'm not going to define  what it means to be reasonable and  that's in a sense an informal notion  but  among all of those reasonable models  you're going to get the same class p  the other thing that makes p important  is that  p roughly corresponds  to  the problems that you can solve in some  reasonable practical sense  not exactly you know problems that  require into the hundredth time  um you could argue  cannot be uh solved in any reasonable  sense um  but  uh you know if you think about it for  example from the perspective of  cryptography  um  uh cryptographic codes that people come  up with are typically designed to  require or or the hope is that they  require an exponential amount of effort  to crack if someone found even an into  the hundredth algorithm to crack uh  that would crack a code  people would feel that the code is not  secure even though into the hundredth is  still a large um  so you know it's a it's a rough kind of  test but it's still used as a kind of  litmus litmus test for practical  solvability if you can solve it in  polynomial time you basically figured  out how to avoid  um  large searches  if you can solve problems in in  polynomial time we'll say more about  that later  but you know  what i want to bring out here is that we  have combined here in the class p  something that's mathematically nice  mathematically element elegant with  something that's practically relevant  and when you have a combination of the  two then you know you have a winner then  you know how you have a concept that's  going to make a difference and that's  been true for the class b this has been  very influential  within and without  the theory of computation  um  so let's look at an example  um let's let's define a new language we  haven't seen before though it's similar  to  procedures that we've looked at before  the path language  which is where i'm going to give you a  graph g  two nodes in the graph s and t  um  where i'm thinking of g is a directed  graph so directed means that the um  the connections between the nodes in g  are going to be  redirected and that they have arrows on  them they're not just lines but they  have an orientation with an arrow  um  so g is a directed graph that has a path  from s to t that respects the directions  um so such a i think i might even have a  picture here yeah so imagine here here  is your graph if you can see it there  are little arrows connecting the uh the  nodes and i want to know is there a path  from the node s to the node t  so that is  you know a picture of a problem in you  know an instance of a graph of a path  problem  and i want to find uh an algorithm for  that and i can show that there is an  algorithm that operates in polynomial  time  for this uh path problem  okay and the algorithm uh you know any  of the standard  searching algorithms  uh would work here but let's just for  completeness sake um include the  breadth first search algorithm that we  have explored previously when we talked  about finite automata um so we'll mark s  and they'll keep repeating until nothing  new is new is marked and we'll mark all  of the nodes that were reachable  by a single arrow from a previously  marked node  um and then cft is marked  after you could mark everything you can  get to so you're going to mark  let's see pictorially i think i have  this in indicated yeah you're going to  mark all of the things that are  reachable from the start not from the  from the node s  and then see  after you can't mark anything new  whether the node is marked and if it is  you'll accept if it is not you reject  okay um  now we can analyze this too um and i'm  not going to be spending a lot of time  analyzing algorithms here um but let's  just uh do the do it kind of this one  time uh again  um  we're gonna we're get we're doing a  bunch of iterations here so gonna be  repeating until nothing new is marked so  each time we mark something new we can  only do that at most end times  at which point we've marked everything  so the number of iterations here is  going to be at most n  and now for each each time we've marked  something we have to  um  look at all of the previously marked  nodes  um and see which things they point at to  mark them too  so this is going to be an inner loop  which again has at most n iterations  because it's going to all of the  previously marked nodes and then once we  have  that we can scan g to see  to mark all of the  new  uh  all of the nodes which we have not yet  marked uh whether they're connected with  a previously marked node by an edge  and i'm being generous here because i  don't really care  um this can be done in the most n  squared steps on a one-take turing  machine i'm not going to describe the  implementation but you know i'll leave  it to you as an exercise but this is  straightforward  so the total number of steps here would  be n iterations times n iterations times  and squared so you're going to be at  most into the fourth steps needed so  this is a polynomial algorithm and  whether i ended up with n to the fourth  or n to the fifth or n cubed i don't  really care um because i'm just trying  to illustrate um that  the total is polynomial  and that's all i'm going to be typically  asking you to do um so to show  polynomial time what i'd be asking you  to do is to show that each stage  each stage of this algorithm  should be clearly polynomial  and that the total number of stages i'm  sorry this should say stages here should  be polynomial  so each stage is polynomial and after  you're doing all the iterations the  total number of stages that are executed  is polynomial and so therefore all  together the total running time the  total number of steps is going to be  polynomial  okay so that's um  the  uh the way  we would write up um polynomial  algorithms in this class um  so  let's see if there's any questions here  i don't want to  get too far ahead of people  um  let's see  um  yes in this theorem i'm talking about  one tape turing machines  um because we're defining everything in  terms of one type machines but now at  this point you know we're talking about  polynomial time we can you know my  analysis is based on one tape turning  machines  but in general you know you could use  any reasonable deterministic model on  which to carry out your analysis uh  because they're all polynomially  equivalent so from the perspective of  coming up with  showing that a problem is in polynomial  time isn't p  um you can use any of the models um  you know that you that you wish that for  convenient okay oh that's a good  question what is n thank you for asking  that question  n  is always going to be  reserved  to indicate the length of the input  so here  n  is going to be when we encode g s and t  and here also i haven't said this but  i'm assuming that  the encoding that you're using is also  somehow reasonable you know i think  we'll talk a little bit about more about  that in the next lecture um which is  going to be after the midterm  um  but you know you can  cause problems if you intentionally try  to come up with nasty encodings which  will you know  represent things uh  with unnecessarily many characters but  um  you know if if you try to be reasonable  then  um  you just use any one of those encodings  and you'll be fine  um so yeah  um so n is the length of the  representation of the input  um  let's see  someone's trying to you're trying to dig  into the my the actual how this is  running here scan g to mark all y where  x y is an edge  i'm saying that you can do an n squared  steps  um  uh  um  you know  if you have x  you can mark it in a certain place on  the tape  and then as you're going through every  other node every other edge um you can  you can go back and compare x with the x  of the edge  and then c  and then find the y  i mean it's just going to be too messy  to talk about here i mean i i'll leave  it this is an exercise to you i'm not  going to try to fumble my way through  explaining why you can do this in n  squared steps but it's it's not hard  um uh  you guys are all like  hardcore algorithms folks you want to  know the algorithms for this i'm not  going to do that sorry uh  high level picture here um  you know so so you know  if you want  you know if you want to look at  detailed analyses this is not the right  course for you um  can k equal n  what is k  um  um  [Music]  no  this if you're talking about this k here  k cannot equal n we're not looking at n  to the n um this is these are all fixed  k  um it's like n squared and q but not  into the end is going to be an  exponential bound and so that's not  going to be included within uh this  union  okay so we're near the end of the hour  um  i'm gonna  introduce one last language here  um called handpath  and the handpath problem  is um  i'm going to ask now again for a path  from s to t  but now a different kind of a path one  that goes through every node of g along  the way  um so i'm looking for a path that goes  that hits every note of g not just the  shortest most direct path but in a sense  the most indirect pair the longest path  that goes through from from s to t  that visits everything else along the  way um  a path that  of that kind that hits every node of the  graph is called the hamiltonian path  um  because the mathematician hamilton  studied um those made some definitions  about that i'm not gonna  i don't actually know the history there  but i just know they're called  hamiltonian pets  um  so here's a picture i want to get from  snt but i want to sort of pick up  everything else along the way  so as you remember the path problem  itself can be solved in p  and what i'd like to know  can the simple modification where i'm  asking you to visit everything else  along the way is that problem also in p  okay and i'm going to put it pose this  as a checking for you  um but actually before i get to that let  me give you an algorithm for handpath um  that doesn't work to show it's in p  um because it's exponential  so  here's an algorithm  for handpath  um  uh where let's m be the number of nodes  in g um  and what i'm going to do is i'm going to  try every can every  possible path in g  um and see if it's actually works as a  hamiltonian path  and except if it is  um  and then  if all paths fail uh then i'll reject  so i'm going to try every possible  routing through g if you want to think  about it  think of  m as every possible permutation of the  nodes of g and then you're going to see  whether that's actually constitutes a  path in g  um that takes you from s to t  and goes through all of the nodes um  so this algorithm would work  this would give you  a correct algorithm for the hand path  problem the problem is  the difficulty is that there are so many  possible paths that it's going to take  you an exponential number of steps to  execute this algorithm it's not a  polynomial time algorithm because there  are many possible pairs that you could  go through um if you're looking at you  know with a very crude bound but you  really can't improve that significantly  there would be um m-factorial which is  going to be  you know much greater than 2 to the m  paths of length m um so the algorithm is  going to run for exponential time and  not polynomial time  so my question for you is i'm going to  pose it as a check-in problem is whether  you could actually do this problem in  polynomial time  so why don't you think about that as i'm  setting up the question  so take the hand path problem  just like the path problem which i  described with that marking algorithm  but now you want to  hit every node along the way  can you show that problem is solvable in  p  and there's a whole range of  possibilities here where either the  answer is yes you can see what the  polynomial time algorithm is  uh  to  uh to definitely know where you can  prove there is no such polynomial time  algorithm  and i'll put this as a checking for you  and see i'm curious to see what you come  up with  most people are getting it wrong  well i mean wrong i don't you know i'm  not clear what wrong is here  um  okay are we done  please uh check something  you can see a few of you have not  answered but  the poll is running out  okay  time is up  um  so  uh in fact  you know as as i think many of you know  but not all of you uh this is an  unsolved problem this is a very famous  unsolved problem which is equivalent to  the p versus np problem that we're going  to be talking about very soon which  among other things would be worth a  million dollars if you solve it  so for those of you who have answered a  or e please talk to me after lecture um  and maybe we can work on it together  um  [Music]  uh no so  um  yeah i  i think most people would believe that  the answer is no um  uh but no one knows how to prove it at  this time uh so i'm interested in the  uh folks who have uh come up with uh  what they think are solutions um and i  should say that there are some folks who  um believe that there might be other  outcomes besides uh just a simple note  um  which might be proven eventually  um  so we're going to talk more about this  um but this is the answer to the  question just for your uh just to make  sure you understand is that it's an  unsolved problem right now so we don't  we don't know uh  definitely yes and definitely no at  least according to the state of  knowledge  of which i'm aware  is not um are not correct answers um but  any of the others  well who knows  um so  i think that's the end of what i had to  say for today  we covered complexity theory as an  introduction looked at different  possible models focused on the one tape  model  introduced based on the one tape model  these uh complexity classes the class p  and we  showed an example of this path problem  being in p  talked also about this hand path problem  which we'll talk about more um  after the midterm  okay so um i'll stick around for a few  minutes uh if you have any further  questions otherwise um  uh  okay so let me just take questions here  um  somebody's asking me about my personal  opinion on p versus np  um  my personal opinion on people versus np  is that p is not equal to np  and that we will prove it someday  when i was a graduate student back in  the mid 70s  i thought it would be solved by now and  in fact i made a bet  with len adelman  who i subsequently ended up becoming the  a of the rsa code  um that we would solve it by the year  2000 and i bet  what was then a lot of money for me  which was an ounce of gold  which i didn't end up then which i did  end up paying off to lend in the year  2000 so  i'm not making any more bets but i still  believe that it will be solved uh  hopefully i'll get a chance to see the  solution  um i've spent a lot of time thinking  about it myself  you know  obviously i haven't solved it otherwise  we would know but um  uh  you know uh hopefully somebody will  i'm getting asked a question here that's  kind of an interesting question but i  don't really know  um  i'm sure i understand it what's the  largest possible run time of a decidal  decidable problem  what is the larger largest decidable  runtime so you  anything that i can describe  can be um  there are going to be  algorithms  that run for longer  uh  you can  define  an algorithm you can define a runtime  which would in a sense  beats all other runtimes  which  so that anything any runtime is going to  be um  dominated by that  extremely slow runtime but it's not  something that one can describe  um i can describe it to you by perceive  mathematical procedure but there's it's  not going to be something like you know  uh  2 to the 2 to the n  somebody's here proposing  they prepare a solution to the hand path  problem  i presume in polynomial time  why is the following flawed if s goes  through all nodes and ends up at t  well s will not you mean you mean i  presume you mean starting at s if we end  up going through all nodes and  and end of t  um the proposal is a little complicated  here  basically you you want to if if i can  try to rephrase it you want to try to  from every possible node you want to try  to calculate a path to t  and also a path from s to that node um  and you can do that for all possible  nodes but there's no way to really  combine them into a single path  that visits all you know solving for  each node separately is not going to do  the trick  because you have to somehow combine all  that information into a single path  just one path that goes from s to t and  visits all the other nodes along the way  and that that is not uh  i i don't see how what your proposal um  how that's how that's going to actually  work  professor spitzer yes  um a question on  the um  like we were kind of talking about  earlier but like what we talked about  today was defined for the one tape turn  machines correct  yep  so um  and you said like we could have applied  for the multi multitape ones but um  are  like  is  well i don't know if it's we talked  about earlier but if something  is accepted by the one tape turn machine  like can be applied to the multi-tape  turn machine and vice versa like are  they interchangeable like that  well they're interchangeable only in the  sense that  the amount of time that you would need  it's a different machine if you have a  multi-tape turing machine for some  language  um you can convert it to a one-tape  turing machine using a procedure that we  described it earlier in the term  and she'll get a different machine it's  going to run for a different amount of  time  by using their procedure the point is is  that the amount of time that the  one-take turing machine is going to run  for is not that much  worse  than the multi-tape turing machine's  time so if the multi-tape turning  machine's time was  you know  uh end cubed  the one tape turning machines uh is  going to be the square of that so it's  going to be into the sixth okay but it's  not going to be it's not going from  multi-tape to one tape is not going to  convert you from polynomial to  exponential that's that's the only point  i'm trying to make it's going to convert  from one polynomial to a somewhat bigger  polynomial but it's still going to leave  you polynomial  [Music]  i don't know you don't seem to  i i can't see your face no no yeah i  guess i guess my opinion is that like  especially when we were talking about  earlier um when you're bringing it up it  just seemed like you could just turn  anything to a multi-tape turn machine  and like completely like cut the timeout  like if i had like something to n log n  like if i do it in a multi-tape turn  machine i have it in like  um like big o of n  yeah i mean  like it just seemed like the multi-tape  was so much more powerful but then  i guess  i guess not with the like with the  explanations and like the  the models we were talking about today  yeah i would not say the multi  multi-tape turing machines are still  pretty limited in their capabilities and  they're  um  you know don't forget when you have a  multi-tape turing machine you have only  a fixed number of tapes i mean you could  also define variations of multi-tape  turing machines that have increasing  number of tapes  as the input uh  you know under  you know either under program control it  can launch new tapes or it could um  um you know just have more tapes  depending upon the size of the input  that would also be a possibility that's  not the model that we have defined but  you could define a model like that  um  but as long as the total amount of work  being done by the machine at any step  it's going to be a polynomial amount of  work  um  then you can convert it to a one tape  turing machine uh with only a polynomial  increase in the bound all right so  you know you want to be careful of  machines you know you know one one thing  i meant to say but didn't say you know  so he would be an unreasonable model  um which you might think of as a  plausible model but it's not going to be  a reasonable model from our standpoint  and that would be a model um for example  that can do  um  full precision say integer  arithmetic um  uh  with a unit cost per operation so each  operation counts as a costs one  um  but i'm going to allow you to do for  example  you know addition and multiplication  uh the the thing that's bad about that  um in terms of being uh unreasonable  is that  uh  after k step each time you do a step  you could double the size of the integer  by by squaring it  yeah so after k steps you could have an  integer which is 2 to the k long  and now doing operations there is going  to involve an exponential amount of work  even in any reasonable sense  um you know in a theoretical sense and  also in a practical sense  so um  you know a model that operates like that  is not going to be able to convert to a  one-tape turing machine with only a  polynomial  increase because it's doing an  exponential amount of work potentially  you know uh within a polynomial number  of steps  um  so that's within a linear number of  steps within n steps  so that's um an example of an  uh  you know of an unreasonable  deterministic model yeah  thank you  sure  so i'm just curious some idea just  occurred to me um  i guess like if you had an oracle  turing machine uh basically just so that  you could look at a uh  a turing machine description and decide  whether it's a decider or not yeah um  then it'd be a little bit interesting to  think about uh  what happens if you look at  all uh you know if you have a  description of a pair of a turing  machine and an input string and you can  look at for all size n  descriptions of a pair uh like what's  the most steps that it takes  uh  for a such a machine  to converge  so then so then you'd have combined  turing machine and input stream  descriptions  uh where you can look at what's the  longest it takes to converge it's just i  don't know just a random thought that  occurred  yeah so we we actually we're gonna we we  will talk about oracle turing machines  uh later on in the term um  you know these are machines that have  access to sort of free information um  and that actually turns out to be uh  i mean  there's some interesting interesting  things you can say about what what  happens when you're providing a machine  with in a sense information for free  that  you know  you might otherwise want to charge it  for actually computing that information  but let's just say we're going to allow  it to  uh get that information without being  charged  and then how does that affect the  complexity of other problems for example  um  and so uh we will talk about that later  but  too much if i think of a digression at  this moment to try to define all that um  but happy to chat with you about it on  piazza if you if you want to raise a  question there i think  sure oh no problem  um so he's asking me about strategies  for solving the p versus np problem you  know we will talk also talk about that a  little later in the term as well  but  you know asking you know clearly  you know it seems beyond the reach of  our present techniques  to be able to prove that some problems  you know really take a long time like  the hamiltonian path problem seems like  you know there's nothing really you can  do that's significantly better than  trying all possibilities as i described  in that exponential algorithm you know  on the last slide uh but  um  how do you prove that nobody knows um so  um  lots of people have tried  including yours truly i mean i spent a  lot of time thinking about it having  succeeded with it but um  there is uh  um  you know  somebody i i think somebody someday  someday somebody will come up with the  the new idea that's needed to solve it  but it's going to clearly take some sort  of a breakthrough some sort of a new  idea it's not just going to be a  combination of existing ideas  the existing methods  um  i think we're about 10 minutes past a  few of you are still here i'm going to  say goodbye to you folks and um  um  shortly i'm going to join my tas for our  t weekly ta meeting so see you guys  thanks for being here

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so welcome everybody welcome back  uh let's get started with today's  lecture  we  um  where were we  on tuesday  we  covered  uh the main topic was the recursion  theorem  which allows  programs to self-reference  and we saw some applications of that  too so  we gave a new proof that atm is  undecidable  we looked at this language of minimal  turing machine descriptions  and we had a short digression into  mathematical logic where we looked at  how one shows that there are true but  unprovable  statements in any uh reasonable formal  system  okay so today we're i'm gonna shift  gears entirely um and we're moving into  the second half of the course uh  where we are beginning a study of  uh computational complexity theory  and uh we'll  say a little bit about that during the  course of the lecture  um of course but um you know the main  things that we're going to cover  in terms of content that you will need  um is defining the  complexity classes and the class p  um so and we'll prove a few theorems  along the way but that's the main  objective of today's lecture  computability theory which was the  subject of the first half of the course  and which is what going what the midterm  exam is going to cover  um  was a subject that was an active area of  of mathematical uh study  in the first part of the 20th century i  really got  it really dates back um into the late  19th century in fact where people were  trying to figure out how to formalize  mathematical reasoning um  [Music]  but that really really got going um in  the 1930s with the work of girdle and  church and turing um  who um  really formalized for the first time  what we mean by algorithm and that that  allowed the study of algorithms to  really get started um with and it had  its impact as i mentioned on the actual  uh design building and thinking about  real computers  um  [Music]  the main question if you kind of boil  the subject down to a a single question  is is some language decidable or not  um  in complexity theory which got started  kind of  when uh computability theory more or  less wrapped up as a subject largely  because they answered many of the  questions that they they answered pretty  much all of the questions that they were  asking so there really aren't uh  interesting unsolved questions left in  that field  and you really need mathematical  questions to keep a subject alive  unsolvable questions so complexity  theory got its start in the 1960s and it  continues on as an activator area of  research to the present day  um  and i guess if you could boil it down it  would be is is a language decidable  with some restriction on the resources  such as the amount of time or memory or  some other or some other kinds of  resources that you might provide you  know randomness and so on although those  are followed within the area of  computational complexity theory  so let's get ourselves started with an  example  here is a language that we've looked at  in the past  a to the k b to the k  and um let's look at it from now from  the perspective of complexity  all of the languages that we're going to  be studying in complexity are all going  to be decidable languages um so the  question of undecidability  undecidability in complexity theory is  not really of interest it's all  decidable languages but the question is  how decidable  you know what sort of resources do you  need to do the deciding  um  so  um for this language a how many steps  are needed  well we're going to spend a little time  just kind of  setting up  the definitions of the subject um and  and motivating them so in the in this  for this language a when i ask how many  steps are needed well it's going to  depend upon which input you have in mind  you know some inputs might require more  steps than others  so the way we're going to set the  subject up which is the the the standard  way that uh people in this field look at  it um and i think you know that applies  to you know  lots of  examples outside as well  um is that we're going to kind of group  all of the inputs of the same length  together  and look at the maximum  uh  uh cost the maximum number of steps you  need to solve any one of those inputs of  a given length and we'll do that for  each length  and the way we're going to frame it is  in terms of giving a maximum or what's  called an upper bound  on  the amount of time that you need  uh to solve all of those inputs of  length n  that's what's uh sometimes called worst  case complexity i'm sure many of you  seen this already but just to make sure  we're all together on this  um uh you might contrast that for  example with what's called average case  complexity where instead of looking at  the most difficult case among all  inputs of length n you take the average  of all inputs of length n and then you  have to then it's a little bit more  complicated because then you need to  have a probability distribution on those  inputs and so on we're not going to look  at it in this course from that  perspective we're only going to be  looking at what's called worst case  complexity  um  so let's um  begin then uh uh by taking uh looking at  this in more detail and uh  taking as our  starting point the theorem that says  that on a one tape turing machine  which is deciding this language a  you know a to the k b to the k  um you can do this on a one tape turing  machine um  m we're calling it  in at most  some constant times n squared steps  for any input of length n  where  the constant is going to be fixed  independent of it  okay  so  this is going to be  you know  having a constant in in in in  factor in the complexity is going to  come all come up often and so instead of  saying this over and over again we're  going to use the notation um  that  m uses order in squared steps um i'm  sure many of you seen that terminology  as well but just just for the purposes  of making sure we're all together on  that there is this big o and little o  notation i'm expecting i'm expecting you  to be familiar with that  big o is when you apply applied to  functions as it's done  you say uh f is big o of g as  for two functions f and g  it's basically if f is less than or  equal to g  if you're ignoring constant factors  and you say f is little o of g  if f is strictly less than g if you're  ignoring constant factors that's kind of  one sort of informal way of looking at  it the precise definition is given up  there on the slide and if you haven't  seen it before make sure you look at it  in the book where it's all carefully  described and defined  so that you're comfortable with these  notions because it's really we're going  to be using this without any further  discussion uh from here on  okay  so let's get to the proof then of this  theorem that you can do uh the language  a in order n squared steps  not super hard if i think if i ask you  to come up with an algorithm to solve a  this would be the algorithm that you  would find basically  first you would start off by scanning  the input  w to make sure it's of the right form  so a run of a's followed by a run of b's  of some lengths  um and if it's not of that form  uh then you're going to reject right  away  the next thing you'll do is then go  through a repeat loop  um in the turing machine  and you know if you imagine here is your  machine here is the input w you're going  to go through that repeating repeatedly  of course you can do this in a number of  different ways but here's the way i i'm  i have in mind for you uh on this slide  anyway  we're going to scan the entire tape  crossing off one  a single a and a single b  um on a  on a scan  and then you're going to keep doing that  until you've crossed off everything  um  unless you run out of a's or you run out  of b's  in that case you're going to reject  um  if you run out of a's or b's before you  run out of uh the other type  then you know that you started out with  an unequal number and so the machine is  going to reject  if you've managed to cross them all off  without running out of one before the  other then you'll accept  i know this is kind of obvious but i  think it's important to get us all  together on this at the beginning  so here's a little um animation of the  turing machine doing that  i'm not showing the motion of the head  but imagine the head scanning back and  forth crossing off these a's and b's  one a and one b on each pass  until they're all crossed off and then  it accepts  unless of course it runs out of a's or  b's before the other type then it  rejects  okay  now let's do a very quick informal  analysis of how much time this has taken  so the very first stage i'm calling each  of these things stages of the turing  machine to distinct  distinguish them from the steps of the  turing machine which the individual  transition function moves  so this is the entire stage of the  machine  the very first stage takes order n steps  because you have to make a scan across  the input  and then you know i'm not giving all the  full detail of course you're going to  scan and then you're going to return the  head back to its starting position i'm  not that's just going to be an extra  factor of n an extra an extra  an extra n  um so that's what we're talking about  being order n steps for the very first  stage um  and then as you do through goes with the  repeat loop each time you go through the  repeat loop you're gonna cross off  one a and one b  so  um that's gonna mean you're gonna have  to do this roughly order you're gonna  have to do this order n times in order  to cross off all the a's and b's  so there's gonna be edit order and  iterations of this repeat loop  each one of them is going to again  require a state a scan so that's order n  steps uh for each one of the iterations  um so adding that all up  um  the very top row gives us the order n  and then we have the order and  iterations times order n steps is order  n squared steps  and so the sum of these two is order n  square order n squared steps  due to the nature of arithmetic when you  have the big o notation  the dominant term  overrides all of the others  so  that's  in a nutshell  how we well this is our very first  example of analyzing the turing machine  algorithm  for a language okay so let me  ask you now whether there is some other  turing machine some other one tape  turing machine which can do better  than this turing machine um does in  terms of how much time it takes  so one idea that you might have  instead of crossing off a single a or in  the single b maybe you can cross off um  two a's and two b's or ten a's and ten  b's well that'll cut down the running  time by a factor of ten  but from the standpoint of this theorem  it's still going to be an order n  squared algorithm  um and so that's not gonna that really  doesn't change  um  the amount of time used from from the  perspective of um from our perspective  where we're going to be ignoring the  constant factors on the running time  um  so i ask you here can you do better than  just improving things by a constant  factor  there we go  okay  so here's a check-in on this on this uh  on this problem on the problem a solving  deciding a on a one tape turing machine  can we do better  than  uh order n squared as i just described  in that uh  in that  theorem and that algorithm for that  theorem or um you know can you get it  down to n log n  um or maybe you can get it down to order  n  um what do you think  uh obviously you know we're  we're just getting started but  just um just make your best guess  and let me just post that for you as a  poll  what's most important to me is that you  understand  the terminology that we're using and the  way we're discussing with the way we're  talking about it because that's going to  be setting setting us up for the  definitions that we're going to come a  little bit later in the lecture  okay  so um i'm going to close the poll  um uh we're kind of all over the place  on it but that's that's that that's good  um  uh  since we haven't really covered this  material yet material yet  in fact b is the correct answer we can  improve this algorithm down to order n  log n  but not all the way down to order n um  so let me get let me show you how you do  it in order n log n on the next slide  and  so here  is a one tape turing machine that can  decide a by using  only order and log n steps  instead of order and squared steps so  this is a significant improvement um  so  i'm just i'll describe the turing  machine here again is the picture of the  machine on an input um  and the very first thing i need to say  is that we're going to scan to make sure  the input is of the right form  and now we're going to again it's going  to be  making repeated passes over the input  but we're going to do it a little  differently now  instead of crossing off a single a and a  single b or some fixed number of a's and  a fixed number of b's we're going to  cross off every other a  and every other b  in in that way we're going to  essentially cut the number of a's and  b's in half  and that's why the number of iterations  is only going to be a log instead of um  linear  okay so we're going to cross off every  other a and every other b  um  and at the same time we're going to keep  track  of the parity the even-odd parity of the  number of a's  um that we've seen and the s and the  parity of the number of b's that we've  seen um that have not yet been crossed  off  um and we're going to compare those  parodies to make sure that they agree  if they ever disagree  we know we started off with different  numbers of a's and b's  um  and i'll i'll illustrate this in a  second just to make sure you where um  we understand this algorithm  okay so  i'm going to write down as a little  table of the parities that we've seen  and i'm going to illustrate the  algorithm with a little animation  okay so again we're now going to  scan across crossing off every other a  and then every other b  but now but before we get to the b's we  observe as we cross these off that we  had six a's now i'm not saying we count  them we just keep track of the even odd  parity that can be done in the finite  control of the machine  counting them would be more complicated  but karen but just keeping track of the  parity is not is is something that the  finite automaton could do  so the parity in this case because they  were six is going to be even  now we cross off the b's  same even parity now we're going to  return the head back to the beginning  i'm obviously not showing the head  moving here we return the head back to  the big beginning and now we scan across  again crossing off every other remaining  a and counting the parity of the  remaining a's  so here now  it's going to be this one and this one  are going to get crossed off  and there were three a's so that was odd  parity  and the same for the b's  three b's odd parity  and now we return our head back to the  beginning  cross again off uh every other a and  every other b  every a  so that was odd parry there was just one  crossing off the b  odd parity because it's just one they  all agree  um so the machine is going to accept  um because everything is now crossed off  and and the parity is agreed along the  way  let me just say for a second you know  obviously if you ever get disagreement  on the parities then you know that the  two uh  the number of a's and number b's had to  disagree  um  but how do we know that if the parities  always agree that we actually did start  out with the same number of a's as b's  and that you could see that in a number  of different ways but perhaps you know  uh a cute way to see it  is there is actually if you look at  these parodies here the sequence of  parodies actually in reverse  so if you let's say odd odd even um and  you look at the binary representation of  the number of a's which is six so the  binary representation would be one one  zero  the fact that you get odd odd even and  one one zero  is not a coincidence in fact the  sequence of parodies in reverse that you  get  is um  exactly  um the same as the binary representation  you'd have to  you know confirm that with a little  proof it's not hard uh but um that once  you've confirmed that you can see that  um if the  sequence of parities agree then the  numbers have to be the same because the  binary uh representations agreed  okay so now getting to the analysis here  again order n steps to do the check  login iterations  each scan takes order n steps  so the total uh running time here is  going to be order n log n  because it's going to be the log n times  n that's where the log n log n comes  from  now questions you might ask could i do  even better than that can i beat  n log n by more than any constant factor  um so can i do little o  of n log n  um and the answer is no  this is the best possible  a one tape turing machine for this  language um  uh so a one day turing machine cannot  decide a by using little o  of n log n steps  we're not going to prove that  i'm not going to ask you to be  responsible for the proof but in fact um  what you can show is that any language  that you can do on a one tape turing  machine in little o of n log n steps  turns out to be regular  okay so um uh you can prove a rather  strong theorem here not super difficult  to to prove but i don't want to spend a  lot of time on  um  uh proving bounds for turing machines  because you know really the whole  purpose of setting this up using turing  machines is to talk about algorithms in  general algorithms in general i'm not  going to be focusing on the nitty gritty  of turing machines  um  okay uh so um  what is  my um  okay yeah so i wanted to just stop and  make sure that we were  we are together here um so  a brief pause and feel free to send me  any questions  um  okay this is a good question if we can  keep track of parodies why can't we just  keep track of the number of a's or b's  um  well you could keep track of the parity  in the finite memory and so you can do  that with effectively no work  but the finite memory cannot is not  enough for doing an account up to you  know some arbitrarily large number  now you could store the count on the  tape  but that's going to cost you time to  maintain that counter  and so um  that's not going to be  so  simple  as keeping track of the parity  in the finite memory  um  okay  somebody's asking if a star b star is  regular yes a star b star is regular so  what  um but a to the k b to the k which is  our language is not regular uh right so  that's the language we're looking at  um  okay  so  getting questions about what happens  with multiple tapes we'll talk about  that in a second um  yes we could do an order end steps on a  regular computer sure but uh  you know  right for this slide anyway we're  looking at one tape turing machines um  okay getting questions about a big o and  little o um  for something to be little  little o means it's less than any  constant factor times the function um  so you have to look at the definition in  the book i think almost  enough people in the class probably know  this and have seen this already i don't  want to spend everybody's time on it but  uh please review it in the book  somebody's asking if you need to store  the parody on the tape no you can just  throw it in the finite memory um  uh  i mean storing the finite memory seems  to be the simplest thing  okay when we move on please feel free to  ask questions to the tas as well we have  two tas at least here attending uh with  me so  um  good  um  right um  all right so  we cannot do better than a one-tape  turing machine on a one-tape turing  machine and order n log n um and that's  something we can prove that we're not  going to do it here  however if you if you have if you change  the model  for example you use a two-tape turing  machine then yes as a lot of you are  suggesting in the chat you can do better  than that  so  if we now have a two-tape turing machine  or a multi-tape turing machine you can  do it in order n steps and that's  actually the point i i'm really trying  to make here um  so if you have here your um  two tape turing machine  um  then  two tips  same language  now what we're going to do is copy the  a's to the second tape  that we can do on a single pass  um and then once the a's have been  copied to the second tape we can  continue on reading the b's  and match them off with the a's that  appear on the second tape  so in order n steps you can do the  comparison instead of order n log n  steps  and of course if they match you're going  to accept otherwise reject  so let's just see here's a little  animation uh  uh uh  demonstrating that of course it's very  simple  um  so here is that here that if you could  see that it came maybe a little too fast  here let's just show it again here is  the heads moving across and the a is  coming down uh to the bottom  and now the head the upper head is going  to continue on reading the bees the  lower head is going to  go back to the beginning on the a's  and matching off the b's with the a's  and that's how we can verify or check  that they are the same number  okay so now in this case they were the  same number so the machine would accept  if they were the different number the  machine would not accept  um  and the analysis is very simple  um  uh each stage here is going to take a  linear number of steps order n steps  um because it just consists of a single  uh scan there are no uh there are no  loops in this machine  no no no repeat loops  um  okay  so  um  question on this  um  all right  why don't we move on then um  now observe  and the point i'm really trying to make  is that  uh  on a one type turing machine you can do  it in n log n but not any better but on  a two tape turning machine you can do it  in order n  um  so there's a difference  in how  much  time you need to spend how many steps  you need to spend depending upon the  model  and that's significant for us um  so the number of steps depends on the  model one tape turing machine was order  and login multitape was order n  we call them model dependence  if you contrast that with the situation  in the complexities in the computability  section of the course  we had model independence  the choice of the model didn't matter  and that was nice for us because  the theory of decidability didn't depend  upon whether you had a one tape turing  machine or a multi-tape turing machine  it was all the same  set of decidable and recognizable  languages  uh  so we didn't have to worry about which  model we're actually going to work with  we could work with any model even just  an informal model of algorithm  um was going to would be good enough  because we're going to end up  with the same  notion  in the end  now that's  goes away  in complexity theory  now we have  a difference depending upon the model  and from a mathematical standpoint  that's a little less nice  because  um you know which model do you work with  if you want to understand the complexity  of some problem you know that you have  at hand  um now you have to make a choice you're  going to work with the turing machine  around how many tapes or you're going to  look at some other model and you're  going to get different results so it's  somewhat less natural from a  mathematical standpoint just to talk  about the complexity of some problem  but we're going to kind of um  bring back something close enough to  model independence by observing that  even though we don't have uh model  independence as we did in computability  theory we can limit  how much dependence there is  so the amount of dependence is going to  be low  as we will see  provided you stick with a reasonable  class of deterministic models  so the dependence though it does exist  is not going to be um  that much it's going to be polynomial  dependence  and we'll say exactly what that means in  a second  um and from our standpoint that's going  to be uh um a small difference uh a  negligible difference that we're going  to ignore  um  so we're going to focus on questions  that do not depend on the model choice  among these reasonable deterministic  models  now you may say well that's not  interesting from a practical standpoint  because polynomial differences say the  difference between n squared and n cubed  certainly make a difference in practice  but  it really  uh depends  on the  um  the uh  um  you know what what kinds of questions  you're focusing in on so if you want to  look at something that's very uh  a very precise distinction say between  you know n squared and n cubed um then  you might want to focus in on  which model you want to be working with  and that's going to be more the domain  of an algorithms class  but from our standpoint we're going to  be looking at other still important  questions but there are questions that  um don't uh depend upon exactly which  polynomial you're going to have we're  going to be looking more at distinctions  between polynomial and exponential  and still there are important practical  questions that arise in in that  somewhat different um  setting  okay  so with that in mind  we're going to continue to use the one  tape turing machine as our basic model  of complexity um  since the the model um among the  reasonable deterministic models in the  end is not going to matter from the from  the perspective of the kinds of  questions we're going to be asking  um  okay  um  so  with that so  we are going to continue then it's  important to remember that from going  forward we're going to stick with the  one tape turing machine model maybe  that's something you would have expected  us to do anyway but i'm trying to  justify that you know through this  little discussion that we had so far  thus far  so now we're going to start defining  things  with the one tape model in mind  um  so so first of all if you have a turing  machine we're going to say it runs in a  certain amount of time  so if t is some sort of time bound  function like n squared or n log n  we'll say the machine runs in that  amount of time like n squared or n log n  if  that machine m  always halts within that number of steps  on all inputs of length n  so it always holds within t of n steps  on inputs of length n  then we'll say that the machine runs in  t of n time  so in other words if the machine runs in  n squared time  then the machine when you give it an  input of length 10 it's gotta it's got  to be guaranteed to hold within a  hundred steps  uh 10 10 squared 100 steps on every  input of length 10.  um that's what it means  um for the machine to be running in that  much time and it has to do that for  every every n for every input length  and with that we're going to come to the  following definition which is  highlighted in color because it's going  to be um  we're going to be using this definition  throughout the semester so it's  important to understand it  this is the definition of what's called  the time complexity classes  and what i'm going to do  is take some bound  t of n and again think of t of n like a  bound like n squared  so if you have time t of n or like time  n squared  that's going to be the collection of all  languages that you can decide  within time and square or within time t  of n  so in other words it's a collection of  all languages b  such that there's some  one tape turing machine here we're  focusing again on the one tape turning  machine there is some deterministic one  tape turing machine  that decides b  and that machine runs in that amount of  time  so this is a collection of languages  the time collect complexity class is a  set of languages i'm going to draw it  now as a diagram  um  so if you take the language again that  we've been  using as our standard example a to the  kb to the k  um  that's in time and log n as we observed  for uh uh  two slides two slides back or three  slides back  so on a one tape turning machine you can  do this language a in time n log n  so it's in the time complexity class n  log n this this  region here captures all of the  languages  that you can do  in order and log in time  for example  that also includes all of the regular  languages  why is that  well any regular language  can be done on a one tape turing machine  in time order n  because the turing machine only just  needs to scan across doesn't even need  to write just need to scan across from  left to right on the on the tape and in  n steps it has the answer  so all of the regular languages are  actually in time n certainly a subset of  time n log n and these all form a kind  of a hierarchy so if you increase the  bound you can imagine that the collect  class of languages grows  as you  allow the machine to have more and more  steps to do its computing  um so these are all the languages that  you can do in n squared  uh order n square time on a one tape  turing machine  and cube time on a one tape turing  machine and so on  to exponential time two to the end time  on a one tape turning machine these are  all collections of languages  um  getting larger and larger um as we  increase the bound um  so someone is asking kind of is  okay uh let's see let me let me get to  some of these questions here i'll try to  get them in order um  uh  okay um  so somebody says if you have a good  question if you have a regular computer  so an ordinary sort of random access  computer so we'll talk about that in a  second can do it in order and uh can you  do it on a multi-tape during machine  also in order and time actually i don't  know the answer teller offhand i suspect  the answer is no  that  ordinary computers have a random access  capability that turing machines do not  and so that there are going to be some  examples of problems that you can do  with a random with a regular computer  that you cannot do with a multi-tape  turing machine in order end time uh i'd  have to double check that though so  we can and so some question  what what we believe is true and what we  can prove to be true um as we'll see  there are a lot of things that we  believe to be true in this subject that  we don't know how to prove  um  somebody's asking kind of an interesting  sort of more advanced question is there  some function f  uh some some some some some function t  where um it's so big that it so that  time t of n gives you all of the  decidable problems  um  it would be a very big t but the the  answer actually to that question is yes  um but  that's a little bit exotic so let let's  not spend a lot of time on that right  here but i'm happy to talk about that  offline  um  it's a good question here somebody's  asking me does it mean that there are no  languages between order n and order n  log n because i pointed out that  anything below n log n is going to be  regular  and so as soon as you get below n log n  you can do it in order n and yes there  is what's called a gap  between order n and order n log n on a  one take turing machine this you don't  get anything new from order n until you  jump up so from order n to order n log  log n  nothing new shows up  um  so we'll talk about those kinds of  things a little a little bit down the  road when we look at actually the  relationship among these various uh  classes and what we call a hierarchy  theorem which shows you know what how  much bigger do you have to make the  bound in order in order to be sure  you'll get something new  um  all right  [Music]  okay somebody's asking is there a  model which has the same time complexity  as a normal computer  uh  well i mean there's the random access  model which is supposed to capture a  normal computer um  okay so  let me let me um  these are all great questions kind of  more  uh  riffing off of  this into in more into more advanced  directions  uh  let's move on um here's another check-in  um suppose we take this is a little bit  of a  check to see how well your how  comfortable you are with the notions  we've just presented and whether you can  think about some of the arguments that  we've made and apply them in into a new  language so take the language ww reverse  um  you know strings  followed by their  um  uh this followed by themselves backwards  this language via the even length  palindromes if you will  um what's the smallest  uh bound that you need  uh to be able to solve  uh  that language b  and i'll pose it as a  pose that is a question for you  so which which time complexity class  is  um  that language b in  is it time  order n or n log n n squared  so on what do you think  so we're about to come to the coffee  break so uh why don't we um  uh  i'll answer any questions that come up i  think we're  got everybody uh answered so i'm gonna  end the polling okay make sure you're in  if you want to be in  um  so  the correct answer is in fact uh order n  squared  um  you it would be hard for you know you  you know every  reasonable guess here would be order n  log n i mean you can come up with the  same procedure  that the as the one we showed at the  beginning uh the order n squared  procedure for eight to the kb to the k  works for ww reverse as well um you can  just cross off  sweep back and forth crossing off  um  the symbol from w  and going across to the other side  crossing off the symbol from w reverse  and that procedure will give you an n  squared an order n squared algorithm  um  you might imagine you can improve it to  order n log n but you cannot you can  prove that order n squared is the best  possible um  i  i'm a little unhappy that a lot of you  came up with order n frankly because i  already told you that order n is uh  these are just regular languages  anything that you can do in less than  little o of n log n is going to be  regular and we know this language is not  regular so this is a this was  uh  not a good answer  you know so  please pay attention  and um okay  so let us  stop sharing uh i will turn now to the  uh our our  our break  uh for five minutes  and i'm happy to you know try to take  questions along the way as we're um  uh we're waiting for the time to  to end so let's see let me  put this up here  okay  let me try to take some of your  questions  um  so someone has asked me about quantum  computers as reasonable models of  you know  you may say a quantum computer is a  reasonable model of computation  and that's fine i would not say it's a  reasonable model deterministic  computation um  uh at least from our standpoint though  let's not quibble about the words um  i'm not including quantum computers in  um the collection of  um machines that i have in mind right  now when i'm talking about the the the  reasonable models of deterministic  computation that we're going to be we're  going to be discussing  um  uh  let's see  okay a bunch of people apparently are  asking the tas  why all regular languages can be done in  order n  um  [Music]  so  if you  think about a dfa  which you know  processes  a  an input of length n  within steps  and the dfa is um  going to be a type of turing machine  that never writes on its tape  uh so if the dfa can do it in in in n  steps the turing machine can do it n  steps and so therefore every regular  language can be done in order and steps  on a turing machine um  not sure where the confusion is so  please  message me if you're not still not  getting it um  okay somebody's saying why are we using  one-take turing machines instead of  random access wouldn't it be better to  use the random access machines  um if you were using if you're trying to  do algorithms yes  that's a more reasonable model we're  trying to prove things about the  computation and from that standpoint we  want to use as simple as model as  possible  uh trying to prove things  um using random access computers it's  possible be very messy  so that's why we don't use uh random  access machines to prove the kinds of  things we're going to be proving  um about computation  you know that that are really the meat  and potatoes of this course  so uh  i mean there's  compelling reasons why you would want to  use a simple model like a turing machine  but not a  powerful model like a random access  computer  so somebody's asking me does  um  the class  the time uh order and log log n have any  elements yes it has all the regular  languages but nothing else order n log  log n is only read only the regular  languages you have to go all the way up  to or and log in before you get  something non-regular  uh somebody's asking me are we going to  talk about how the random access model  works no  you know there's uh beyond the scope of  this course  outside of what we're going to be doing  we're going to talk about turing  machines  not because we care so much about turing  machines but i'm trying to prove things  about computation and the turing  machines are a convenient vehicle for  doing that um  okay our candle has burned out uh why  don't we return then to um  the next uh slide so everybody come back  so this answers one of the questions i  got on the chat is what's what actually  is the dependency between multi-tape  turing machines and one tape turing  machines can we bound that in general  yes we can  we're going to show  that converting a multi-tape turing  machine to a one-tape turing machine can  at most blow up the amount of time  that's necessary by a squaring  um no i acknowledge it's a lot but it  still allows you um  uh but you know it's still small  compared with an exponential increase  and we're going to be focusing in this  course  on things like the difference between  polynomial and exponential not between  the different not not between the  difference uh of  n not between  not the difference between n squared and  n cubed that's going to be less of a  fact less of an issue from us  uh  okay  um  [Music]  so  the way i'm framing  uh  showing this theorem is that if you have  a multi-tape turing machine that can do  a language in a certain amount of time  then it's in the time complexity class  of that time bound squared  and the way i'm just saying that is  because this is the bound that's  utilizing the one tape model so another  way of saying that is converting  multi-tape to one tape  squares the amount of time you need at  most  so the way we're going to  prove that is  simply by going back and remembering the  conversion that we already presented  from multi-tape to one tape  and observe that if we analyze that  conversion it just uh ends up squaring  the amount of time that the multitape  used  okay so  why is that  so if you remember  let's just make sure we're all  altogether on this the way the single  tape machine s  simulates the multi-tape turing machine  m  is that it takes the contents of each of  m's tapes  you know up to the place where there's  infinitely many blanks obviously you  don't store the infinite part but the  the the the active portion of each of  them's tapes you're going to store them  consecutively in separate blocks on s's  tape on estes only tape  um and now every time m makes one move s  has to scan its entire tape to uh to uh  to  see what's under each of the heads and  to do all the updating  so to simulate one take step of m's  computation s is going to use order t of  n steps  where t of n is the total running time  that m is going to use so why is t of n  steps coming up here well that's because  you have to measure how s is going to  make a scan of course its tape  how big can its tape  be  well m  if it's trying to use as much tape as  possible  can use at most  t of n  um  tape on each of the t event cells on  each of its tapes so all together  there's just going to be some constant  number of times t of n cells on s's tape  do you see that  so each one of these is going to be at  most t of n long so this all together is  going to be order t of n log because you  know what can m do it could send its  head out  say the head on this tape here moving as  fast as possible to the right using as  much tape as it can but you can only use  t event cells in t of n time  so this is going to be ordered t of n  so one step of m's computation is going  to be t of n steps on s's computation  but m itself has t of n steps so it's  going to be t of n times t of n for the  total number of steps that s is going to  end up using  and that's where the squaring comes from  okay  similar results i'm not going to do lots  of simulations of one model by another i  think that you'll get the idea and you  can if you're interested you can uh  study those on your own but you can  convert multi-dimensional turing  machines to one one tape turing machines  um you know one tape ordinary uh  linear  one tape  one dimensional uh machines  and  the bottom line is that among all of the  reasonable models they're all what are  called polynomially related  uh if each can simulate the other with  at most the polynomial overhead  so if one of the machines can use this t  of n time the other machine that's  simulating it would use t to the k of n  time for some k  um that's what it means for the two  machines to be polynomially related and  all reasonable deterministic models are  polynomially related  so as we've already seen one tape in  multi-tape turing machines are  polynomial related because converting  multi-tape to one tape blows you up by  most squaring  so k equals two in this case  um multi-dimensional turing machines  again polynomial related  the random access machine which i'm not  going to define but it's the machine  that you might imagine you know you  would i'm sure they must define in some  form in the algorithms classes  polynomial related cellular automata  which are just arrays of a finite  automata that can communicate with each  other similar similarly  um  you know all the reasonable  deterministic models  um you know again classical models i'm  not talking about quantum computing  are polynomial polynomially related  so we are um  uh that kind of justifies our  uh choice in picking one of them as long  as we're going to ask questions which  don't depend upon the polynomial  so  let's then talk about the class p  so the class p this is an important  definition for us  this is um  the collection of all languages  that you can do  in time into the k  for some k  on a one take turing machine  or as i've written it over here i don't  know if this notation is  um unfamiliar to you but this is like  just a big sum  um but here it's a big union symbol  it's union over all values of k  of the time class n to the k  so this is  time n union to time n squared union  time n cubed union time into the fourth  and so on  um and we call these the polynomial time  decidable languages  okay so we're going to be spending um  you know uh a certain amount of effort  uh  exploring this class p and other similar  classes  somebody's asking why is it a union i'm  not sure how else you would write it so  if somebody hasn't if you have a  proposal for a different way to write it  that's fine but this is you know k this  is for all k  i don't know you know there's if you  only had a limit finite number of k's  you could just take the biggest one but  since it's for all k you need to write  it as a union  um  now i want to argue that the class p  um is an important class  and  you know why is it had so much impact on  the subject  um and in terms of applications as well  um  so one thing is that  the class p is invariant for all  reasonable deterministic models and what  do i mean by that  so we have defined the class p  in terms of these time classes here  which in turn are defined in terms of  the one tape model  so we have defined p by using one tape  turing machines  now if we had to find p  in terms of multi-tape turing machines  we get exactly the same class  because  one tape and multi-tape turing machines  are polynomially related to one another  and since we're  taking the union over all polynomials  that polynomial difference is going to  wash out  um  similarly we could define p  using any of the other reasonable  deterministic models we get exactly the  same class  so in a sense we get back what we um the  situation that we had in computability  theory when  the class of decidable languages  didn't depend on the choice of model  here the class p does not matter  depending upon the choice of reasonable  deterministic model  um  and we also kind of even in the case of  computability theory we have to stick  with kind of reasonable models that  cannot do an infinite amount of work in  one step  um  so  uh  you know when i'm not going to define  what it means to be reasonable and  that's in a sense an informal notion  but  among all of those reasonable models  you're going to get the same class p  the other thing that makes p important  is that  p roughly corresponds  to  the problems that you can solve in some  reasonable practical sense  not exactly you know problems that  require into the hundredth time  um you could argue  cannot be uh solved in any reasonable  sense um  but  uh you know if you think about it for  example from the perspective of  cryptography  um  uh cryptographic codes that people come  up with are typically designed to  require or or the hope is that they  require an exponential amount of effort  to crack if someone found even an into  the hundredth algorithm to crack uh  that would crack a code  people would feel that the code is not  secure even though into the hundredth is  still a large um  so you know it's a it's a rough kind of  test but it's still used as a kind of  litmus litmus test for practical  solvability if you can solve it in  polynomial time you basically figured  out how to avoid  um  large searches  if you can solve problems in in  polynomial time we'll say more about  that later  but you know  what i want to bring out here is that we  have combined here in the class p  something that's mathematically nice  mathematically element elegant with  something that's practically relevant  and when you have a combination of the  two then you know you have a winner then  you know how you have a concept that's  going to make a difference and that's  been true for the class b this has been  very influential  within and without  the theory of computation  um  so let's look at an example  um let's let's define a new language we  haven't seen before though it's similar  to  procedures that we've looked at before  the path language  which is where i'm going to give you a  graph g  two nodes in the graph s and t  um  where i'm thinking of g is a directed  graph so directed means that the um  the connections between the nodes in g  are going to be  redirected and that they have arrows on  them they're not just lines but they  have an orientation with an arrow  um  so g is a directed graph that has a path  from s to t that respects the directions  um so such a i think i might even have a  picture here yeah so imagine here here  is your graph if you can see it there  are little arrows connecting the uh the  nodes and i want to know is there a path  from the node s to the node t  so that is  you know a picture of a problem in you  know an instance of a graph of a path  problem  and i want to find uh an algorithm for  that and i can show that there is an  algorithm that operates in polynomial  time  for this uh path problem  okay and the algorithm uh you know any  of the standard  searching algorithms  uh would work here but let's just for  completeness sake um include the  breadth first search algorithm that we  have explored previously when we talked  about finite automata um so we'll mark s  and they'll keep repeating until nothing  new is new is marked and we'll mark all  of the nodes that were reachable  by a single arrow from a previously  marked node  um and then cft is marked  after you could mark everything you can  get to so you're going to mark  let's see pictorially i think i have  this in indicated yeah you're going to  mark all of the things that are  reachable from the start not from the  from the node s  and then see  after you can't mark anything new  whether the node is marked and if it is  you'll accept if it is not you reject  okay um  now we can analyze this too um and i'm  not going to be spending a lot of time  analyzing algorithms here um but let's  just uh do the do it kind of this one  time uh again  um  we're gonna we're get we're doing a  bunch of iterations here so gonna be  repeating until nothing new is marked so  each time we mark something new we can  only do that at most end times  at which point we've marked everything  so the number of iterations here is  going to be at most n  and now for each each time we've marked  something we have to  um  look at all of the previously marked  nodes  um and see which things they point at to  mark them too  so this is going to be an inner loop  which again has at most n iterations  because it's going to all of the  previously marked nodes and then once we  have  that we can scan g to see  to mark all of the  new  uh  all of the nodes which we have not yet  marked uh whether they're connected with  a previously marked node by an edge  and i'm being generous here because i  don't really care  um this can be done in the most n  squared steps on a one-take turing  machine i'm not going to describe the  implementation but you know i'll leave  it to you as an exercise but this is  straightforward  so the total number of steps here would  be n iterations times n iterations times  and squared so you're going to be at  most into the fourth steps needed so  this is a polynomial algorithm and  whether i ended up with n to the fourth  or n to the fifth or n cubed i don't  really care um because i'm just trying  to illustrate um that  the total is polynomial  and that's all i'm going to be typically  asking you to do um so to show  polynomial time what i'd be asking you  to do is to show that each stage  each stage of this algorithm  should be clearly polynomial  and that the total number of stages i'm  sorry this should say stages here should  be polynomial  so each stage is polynomial and after  you're doing all the iterations the  total number of stages that are executed  is polynomial and so therefore all  together the total running time the  total number of steps is going to be  polynomial  okay so that's um  the  uh the way  we would write up um polynomial  algorithms in this class um  so  let's see if there's any questions here  i don't want to  get too far ahead of people  um  let's see  um  yes in this theorem i'm talking about  one tape turing machines  um because we're defining everything in  terms of one type machines but now at  this point you know we're talking about  polynomial time we can you know my  analysis is based on one tape turning  machines  but in general you know you could use  any reasonable deterministic model on  which to carry out your analysis uh  because they're all polynomially  equivalent so from the perspective of  coming up with  showing that a problem is in polynomial  time isn't p  um you can use any of the models um  you know that you that you wish that for  convenient okay oh that's a good  question what is n thank you for asking  that question  n  is always going to be  reserved  to indicate the length of the input  so here  n  is going to be when we encode g s and t  and here also i haven't said this but  i'm assuming that  the encoding that you're using is also  somehow reasonable you know i think  we'll talk a little bit about more about  that in the next lecture um which is  going to be after the midterm  um  but you know you can  cause problems if you intentionally try  to come up with nasty encodings which  will you know  represent things uh  with unnecessarily many characters but  um  you know if if you try to be reasonable  then  um  you just use any one of those encodings  and you'll be fine  um so yeah  um so n is the length of the  representation of the input  um  let's see  someone's trying to you're trying to dig  into the my the actual how this is  running here scan g to mark all y where  x y is an edge  i'm saying that you can do an n squared  steps  um  uh  um  you know  if you have x  you can mark it in a certain place on  the tape  and then as you're going through every  other node every other edge um you can  you can go back and compare x with the x  of the edge  and then c  and then find the y  i mean it's just going to be too messy  to talk about here i mean i i'll leave  it this is an exercise to you i'm not  going to try to fumble my way through  explaining why you can do this in n  squared steps but it's it's not hard  um uh  you guys are all like  hardcore algorithms folks you want to  know the algorithms for this i'm not  going to do that sorry uh  high level picture here um  you know so so you know  if you want  you know if you want to look at  detailed analyses this is not the right  course for you um  can k equal n  what is k  um  um  [Music]  no  this if you're talking about this k here  k cannot equal n we're not looking at n  to the n um this is these are all fixed  k  um it's like n squared and q but not  into the end is going to be an  exponential bound and so that's not  going to be included within uh this  union  okay so we're near the end of the hour  um  i'm gonna  introduce one last language here  um called handpath  and the handpath problem  is um  i'm going to ask now again for a path  from s to t  but now a different kind of a path one  that goes through every node of g along  the way  um so i'm looking for a path that goes  that hits every note of g not just the  shortest most direct path but in a sense  the most indirect pair the longest path  that goes through from from s to t  that visits everything else along the  way um  a path that  of that kind that hits every node of the  graph is called the hamiltonian path  um  because the mathematician hamilton  studied um those made some definitions  about that i'm not gonna  i don't actually know the history there  but i just know they're called  hamiltonian pets  um  so here's a picture i want to get from  snt but i want to sort of pick up  everything else along the way  so as you remember the path problem  itself can be solved in p  and what i'd like to know  can the simple modification where i'm  asking you to visit everything else  along the way is that problem also in p  okay and i'm going to put it pose this  as a checking for you  um but actually before i get to that let  me give you an algorithm for handpath um  that doesn't work to show it's in p  um because it's exponential  so  here's an algorithm  for handpath  um  uh where let's m be the number of nodes  in g um  and what i'm going to do is i'm going to  try every can every  possible path in g  um and see if it's actually works as a  hamiltonian path  and except if it is  um  and then  if all paths fail uh then i'll reject  so i'm going to try every possible  routing through g if you want to think  about it  think of  m as every possible permutation of the  nodes of g and then you're going to see  whether that's actually constitutes a  path in g  um that takes you from s to t  and goes through all of the nodes um  so this algorithm would work  this would give you  a correct algorithm for the hand path  problem the problem is  the difficulty is that there are so many  possible paths that it's going to take  you an exponential number of steps to  execute this algorithm it's not a  polynomial time algorithm because there  are many possible pairs that you could  go through um if you're looking at you  know with a very crude bound but you  really can't improve that significantly  there would be um m-factorial which is  going to be  you know much greater than 2 to the m  paths of length m um so the algorithm is  going to run for exponential time and  not polynomial time  so my question for you is i'm going to  pose it as a check-in problem is whether  you could actually do this problem in  polynomial time  so why don't you think about that as i'm  setting up the question  so take the hand path problem  just like the path problem which i  described with that marking algorithm  but now you want to  hit every node along the way  can you show that problem is solvable in  p  and there's a whole range of  possibilities here where either the  answer is yes you can see what the  polynomial time algorithm is  uh  to  uh to definitely know where you can  prove there is no such polynomial time  algorithm  and i'll put this as a checking for you  and see i'm curious to see what you come  up with  most people are getting it wrong  well i mean wrong i don't you know i'm  not clear what wrong is here  um  okay are we done  please uh check something  you can see a few of you have not  answered but  the poll is running out  okay  time is up  um  so  uh in fact  you know as as i think many of you know  but not all of you uh this is an  unsolved problem this is a very famous  unsolved problem which is equivalent to  the p versus np problem that we're going  to be talking about very soon which  among other things would be worth a  million dollars if you solve it  so for those of you who have answered a  or e please talk to me after lecture um  and maybe we can work on it together  um  [Music]  uh no so  um  yeah i  i think most people would believe that  the answer is no um  uh but no one knows how to prove it at  this time uh so i'm interested in the  uh folks who have uh come up with uh  what they think are solutions um and i  should say that there are some folks who  um believe that there might be other  outcomes besides uh just a simple note  um  which might be proven eventually  um  so we're going to talk more about this  um but this is the answer to the  question just for your uh just to make  sure you understand is that it's an  unsolved problem right now so we don't  we don't know uh  definitely yes and definitely no at  least according to the state of  knowledge  of which i'm aware  is not um are not correct answers um but  any of the others  well who knows  um so  i think that's the end of what i had to  say for today  we covered complexity theory as an  introduction looked at different  possible models focused on the one tape  model  introduced based on the one tape model  these uh complexity classes the class p  and we  showed an example of this path problem  being in p  talked also about this hand path problem  which we'll talk about more um  after the midterm  okay so um i'll stick around for a few  minutes uh if you have any further  questions otherwise um  uh  okay so let me just take questions here  um  somebody's asking me about my personal  opinion on p versus np  um  my personal opinion on people versus np  is that p is not equal to np  and that we will prove it someday  when i was a graduate student back in  the mid 70s  i thought it would be solved by now and  in fact i made a bet  with len adelman  who i subsequently ended up becoming the  a of the rsa code  um that we would solve it by the year  2000 and i bet  what was then a lot of money for me  which was an ounce of gold  which i didn't end up then which i did  end up paying off to lend in the year  2000 so  i'm not making any more bets but i still  believe that it will be solved uh  hopefully i'll get a chance to see the  solution  um i've spent a lot of time thinking  about it myself  you know  obviously i haven't solved it otherwise  we would know but um  uh  you know uh hopefully somebody will  i'm getting asked a question here that's  kind of an interesting question but i  don't really know  um  i'm sure i understand it what's the  largest possible run time of a decidal  decidable problem  what is the larger largest decidable  runtime so you  anything that i can describe  can be um  there are going to be  algorithms  that run for longer  uh  you can  define  an algorithm you can define a runtime  which would in a sense  beats all other runtimes  which  so that anything any runtime is going to  be um  dominated by that  extremely slow runtime but it's not  something that one can describe  um i can describe it to you by perceive  mathematical procedure but there's it's  not going to be something like you know  uh  2 to the 2 to the n  somebody's here proposing  they prepare a solution to the hand path  problem  i presume in polynomial time  why is the following flawed if s goes  through all nodes and ends up at t  well s will not you mean you mean i  presume you mean starting at s if we end  up going through all nodes and  and end of t  um the proposal is a little complicated  here  basically you you want to if if i can  try to rephrase it you want to try to  from every possible node you want to try  to calculate a path to t  and also a path from s to that node um  and you can do that for all possible  nodes but there's no way to really  combine them into a single path  that visits all you know solving for  each node separately is not going to do  the trick  because you have to somehow combine all  that information into a single path  just one path that goes from s to t and  visits all the other nodes along the way  and that that is not uh  i i don't see how what your proposal um  how that's how that's going to actually  work  professor spitzer yes  um a question on  the um  like we were kind of talking about  earlier but like what we talked about  today was defined for the one tape turn  machines correct  yep  so um  and you said like we could have applied  for the multi multitape ones but um  are  like  is  well i don't know if it's we talked  about earlier but if something  is accepted by the one tape turn machine  like can be applied to the multi-tape  turn machine and vice versa like are  they interchangeable like that  well they're interchangeable only in the  sense that  the amount of time that you would need  it's a different machine if you have a  multi-tape turing machine for some  language  um you can convert it to a one-tape  turing machine using a procedure that we  described it earlier in the term  and she'll get a different machine it's  going to run for a different amount of  time  by using their procedure the point is is  that the amount of time that the  one-take turing machine is going to run  for is not that much  worse  than the multi-tape turing machine's  time so if the multi-tape turning  machine's time was  you know  uh end cubed  the one tape turning machines uh is  going to be the square of that so it's  going to be into the sixth okay but it's  not going to be it's not going from  multi-tape to one tape is not going to  convert you from polynomial to  exponential that's that's the only point  i'm trying to make it's going to convert  from one polynomial to a somewhat bigger  polynomial but it's still going to leave  you polynomial  [Music]  i don't know you don't seem to  i i can't see your face no no yeah i  guess i guess my opinion is that like  especially when we were talking about  earlier um when you're bringing it up it  just seemed like you could just turn  anything to a multi-tape turn machine  and like completely like cut the timeout  like if i had like something to n log n  like if i do it in a multi-tape turn  machine i have it in like  um like big o of n  yeah i mean  like it just seemed like the multi-tape  was so much more powerful but then  i guess  i guess not with the like with the  explanations and like the  the models we were talking about today  yeah i would not say the multi  multi-tape turing machines are still  pretty limited in their capabilities and  they're  um  you know don't forget when you have a  multi-tape turing machine you have only  a fixed number of tapes i mean you could  also define variations of multi-tape  turing machines that have increasing  number of tapes  as the input uh  you know under  you know either under program control it  can launch new tapes or it could um  um you know just have more tapes  depending upon the size of the input  that would also be a possibility that's  not the model that we have defined but  you could define a model like that  um  but as long as the total amount of work  being done by the machine at any step  it's going to be a polynomial amount of  work  um  then you can convert it to a one tape  turing machine uh with only a polynomial  increase in the bound all right so  you know you want to be careful of  machines you know you know one one thing  i meant to say but didn't say you know  so he would be an unreasonable model  um which you might think of as a  plausible model but it's not going to be  a reasonable model from our standpoint  and that would be a model um for example  that can do  um  full precision say integer  arithmetic um  uh  with a unit cost per operation so each  operation counts as a costs one  um  but i'm going to allow you to do for  example  you know addition and multiplication  uh the the thing that's bad about that  um in terms of being uh unreasonable  is that  uh  after k step each time you do a step  you could double the size of the integer  by by squaring it  yeah so after k steps you could have an  integer which is 2 to the k long  and now doing operations there is going  to involve an exponential amount of work  even in any reasonable sense  um you know in a theoretical sense and  also in a practical sense  so um  you know a model that operates like that  is not going to be able to convert to a  one-tape turing machine with only a  polynomial  increase because it's doing an  exponential amount of work potentially  you know uh within a polynomial number  of steps  um  so that's within a linear number of  steps within n steps  so that's um an example of an  uh  you know of an unreasonable  deterministic model yeah  thank you  sure  so i'm just curious some idea just  occurred to me um  i guess like if you had an oracle  turing machine uh basically just so that  you could look at a uh  a turing machine description and decide  whether it's a decider or not yeah um  then it'd be a little bit interesting to  think about uh  what happens if you look at  all uh you know if you have a  description of a pair of a turing  machine and an input string and you can  look at for all size n  descriptions of a pair uh like what's  the most steps that it takes  uh  for a such a machine  to converge  so then so then you'd have combined  turing machine and input stream  descriptions  uh where you can look at what's the  longest it takes to converge it's just i  don't know just a random thought that  occurred  yeah so we we actually we're gonna we we  will talk about oracle turing machines  uh later on in the term um  you know these are machines that have  access to sort of free information um  and that actually turns out to be uh  i mean  there's some interesting interesting  things you can say about what what  happens when you're providing a machine  with in a sense information for free  that  you know  you might otherwise want to charge it  for actually computing that information  but let's just say we're going to allow  it to  uh get that information without being  charged  and then how does that affect the  complexity of other problems for example  um  and so uh we will talk about that later  but  too much if i think of a digression at  this moment to try to define all that um  but happy to chat with you about it on  piazza if you if you want to raise a  question there i think  sure oh no problem  um so he's asking me about strategies  for solving the p versus np problem you  know we will talk also talk about that a  little later in the term as well  but  you know asking you know clearly  you know it seems beyond the reach of  our present techniques  to be able to prove that some problems  you know really take a long time like  the hamiltonian path problem seems like  you know there's nothing really you can  do that's significantly better than  trying all possibilities as i described  in that exponential algorithm you know  on the last slide uh but  um  how do you prove that nobody knows um so  um  lots of people have tried  including yours truly i mean i spent a  lot of time thinking about it having  succeeded with it but um  there is uh  um  you know  somebody i i think somebody someday  someday somebody will come up with the  the new idea that's needed to solve it  but it's going to clearly take some sort  of a breakthrough some sort of a new  idea it's not just going to be a  combination of existing ideas  the existing methods  um  i think we're about 10 minutes past a  few of you are still here i'm going to  say goodbye to you folks and um  um  shortly i'm going to join my tas for our  t weekly ta meeting so see you guys  thanks for being here
 

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00:07:06,070 --> 00:07:06,080

 

216
00:07:06,080 --> 00:07:08,629


217
00:07:08,629 --> 00:07:10,150

 

218
00:07:10,150 --> 00:07:14,629

 

219
00:07:14,629 --> 00:07:18,390

 

220
00:07:18,390 --> 00:07:19,990

 

221
00:07:19,990 --> 00:07:22,629

 

222
00:07:22,629 --> 00:07:23,990

 

223
00:07:23,990 --> 00:07:26,230

 

224
00:07:26,230 --> 00:07:26,240

 

225
00:07:26,240 --> 00:07:27,270


226
00:07:27,270 --> 00:07:30,710

 

227
00:07:30,710 --> 00:07:32,390

 

228
00:07:32,390 --> 00:07:34,550

 

229
00:07:34,550 --> 00:07:36,230

 

230
00:07:36,230 --> 00:07:38,469

 

231
00:07:38,469 --> 00:07:41,350

 

232
00:07:41,350 --> 00:07:43,350

 

233
00:07:43,350 --> 00:07:46,390

 

234
00:07:46,390 --> 00:07:48,629

 

235
00:07:48,629 --> 00:07:53,029

 

236
00:07:53,029 --> 00:07:55,029

 

237
00:07:55,029 --> 00:07:57,350

 

238
00:07:57,350 --> 00:07:58,869

 

239
00:07:58,869 --> 00:08:01,589

 

240
00:08:01,589 --> 00:08:04,390

 

241
00:08:04,390 --> 00:08:07,029

 

242
00:08:07,029 --> 00:08:08,790

 

243
00:08:08,790 --> 00:08:10,469

 

244
00:08:10,469 --> 00:08:12,550

 

245
00:08:12,550 --> 00:08:14,309

 

246
00:08:14,309 --> 00:08:17,430

 

247
00:08:17,430 --> 00:08:19,350

 

248
00:08:19,350 --> 00:08:22,230

 

249
00:08:22,230 --> 00:08:23,350

 

250
00:08:23,350 --> 00:08:25,189

 

251
00:08:25,189 --> 00:08:27,189

 

252
00:08:27,189 --> 00:08:31,110

 

253
00:08:31,110 --> 00:08:31,120

 

254
00:08:31,120 --> 00:08:32,310


255
00:08:32,310 --> 00:08:34,310

 

256
00:08:34,310 --> 00:08:36,790

 

257
00:08:36,790 --> 00:08:39,509

 

258
00:08:39,509 --> 00:08:41,750

 

259
00:08:41,750 --> 00:08:43,589

 

260
00:08:43,589 --> 00:08:44,870

 

261
00:08:44,870 --> 00:08:46,630

 

262
00:08:46,630 --> 00:08:48,470

 

263
00:08:48,470 --> 00:08:49,670

 

264
00:08:49,670 --> 00:08:53,110

 

265
00:08:53,110 --> 00:08:56,230

 

266
00:08:56,230 --> 00:08:57,910

 

267
00:08:57,910 --> 00:09:00,150

 

268
00:09:00,150 --> 00:09:01,750

 

269
00:09:01,750 --> 00:09:01,760

 

270
00:09:01,760 --> 00:09:03,350


271
00:09:03,350 --> 00:09:05,110

 

272
00:09:05,110 --> 00:09:06,710

 

273
00:09:06,710 --> 00:09:09,110

 

274
00:09:09,110 --> 00:09:11,190

 

275
00:09:11,190 --> 00:09:13,430

 

276
00:09:13,430 --> 00:09:16,949

 

277
00:09:16,949 --> 00:09:18,230

 

278
00:09:18,230 --> 00:09:20,630

 

279
00:09:20,630 --> 00:09:22,949

 

280
00:09:22,949 --> 00:09:22,959

 

281
00:09:22,959 --> 00:09:24,630


282
00:09:24,630 --> 00:09:27,110

 

283
00:09:27,110 --> 00:09:29,190

 

284
00:09:29,190 --> 00:09:31,910

 

285
00:09:31,910 --> 00:09:33,590

 

286
00:09:33,590 --> 00:09:34,949

 

287
00:09:34,949 --> 00:09:36,710

 

288
00:09:36,710 --> 00:09:38,870

 

289
00:09:38,870 --> 00:09:38,880

 

290
00:09:38,880 --> 00:09:40,150


291
00:09:40,150 --> 00:09:42,630

 

292
00:09:42,630 --> 00:09:44,070

 

293
00:09:44,070 --> 00:09:46,389

 

294
00:09:46,389 --> 00:09:46,399

 

295
00:09:46,399 --> 00:09:47,509


296
00:09:47,509 --> 00:09:49,350

 

297
00:09:49,350 --> 00:09:52,310

 

298
00:09:52,310 --> 00:09:54,070

 

299
00:09:54,070 --> 00:09:56,470

 

300
00:09:56,470 --> 00:09:59,990

 

301
00:09:59,990 --> 00:10:02,550

 

302
00:10:02,550 --> 00:10:04,230

 

303
00:10:04,230 --> 00:10:06,310

 

304
00:10:06,310 --> 00:10:08,310

 

305
00:10:08,310 --> 00:10:10,069

 

306
00:10:10,069 --> 00:10:11,910

 

307
00:10:11,910 --> 00:10:14,389

 

308
00:10:14,389 --> 00:10:19,350

 

309
00:10:19,350 --> 00:10:21,269

 

310
00:10:21,269 --> 00:10:23,110

 

311
00:10:23,110 --> 00:10:25,910

 

312
00:10:25,910 --> 00:10:28,310

 

313
00:10:28,310 --> 00:10:29,750

 

314
00:10:29,750 --> 00:10:30,949

 

315
00:10:30,949 --> 00:10:33,269

 

316
00:10:33,269 --> 00:10:34,790

 

317
00:10:34,790 --> 00:10:34,800

 

318
00:10:34,800 --> 00:10:36,310


319
00:10:36,310 --> 00:10:36,320

 

320
00:10:36,320 --> 00:10:38,389


321
00:10:38,389 --> 00:10:40,069

 

322
00:10:40,069 --> 00:10:42,470

 

323
00:10:42,470 --> 00:10:44,470

 

324
00:10:44,470 --> 00:10:46,630

 

325
00:10:46,630 --> 00:10:48,470

 

326
00:10:48,470 --> 00:10:50,470

 

327
00:10:50,470 --> 00:10:52,790

 

328
00:10:52,790 --> 00:10:54,870

 

329
00:10:54,870 --> 00:10:56,630

 

330
00:10:56,630 --> 00:10:56,640

 

331
00:10:56,640 --> 00:10:57,990


332
00:10:57,990 --> 00:11:01,350

 

333
00:11:01,350 --> 00:11:02,790

 

334
00:11:02,790 --> 00:11:04,870

 

335
00:11:04,870 --> 00:11:06,630

 

336
00:11:06,630 --> 00:11:07,910

 

337
00:11:07,910 --> 00:11:09,750

 

338
00:11:09,750 --> 00:11:11,990

 

339
00:11:11,990 --> 00:11:13,590

 

340
00:11:13,590 --> 00:11:16,949

 

341
00:11:16,949 --> 00:11:18,389

 

342
00:11:18,389 --> 00:11:19,829

 

343
00:11:19,829 --> 00:11:21,670

 

344
00:11:21,670 --> 00:11:23,750

 

345
00:11:23,750 --> 00:11:25,509

 

346
00:11:25,509 --> 00:11:27,350

 

347
00:11:27,350 --> 00:11:29,590

 

348
00:11:29,590 --> 00:11:31,269

 

349
00:11:31,269 --> 00:11:31,279

 

350
00:11:31,279 --> 00:11:32,069


351
00:11:32,069 --> 00:11:34,150

 

352
00:11:34,150 --> 00:11:36,310

 

353
00:11:36,310 --> 00:11:37,829

 

354
00:11:37,829 --> 00:11:39,990

 

355
00:11:39,990 --> 00:11:41,269

 

356
00:11:41,269 --> 00:11:43,590

 

357
00:11:43,590 --> 00:11:44,870

 

358
00:11:44,870 --> 00:11:47,190

 

359
00:11:47,190 --> 00:11:50,629

 

360
00:11:50,629 --> 00:11:53,350

 

361
00:11:53,350 --> 00:11:53,360

 

362
00:11:53,360 --> 00:11:54,629


363
00:11:54,629 --> 00:11:57,509

 

364
00:11:57,509 --> 00:11:58,710

 

365
00:11:58,710 --> 00:12:00,870

 

366
00:12:00,870 --> 00:12:03,190

 

367
00:12:03,190 --> 00:12:06,870

 

368
00:12:06,870 --> 00:12:09,190

 

369
00:12:09,190 --> 00:12:11,509

 

370
00:12:11,509 --> 00:12:13,430

 

371
00:12:13,430 --> 00:12:15,430

 

372
00:12:15,430 --> 00:12:19,110

 

373
00:12:19,110 --> 00:12:19,120

 

374
00:12:19,120 --> 00:12:20,629


375
00:12:20,629 --> 00:12:20,639

 

376
00:12:20,639 --> 00:12:21,590


377
00:12:21,590 --> 00:12:23,030

 

378
00:12:23,030 --> 00:12:26,310

 

379
00:12:26,310 --> 00:12:28,470

 

380
00:12:28,470 --> 00:12:28,480

 

381
00:12:28,480 --> 00:12:29,590


382
00:12:29,590 --> 00:12:32,870

 

383
00:12:32,870 --> 00:12:35,030

 

384
00:12:35,030 --> 00:12:36,710

 

385
00:12:36,710 --> 00:12:39,350

 

386
00:12:39,350 --> 00:12:42,310

 

387
00:12:42,310 --> 00:12:44,550

 

388
00:12:44,550 --> 00:12:47,350

 

389
00:12:47,350 --> 00:12:49,110

 

390
00:12:49,110 --> 00:12:52,310

 

391
00:12:52,310 --> 00:12:54,870

 

392
00:12:54,870 --> 00:12:56,790

 

393
00:12:56,790 --> 00:12:58,870

 

394
00:12:58,870 --> 00:13:00,870

 

395
00:13:00,870 --> 00:13:02,150

 

396
00:13:02,150 --> 00:13:03,590

 

397
00:13:03,590 --> 00:13:06,069

 

398
00:13:06,069 --> 00:13:07,509

 

399
00:13:07,509 --> 00:13:07,519

 

400
00:13:07,519 --> 00:13:08,550


401
00:13:08,550 --> 00:13:11,509

 

402
00:13:11,509 --> 00:13:14,629

 

403
00:13:14,629 --> 00:13:16,389

 

404
00:13:16,389 --> 00:13:18,829

 

405
00:13:18,829 --> 00:13:18,839

 

406
00:13:18,839 --> 00:13:20,389


407
00:13:20,389 --> 00:13:23,670

 

408
00:13:23,670 --> 00:13:25,269

 

409
00:13:25,269 --> 00:13:25,279

 

410
00:13:25,279 --> 00:13:26,550


411
00:13:26,550 --> 00:13:27,829

 

412
00:13:27,829 --> 00:13:27,839

 

413
00:13:27,839 --> 00:13:28,949


414
00:13:28,949 --> 00:13:31,269

 

415
00:13:31,269 --> 00:13:34,389

 

416
00:13:34,389 --> 00:13:36,470

 

417
00:13:36,470 --> 00:13:38,310

 

418
00:13:38,310 --> 00:13:38,320

 

419
00:13:38,320 --> 00:13:39,990


420
00:13:39,990 --> 00:13:42,550

 

421
00:13:42,550 --> 00:13:43,829

 

422
00:13:43,829 --> 00:13:44,870

 

423
00:13:44,870 --> 00:13:46,230

 

424
00:13:46,230 --> 00:13:49,189

 

425
00:13:49,189 --> 00:13:50,870

 

426
00:13:50,870 --> 00:13:53,670

 

427
00:13:53,670 --> 00:13:53,680

 

428
00:13:53,680 --> 00:13:54,470


429
00:13:54,470 --> 00:13:56,230

 

430
00:13:56,230 --> 00:13:58,710

 

431
00:13:58,710 --> 00:14:00,550

 

432
00:14:00,550 --> 00:14:04,069

 

433
00:14:04,069 --> 00:14:06,069

 

434
00:14:06,069 --> 00:14:06,079

 

435
00:14:06,079 --> 00:14:09,030


436
00:14:09,030 --> 00:14:10,389

 

437
00:14:10,389 --> 00:14:10,399

 

438
00:14:10,399 --> 00:14:11,829


439
00:14:11,829 --> 00:14:13,509

 

440
00:14:13,509 --> 00:14:14,710

 

441
00:14:14,710 --> 00:14:17,590

 

442
00:14:17,590 --> 00:14:19,910

 

443
00:14:19,910 --> 00:14:21,189

 

444
00:14:21,189 --> 00:14:23,910

 

445
00:14:23,910 --> 00:14:23,920

 

446
00:14:23,920 --> 00:14:24,949


447
00:14:24,949 --> 00:14:28,870

 

448
00:14:28,870 --> 00:14:31,509

 

449
00:14:31,509 --> 00:14:33,750

 

450
00:14:33,750 --> 00:14:33,760

 

451
00:14:33,760 --> 00:14:35,350


452
00:14:35,350 --> 00:14:35,360

 

453
00:14:35,360 --> 00:14:36,790


454
00:14:36,790 --> 00:14:38,230

 

455
00:14:38,230 --> 00:14:40,470

 

456
00:14:40,470 --> 00:14:42,710

 

457
00:14:42,710 --> 00:14:45,350

 

458
00:14:45,350 --> 00:14:46,470

 

459
00:14:46,470 --> 00:14:48,870

 

460
00:14:48,870 --> 00:14:50,550

 

461
00:14:50,550 --> 00:14:53,990

 

462
00:14:53,990 --> 00:14:54,000

 

463
00:14:54,000 --> 00:14:57,430


464
00:14:57,430 --> 00:14:58,389

 

465
00:14:58,389 --> 00:15:00,230

 

466
00:15:00,230 --> 00:15:02,629

 

467
00:15:02,629 --> 00:15:05,269

 

468
00:15:05,269 --> 00:15:07,750

 

469
00:15:07,750 --> 00:15:13,670

 

470
00:15:13,670 --> 00:15:13,680

 

471
00:15:13,680 --> 00:15:14,870


472
00:15:14,870 --> 00:15:16,310

 

473
00:15:16,310 --> 00:15:18,470

 

474
00:15:18,470 --> 00:15:21,750

 

475
00:15:21,750 --> 00:15:23,990

 

476
00:15:23,990 --> 00:15:25,670

 

477
00:15:25,670 --> 00:15:28,470

 

478
00:15:28,470 --> 00:15:29,829

 

479
00:15:29,829 --> 00:15:30,710

 

480
00:15:30,710 --> 00:15:33,350

 

481
00:15:33,350 --> 00:15:34,230

 

482
00:15:34,230 --> 00:15:36,710

 

483
00:15:36,710 --> 00:15:38,470

 

484
00:15:38,470 --> 00:15:40,470

 

485
00:15:40,470 --> 00:15:41,990

 

486
00:15:41,990 --> 00:15:44,629

 

487
00:15:44,629 --> 00:15:46,470

 

488
00:15:46,470 --> 00:15:47,670

 

489
00:15:47,670 --> 00:15:49,749

 

490
00:15:49,749 --> 00:15:51,829

 

491
00:15:51,829 --> 00:15:54,550

 

492
00:15:54,550 --> 00:15:57,509

 

493
00:15:57,509 --> 00:15:57,519

 

494
00:15:57,519 --> 00:15:59,110


495
00:15:59,110 --> 00:16:00,710

 

496
00:16:00,710 --> 00:16:03,030

 

497
00:16:03,030 --> 00:16:03,040

 

498
00:16:03,040 --> 00:16:05,269


499
00:16:05,269 --> 00:16:07,110

 

500
00:16:07,110 --> 00:16:07,120

 

501
00:16:07,120 --> 00:16:08,470


502
00:16:08,470 --> 00:16:11,910

 

503
00:16:11,910 --> 00:16:13,430

 

504
00:16:13,430 --> 00:16:15,829

 

505
00:16:15,829 --> 00:16:17,990

 

506
00:16:17,990 --> 00:16:20,150

 

507
00:16:20,150 --> 00:16:20,160

 

508
00:16:20,160 --> 00:16:22,069


509
00:16:22,069 --> 00:16:23,749

 

510
00:16:23,749 --> 00:16:26,310

 

511
00:16:26,310 --> 00:16:27,910

 

512
00:16:27,910 --> 00:16:30,150

 

513
00:16:30,150 --> 00:16:32,389

 

514
00:16:32,389 --> 00:16:32,399

 

515
00:16:32,399 --> 00:16:33,670


516
00:16:33,670 --> 00:16:35,590

 

517
00:16:35,590 --> 00:16:38,150

 

518
00:16:38,150 --> 00:16:40,389

 

519
00:16:40,389 --> 00:16:42,069

 

520
00:16:42,069 --> 00:16:43,509

 

521
00:16:43,509 --> 00:16:45,910

 

522
00:16:45,910 --> 00:16:47,189

 

523
00:16:47,189 --> 00:16:49,430

 

524
00:16:49,430 --> 00:16:52,470

 

525
00:16:52,470 --> 00:16:56,790

 

526
00:16:56,790 --> 00:16:58,470

 

527
00:16:58,470 --> 00:17:01,189

 

528
00:17:01,189 --> 00:17:03,590

 

529
00:17:03,590 --> 00:17:05,510

 

530
00:17:05,510 --> 00:17:07,510

 

531
00:17:07,510 --> 00:17:09,510

 

532
00:17:09,510 --> 00:17:11,029

 

533
00:17:11,029 --> 00:17:12,870

 

534
00:17:12,870 --> 00:17:14,390

 

535
00:17:14,390 --> 00:17:16,789

 

536
00:17:16,789 --> 00:17:18,870

 

537
00:17:18,870 --> 00:17:21,029

 

538
00:17:21,029 --> 00:17:23,750

 

539
00:17:23,750 --> 00:17:28,069

 

540
00:17:28,069 --> 00:17:30,150

 

541
00:17:30,150 --> 00:17:31,590

 

542
00:17:31,590 --> 00:17:33,350

 

543
00:17:33,350 --> 00:17:35,190

 

544
00:17:35,190 --> 00:17:37,430

 

545
00:17:37,430 --> 00:17:39,430

 

546
00:17:39,430 --> 00:17:41,270

 

547
00:17:41,270 --> 00:17:42,549

 

548
00:17:42,549 --> 00:17:44,070

 

549
00:17:44,070 --> 00:17:45,990

 

550
00:17:45,990 --> 00:17:48,230

 

551
00:17:48,230 --> 00:17:50,630

 

552
00:17:50,630 --> 00:17:50,640

 

553
00:17:50,640 --> 00:17:51,669


554
00:17:51,669 --> 00:17:54,549

 

555
00:17:54,549 --> 00:17:56,710

 

556
00:17:56,710 --> 00:17:58,070

 

557
00:17:58,070 --> 00:17:58,080

 

558
00:17:58,080 --> 00:17:59,350


559
00:17:59,350 --> 00:18:01,510

 

560
00:18:01,510 --> 00:18:02,950

 

561
00:18:02,950 --> 00:18:04,710

 

562
00:18:04,710 --> 00:18:07,510

 

563
00:18:07,510 --> 00:18:09,110

 

564
00:18:09,110 --> 00:18:11,270

 

565
00:18:11,270 --> 00:18:12,390

 

566
00:18:12,390 --> 00:18:16,230

 

567
00:18:16,230 --> 00:18:19,029

 

568
00:18:19,029 --> 00:18:20,870

 

569
00:18:20,870 --> 00:18:20,880

 

570
00:18:20,880 --> 00:18:23,190


571
00:18:23,190 --> 00:18:25,029

 

572
00:18:25,029 --> 00:18:26,710

 

573
00:18:26,710 --> 00:18:28,549

 

574
00:18:28,549 --> 00:18:30,150

 

575
00:18:30,150 --> 00:18:31,830

 

576
00:18:31,830 --> 00:18:31,840

 

577
00:18:31,840 --> 00:18:33,029


578
00:18:33,029 --> 00:18:33,039

 

579
00:18:33,039 --> 00:18:33,990


580
00:18:33,990 --> 00:18:36,150

 

581
00:18:36,150 --> 00:18:38,150

 

582
00:18:38,150 --> 00:18:40,549

 

583
00:18:40,549 --> 00:18:42,310

 

584
00:18:42,310 --> 00:18:44,470

 

585
00:18:44,470 --> 00:18:46,630

 

586
00:18:46,630 --> 00:18:48,230

 

587
00:18:48,230 --> 00:18:49,990

 

588
00:18:49,990 --> 00:18:52,230

 

589
00:18:52,230 --> 00:18:56,310

 

590
00:18:56,310 --> 00:18:58,870

 

591
00:18:58,870 --> 00:19:01,590

 

592
00:19:01,590 --> 00:19:03,990

 

593
00:19:03,990 --> 00:19:04,000

 

594
00:19:04,000 --> 00:19:05,029


595
00:19:05,029 --> 00:19:07,110

 

596
00:19:07,110 --> 00:19:08,549

 

597
00:19:08,549 --> 00:19:10,789

 

598
00:19:10,789 --> 00:19:12,710

 

599
00:19:12,710 --> 00:19:12,720

 

600
00:19:12,720 --> 00:19:14,310


601
00:19:14,310 --> 00:19:15,909

 

602
00:19:15,909 --> 00:19:15,919

 

603
00:19:15,919 --> 00:19:17,029


604
00:19:17,029 --> 00:19:19,590

 

605
00:19:19,590 --> 00:19:21,270

 

606
00:19:21,270 --> 00:19:22,549

 

607
00:19:22,549 --> 00:19:26,070

 

608
00:19:26,070 --> 00:19:28,390

 

609
00:19:28,390 --> 00:19:30,310

 

610
00:19:30,310 --> 00:19:32,710

 

611
00:19:32,710 --> 00:19:34,230

 

612
00:19:34,230 --> 00:19:37,590

 

613
00:19:37,590 --> 00:19:40,310

 

614
00:19:40,310 --> 00:19:43,110

 

615
00:19:43,110 --> 00:19:45,029

 

616
00:19:45,029 --> 00:19:47,830

 

617
00:19:47,830 --> 00:19:50,310

 

618
00:19:50,310 --> 00:19:52,470

 

619
00:19:52,470 --> 00:19:54,630

 

620
00:19:54,630 --> 00:19:56,470

 

621
00:19:56,470 --> 00:19:56,480

 

622
00:19:56,480 --> 00:19:58,549


623
00:19:58,549 --> 00:20:00,630

 

624
00:20:00,630 --> 00:20:03,190

 

625
00:20:03,190 --> 00:20:06,070

 

626
00:20:06,070 --> 00:20:08,470

 

627
00:20:08,470 --> 00:20:10,230

 

628
00:20:10,230 --> 00:20:12,470

 

629
00:20:12,470 --> 00:20:14,630

 

630
00:20:14,630 --> 00:20:16,630

 

631
00:20:16,630 --> 00:20:18,870

 

632
00:20:18,870 --> 00:20:22,230

 

633
00:20:22,230 --> 00:20:24,789

 

634
00:20:24,789 --> 00:20:27,510

 

635
00:20:27,510 --> 00:20:29,190

 

636
00:20:29,190 --> 00:20:30,310

 

637
00:20:30,310 --> 00:20:34,070

 

638
00:20:34,070 --> 00:20:36,870

 

639
00:20:36,870 --> 00:20:38,710

 

640
00:20:38,710 --> 00:20:41,750

 

641
00:20:41,750 --> 00:20:45,909

 

642
00:20:45,909 --> 00:20:49,029

 

643
00:20:49,029 --> 00:20:51,029

 

644
00:20:51,029 --> 00:20:53,430

 

645
00:20:53,430 --> 00:20:55,029

 

646
00:20:55,029 --> 00:20:55,039

 

647
00:20:55,039 --> 00:20:56,310


648
00:20:56,310 --> 00:20:58,149

 

649
00:20:58,149 --> 00:20:59,590

 

650
00:20:59,590 --> 00:21:01,430

 

651
00:21:01,430 --> 00:21:03,110

 

652
00:21:03,110 --> 00:21:04,950

 

653
00:21:04,950 --> 00:21:07,110

 

654
00:21:07,110 --> 00:21:08,630

 

655
00:21:08,630 --> 00:21:08,640

 

656
00:21:08,640 --> 00:21:11,029


657
00:21:11,029 --> 00:21:13,669

 

658
00:21:13,669 --> 00:21:14,950

 

659
00:21:14,950 --> 00:21:21,510

 

660
00:21:21,510 --> 00:21:23,830

 

661
00:21:23,830 --> 00:21:25,590

 

662
00:21:25,590 --> 00:21:29,029

 

663
00:21:29,029 --> 00:21:31,350

 

664
00:21:31,350 --> 00:21:34,230

 

665
00:21:34,230 --> 00:21:34,240

 

666
00:21:34,240 --> 00:21:38,070


667
00:21:38,070 --> 00:21:39,909

 

668
00:21:39,909 --> 00:21:41,990

 

669
00:21:41,990 --> 00:21:46,230

 

670
00:21:46,230 --> 00:21:46,240

 

671
00:21:46,240 --> 00:21:47,110


672
00:21:47,110 --> 00:21:49,029

 

673
00:21:49,029 --> 00:21:51,190

 

674
00:21:51,190 --> 00:21:53,750

 

675
00:21:53,750 --> 00:21:55,510

 

676
00:21:55,510 --> 00:21:57,190

 

677
00:21:57,190 --> 00:21:59,830

 

678
00:21:59,830 --> 00:22:01,590

 

679
00:22:01,590 --> 00:22:01,600

 

680
00:22:01,600 --> 00:22:02,789


681
00:22:02,789 --> 00:22:04,789

 

682
00:22:04,789 --> 00:22:06,630

 

683
00:22:06,630 --> 00:22:09,510

 

684
00:22:09,510 --> 00:22:13,430

 

685
00:22:13,430 --> 00:22:13,440

 

686
00:22:13,440 --> 00:22:14,470


687
00:22:14,470 --> 00:22:14,480

 

688
00:22:14,480 --> 00:22:15,909


689
00:22:15,909 --> 00:22:18,630

 

690
00:22:18,630 --> 00:22:21,430

 

691
00:22:21,430 --> 00:22:21,440

 

692
00:22:21,440 --> 00:22:22,950


693
00:22:22,950 --> 00:22:22,960

 

694
00:22:22,960 --> 00:22:25,110


695
00:22:25,110 --> 00:22:26,710

 

696
00:22:26,710 --> 00:22:29,110

 

697
00:22:29,110 --> 00:22:29,120

 

698
00:22:29,120 --> 00:22:30,390


699
00:22:30,390 --> 00:22:32,549

 

700
00:22:32,549 --> 00:22:35,190

 

701
00:22:35,190 --> 00:22:37,190

 

702
00:22:37,190 --> 00:22:37,200

 

703
00:22:37,200 --> 00:22:39,590


704
00:22:39,590 --> 00:22:39,600

 

705
00:22:39,600 --> 00:22:40,390


706
00:22:40,390 --> 00:22:40,400

 

707
00:22:40,400 --> 00:22:42,070


708
00:22:42,070 --> 00:22:43,750

 

709
00:22:43,750 --> 00:22:45,669

 

710
00:22:45,669 --> 00:22:48,789

 

711
00:22:48,789 --> 00:22:50,549

 

712
00:22:50,549 --> 00:22:53,669

 

713
00:22:53,669 --> 00:22:55,830

 

714
00:22:55,830 --> 00:22:57,590

 

715
00:22:57,590 --> 00:23:04,230

 

716
00:23:04,230 --> 00:23:06,950

 

717
00:23:06,950 --> 00:23:10,390

 

718
00:23:10,390 --> 00:23:12,149

 

719
00:23:12,149 --> 00:23:14,070

 

720
00:23:14,070 --> 00:23:18,070

 

721
00:23:18,070 --> 00:23:19,909

 

722
00:23:19,909 --> 00:23:22,390

 

723
00:23:22,390 --> 00:23:24,070

 

724
00:23:24,070 --> 00:23:25,830

 

725
00:23:25,830 --> 00:23:27,990

 

726
00:23:27,990 --> 00:23:29,990

 

727
00:23:29,990 --> 00:23:31,350

 

728
00:23:31,350 --> 00:23:33,110

 

729
00:23:33,110 --> 00:23:37,110

 

730
00:23:37,110 --> 00:23:37,120

 

731
00:23:37,120 --> 00:23:38,710


732
00:23:38,710 --> 00:23:40,230

 

733
00:23:40,230 --> 00:23:42,390

 

734
00:23:42,390 --> 00:23:45,350

 

735
00:23:45,350 --> 00:23:47,750

 

736
00:23:47,750 --> 00:23:50,310

 

737
00:23:50,310 --> 00:23:51,430

 

738
00:23:51,430 --> 00:23:51,440

 

739
00:23:51,440 --> 00:23:55,110


740
00:23:55,110 --> 00:23:55,120

 

741
00:23:55,120 --> 00:23:56,710


742
00:23:56,710 --> 00:23:56,720

 

743
00:23:56,720 --> 00:24:00,549


744
00:24:00,549 --> 00:24:02,950

 

745
00:24:02,950 --> 00:24:04,710

 

746
00:24:04,710 --> 00:24:06,950

 

747
00:24:06,950 --> 00:24:08,390

 

748
00:24:08,390 --> 00:24:12,149

 

749
00:24:12,149 --> 00:24:13,669

 

750
00:24:13,669 --> 00:24:14,870

 

751
00:24:14,870 --> 00:24:16,630

 

752
00:24:16,630 --> 00:24:17,669

 

753
00:24:17,669 --> 00:24:19,190

 

754
00:24:19,190 --> 00:24:21,190

 

755
00:24:21,190 --> 00:24:23,430

 

756
00:24:23,430 --> 00:24:24,630

 

757
00:24:24,630 --> 00:24:24,640

 

758
00:24:24,640 --> 00:24:26,549


759
00:24:26,549 --> 00:24:28,310

 

760
00:24:28,310 --> 00:24:30,149

 

761
00:24:30,149 --> 00:24:32,549

 

762
00:24:32,549 --> 00:24:34,789

 

763
00:24:34,789 --> 00:24:36,870

 

764
00:24:36,870 --> 00:24:39,510

 

765
00:24:39,510 --> 00:24:41,350

 

766
00:24:41,350 --> 00:24:41,360

 

767
00:24:41,360 --> 00:24:42,710


768
00:24:42,710 --> 00:24:42,720

 

769
00:24:42,720 --> 00:24:44,789


770
00:24:44,789 --> 00:24:46,390

 

771
00:24:46,390 --> 00:24:48,230

 

772
00:24:48,230 --> 00:24:49,830

 

773
00:24:49,830 --> 00:24:51,510

 

774
00:24:51,510 --> 00:24:54,390

 

775
00:24:54,390 --> 00:24:56,310

 

776
00:24:56,310 --> 00:24:58,390

 

777
00:24:58,390 --> 00:25:01,350

 

778
00:25:01,350 --> 00:25:03,430

 

779
00:25:03,430 --> 00:25:05,110

 

780
00:25:05,110 --> 00:25:07,510

 

781
00:25:07,510 --> 00:25:09,590

 

782
00:25:09,590 --> 00:25:09,600

 

783
00:25:09,600 --> 00:25:12,470


784
00:25:12,470 --> 00:25:13,750

 

785
00:25:13,750 --> 00:25:16,230

 

786
00:25:16,230 --> 00:25:17,350

 

787
00:25:17,350 --> 00:25:18,950

 

788
00:25:18,950 --> 00:25:20,549

 

789
00:25:20,549 --> 00:25:22,230

 

790
00:25:22,230 --> 00:25:22,240

 

791
00:25:22,240 --> 00:25:23,190


792
00:25:23,190 --> 00:25:23,200

 

793
00:25:23,200 --> 00:25:24,950


794
00:25:24,950 --> 00:25:27,269

 

795
00:25:27,269 --> 00:25:29,510

 

796
00:25:29,510 --> 00:25:30,950

 

797
00:25:30,950 --> 00:25:33,190

 

798
00:25:33,190 --> 00:25:37,029

 

799
00:25:37,029 --> 00:25:39,350

 

800
00:25:39,350 --> 00:25:41,190

 

801
00:25:41,190 --> 00:25:42,789

 

802
00:25:42,789 --> 00:25:45,350

 

803
00:25:45,350 --> 00:25:47,269

 

804
00:25:47,269 --> 00:25:49,269

 

805
00:25:49,269 --> 00:25:55,029

 

806
00:25:55,029 --> 00:25:56,950

 

807
00:25:56,950 --> 00:25:58,870

 

808
00:25:58,870 --> 00:26:00,149

 

809
00:26:00,149 --> 00:26:01,830

 

810
00:26:01,830 --> 00:26:01,840

 

811
00:26:01,840 --> 00:26:02,710


812
00:26:02,710 --> 00:26:04,789

 

813
00:26:04,789 --> 00:26:04,799

 

814
00:26:04,799 --> 00:26:07,350


815
00:26:07,350 --> 00:26:10,149

 

816
00:26:10,149 --> 00:26:12,470

 

817
00:26:12,470 --> 00:26:14,390

 

818
00:26:14,390 --> 00:26:17,430

 

819
00:26:17,430 --> 00:26:19,110

 

820
00:26:19,110 --> 00:26:21,510

 

821
00:26:21,510 --> 00:26:21,520

 

822
00:26:21,520 --> 00:26:22,789


823
00:26:22,789 --> 00:26:22,799

 

824
00:26:22,799 --> 00:26:23,669


825
00:26:23,669 --> 00:26:23,679

 

826
00:26:23,679 --> 00:26:24,789


827
00:26:24,789 --> 00:26:24,799

 

828
00:26:24,799 --> 00:26:27,510


829
00:26:27,510 --> 00:26:31,190

 

830
00:26:31,190 --> 00:26:31,200

 

831
00:26:31,200 --> 00:26:33,830


832
00:26:33,830 --> 00:26:37,029

 

833
00:26:37,029 --> 00:26:41,269

 

834
00:26:41,269 --> 00:26:43,190

 

835
00:26:43,190 --> 00:26:45,669

 

836
00:26:45,669 --> 00:26:47,110

 

837
00:26:47,110 --> 00:26:47,120

 

838
00:26:47,120 --> 00:26:48,710


839
00:26:48,710 --> 00:26:50,870

 

840
00:26:50,870 --> 00:26:53,430

 

841
00:26:53,430 --> 00:26:55,029

 

842
00:26:55,029 --> 00:26:56,470

 

843
00:26:56,470 --> 00:26:56,480

 

844
00:26:56,480 --> 00:26:57,669


845
00:26:57,669 --> 00:27:00,230

 

846
00:27:00,230 --> 00:27:01,750

 

847
00:27:01,750 --> 00:27:01,760

 

848
00:27:01,760 --> 00:27:02,630


849
00:27:02,630 --> 00:27:04,149

 

850
00:27:04,149 --> 00:27:05,990

 

851
00:27:05,990 --> 00:27:06,000

 

852
00:27:06,000 --> 00:27:07,110


853
00:27:07,110 --> 00:27:10,230

 

854
00:27:10,230 --> 00:27:12,149

 

855
00:27:12,149 --> 00:27:14,070

 

856
00:27:14,070 --> 00:27:16,710

 

857
00:27:16,710 --> 00:27:20,149

 

858
00:27:20,149 --> 00:27:22,470

 

859
00:27:22,470 --> 00:27:24,870

 

860
00:27:24,870 --> 00:27:27,909

 

861
00:27:27,909 --> 00:27:30,630

 

862
00:27:30,630 --> 00:27:33,830

 

863
00:27:33,830 --> 00:27:36,870

 

864
00:27:36,870 --> 00:27:40,310

 

865
00:27:40,310 --> 00:27:41,990

 

866
00:27:41,990 --> 00:27:44,149

 

867
00:27:44,149 --> 00:27:46,549

 

868
00:27:46,549 --> 00:27:49,029

 

869
00:27:49,029 --> 00:27:49,039

 

870
00:27:49,039 --> 00:27:50,549


871
00:27:50,549 --> 00:27:50,559

 

872
00:27:50,559 --> 00:27:51,350


873
00:27:51,350 --> 00:27:52,870

 

874
00:27:52,870 --> 00:27:54,310

 

875
00:27:54,310 --> 00:27:55,990

 

876
00:27:55,990 --> 00:27:58,310

 

877
00:27:58,310 --> 00:27:59,909

 

878
00:27:59,909 --> 00:28:01,750

 

879
00:28:01,750 --> 00:28:03,909

 

880
00:28:03,909 --> 00:28:03,919

 

881
00:28:03,919 --> 00:28:05,029


882
00:28:05,029 --> 00:28:07,269

 

883
00:28:07,269 --> 00:28:09,669

 

884
00:28:09,669 --> 00:28:11,269

 

885
00:28:11,269 --> 00:28:13,510

 

886
00:28:13,510 --> 00:28:14,710

 

887
00:28:14,710 --> 00:28:17,669

 

888
00:28:17,669 --> 00:28:20,310

 

889
00:28:20,310 --> 00:28:23,430

 

890
00:28:23,430 --> 00:28:23,440

 

891
00:28:23,440 --> 00:28:24,950


892
00:28:24,950 --> 00:28:26,870

 

893
00:28:26,870 --> 00:28:28,470

 

894
00:28:28,470 --> 00:28:30,870

 

895
00:28:30,870 --> 00:28:31,990

 

896
00:28:31,990 --> 00:28:33,909

 

897
00:28:33,909 --> 00:28:35,269

 

898
00:28:35,269 --> 00:28:37,430

 

899
00:28:37,430 --> 00:28:38,950

 

900
00:28:38,950 --> 00:28:41,269

 

901
00:28:41,269 --> 00:28:43,350

 

902
00:28:43,350 --> 00:28:44,789

 

903
00:28:44,789 --> 00:28:47,510

 

904
00:28:47,510 --> 00:28:50,630

 

905
00:28:50,630 --> 00:28:53,269

 

906
00:28:53,269 --> 00:28:56,230

 

907
00:28:56,230 --> 00:28:58,310

 

908
00:28:58,310 --> 00:29:00,549

 

909
00:29:00,549 --> 00:29:02,630

 

910
00:29:02,630 --> 00:29:05,110

 

911
00:29:05,110 --> 00:29:06,950

 

912
00:29:06,950 --> 00:29:08,149

 

913
00:29:08,149 --> 00:29:09,430

 

914
00:29:09,430 --> 00:29:11,350

 

915
00:29:11,350 --> 00:29:16,870

 

916
00:29:16,870 --> 00:29:19,190

 

917
00:29:19,190 --> 00:29:21,350

 

918
00:29:21,350 --> 00:29:23,269

 

919
00:29:23,269 --> 00:29:23,279

 

920
00:29:23,279 --> 00:29:24,389


921
00:29:24,389 --> 00:29:25,830

 

922
00:29:25,830 --> 00:29:26,950

 

923
00:29:26,950 --> 00:29:29,590

 

924
00:29:29,590 --> 00:29:32,789

 

925
00:29:32,789 --> 00:29:34,149

 

926
00:29:34,149 --> 00:29:35,430

 

927
00:29:35,430 --> 00:29:35,440

 

928
00:29:35,440 --> 00:29:37,510


929
00:29:37,510 --> 00:29:39,430

 

930
00:29:39,430 --> 00:29:42,070

 

931
00:29:42,070 --> 00:29:43,909

 

932
00:29:43,909 --> 00:29:43,919

 

933
00:29:43,919 --> 00:29:44,870


934
00:29:44,870 --> 00:29:46,549

 

935
00:29:46,549 --> 00:29:48,470

 

936
00:29:48,470 --> 00:29:50,470

 

937
00:29:50,470 --> 00:29:52,470

 

938
00:29:52,470 --> 00:29:56,789

 

939
00:29:56,789 --> 00:29:56,799

 

940
00:29:56,799 --> 00:29:57,669


941
00:29:57,669 --> 00:29:58,870

 

942
00:29:58,870 --> 00:30:01,750

 

943
00:30:01,750 --> 00:30:03,190

 

944
00:30:03,190 --> 00:30:03,200

 

945
00:30:03,200 --> 00:30:04,950


946
00:30:04,950 --> 00:30:11,350

 

947
00:30:11,350 --> 00:30:11,360

 

948
00:30:11,360 --> 00:30:12,310


949
00:30:12,310 --> 00:30:14,549

 

950
00:30:14,549 --> 00:30:17,590

 

951
00:30:17,590 --> 00:30:20,710

 

952
00:30:20,710 --> 00:30:23,350

 

953
00:30:23,350 --> 00:30:26,549

 

954
00:30:26,549 --> 00:30:29,350

 

955
00:30:29,350 --> 00:30:30,950

 

956
00:30:30,950 --> 00:30:32,549

 

957
00:30:32,549 --> 00:30:34,230

 

958
00:30:34,230 --> 00:30:36,310

 

959
00:30:36,310 --> 00:30:38,950

 

960
00:30:38,950 --> 00:30:41,430

 

961
00:30:41,430 --> 00:30:44,549

 

962
00:30:44,549 --> 00:30:46,230

 

963
00:30:46,230 --> 00:30:48,149

 

964
00:30:48,149 --> 00:30:50,389

 

965
00:30:50,389 --> 00:30:52,149

 

966
00:30:52,149 --> 00:30:54,389

 

967
00:30:54,389 --> 00:30:56,549

 

968
00:30:56,549 --> 00:30:56,559

 

969
00:30:56,559 --> 00:30:58,070


970
00:30:58,070 --> 00:30:58,080

 

971
00:30:58,080 --> 00:30:59,750


972
00:30:59,750 --> 00:31:01,990

 

973
00:31:01,990 --> 00:31:04,549

 

974
00:31:04,549 --> 00:31:06,549

 

975
00:31:06,549 --> 00:31:08,549

 

976
00:31:08,549 --> 00:31:11,669

 

977
00:31:11,669 --> 00:31:13,350

 

978
00:31:13,350 --> 00:31:15,350

 

979
00:31:15,350 --> 00:31:16,950

 

980
00:31:16,950 --> 00:31:19,190

 

981
00:31:19,190 --> 00:31:19,200

 

982
00:31:19,200 --> 00:31:21,509


983
00:31:21,509 --> 00:31:21,519

 

984
00:31:21,519 --> 00:31:22,630


985
00:31:22,630 --> 00:31:22,640

 

986
00:31:22,640 --> 00:31:25,110


987
00:31:25,110 --> 00:31:25,120

 

988
00:31:25,120 --> 00:31:25,909


989
00:31:25,909 --> 00:31:27,350

 

990
00:31:27,350 --> 00:31:29,350

 

991
00:31:29,350 --> 00:31:32,630

 

992
00:31:32,630 --> 00:31:34,389

 

993
00:31:34,389 --> 00:31:35,990

 

994
00:31:35,990 --> 00:31:37,110

 

995
00:31:37,110 --> 00:31:38,470

 

996
00:31:38,470 --> 00:31:40,470

 

997
00:31:40,470 --> 00:31:42,710

 

998
00:31:42,710 --> 00:31:45,029

 

999
00:31:45,029 --> 00:31:46,549

 

1000
00:31:46,549 --> 00:31:46,559

 

1001
00:31:46,559 --> 00:31:47,430


1002
00:31:47,430 --> 00:31:50,149

 

1003
00:31:50,149 --> 00:31:50,159

 

1004
00:31:50,159 --> 00:31:50,950


1005
00:31:50,950 --> 00:31:52,710

 

1006
00:31:52,710 --> 00:31:54,950

 

1007
00:31:54,950 --> 00:31:57,430

 

1008
00:31:57,430 --> 00:31:59,269

 

1009
00:31:59,269 --> 00:32:02,870

 

1010
00:32:02,870 --> 00:32:05,029

 

1011
00:32:05,029 --> 00:32:07,909

 

1012
00:32:07,909 --> 00:32:07,919

 

1013
00:32:07,919 --> 00:32:08,870


1014
00:32:08,870 --> 00:32:10,310

 

1015
00:32:10,310 --> 00:32:13,909

 

1016
00:32:13,909 --> 00:32:16,630

 

1017
00:32:16,630 --> 00:32:19,190

 

1018
00:32:19,190 --> 00:32:21,029

 

1019
00:32:21,029 --> 00:32:22,870

 

1020
00:32:22,870 --> 00:32:25,029

 

1021
00:32:25,029 --> 00:32:27,269

 

1022
00:32:27,269 --> 00:32:29,669

 

1023
00:32:29,669 --> 00:32:31,190

 

1024
00:32:31,190 --> 00:32:33,669

 

1025
00:32:33,669 --> 00:32:35,350

 

1026
00:32:35,350 --> 00:32:36,950

 

1027
00:32:36,950 --> 00:32:40,310

 

1028
00:32:40,310 --> 00:32:42,389

 

1029
00:32:42,389 --> 00:32:44,549

 

1030
00:32:44,549 --> 00:32:46,789

 

1031
00:32:46,789 --> 00:32:48,310

 

1032
00:32:48,310 --> 00:32:52,870

 

1033
00:32:52,870 --> 00:32:54,630

 

1034
00:32:54,630 --> 00:32:56,789

 

1035
00:32:56,789 --> 00:32:58,710

 

1036
00:32:58,710 --> 00:32:59,990

 

1037
00:32:59,990 --> 00:33:01,350

 

1038
00:33:01,350 --> 00:33:02,710

 

1039
00:33:02,710 --> 00:33:06,470

 

1040
00:33:06,470 --> 00:33:08,310

 

1041
00:33:08,310 --> 00:33:12,230

 

1042
00:33:12,230 --> 00:33:14,549

 

1043
00:33:14,549 --> 00:33:16,630

 

1044
00:33:16,630 --> 00:33:19,110

 

1045
00:33:19,110 --> 00:33:21,909

 

1046
00:33:21,909 --> 00:33:23,990

 

1047
00:33:23,990 --> 00:33:25,430

 

1048
00:33:25,430 --> 00:33:27,269

 

1049
00:33:27,269 --> 00:33:29,509

 

1050
00:33:29,509 --> 00:33:32,070

 

1051
00:33:32,070 --> 00:33:32,950

 

1052
00:33:32,950 --> 00:33:34,950

 

1053
00:33:34,950 --> 00:33:36,389

 

1054
00:33:36,389 --> 00:33:38,549

 

1055
00:33:38,549 --> 00:33:40,389

 

1056
00:33:40,389 --> 00:33:41,990

 

1057
00:33:41,990 --> 00:33:43,750

 

1058
00:33:43,750 --> 00:33:45,350

 

1059
00:33:45,350 --> 00:33:47,509

 

1060
00:33:47,509 --> 00:33:49,830

 

1061
00:33:49,830 --> 00:33:49,840

 

1062
00:33:49,840 --> 00:33:53,669


1063
00:33:53,669 --> 00:33:58,710

 

1064
00:33:58,710 --> 00:34:00,549

 

1065
00:34:00,549 --> 00:34:01,830

 

1066
00:34:01,830 --> 00:34:03,590

 

1067
00:34:03,590 --> 00:34:03,600

 

1068
00:34:03,600 --> 00:34:04,470


1069
00:34:04,470 --> 00:34:06,470

 

1070
00:34:06,470 --> 00:34:07,430

 

1071
00:34:07,430 --> 00:34:10,069

 

1072
00:34:10,069 --> 00:34:12,550

 

1073
00:34:12,550 --> 00:34:12,560

 

1074
00:34:12,560 --> 00:34:13,510


1075
00:34:13,510 --> 00:34:15,909

 

1076
00:34:15,909 --> 00:34:17,510

 

1077
00:34:17,510 --> 00:34:19,589

 

1078
00:34:19,589 --> 00:34:21,030

 

1079
00:34:21,030 --> 00:34:23,109

 

1080
00:34:23,109 --> 00:34:25,909

 

1081
00:34:25,909 --> 00:34:27,990

 

1082
00:34:27,990 --> 00:34:31,109

 

1083
00:34:31,109 --> 00:34:33,430

 

1084
00:34:33,430 --> 00:34:33,440

 

1085
00:34:33,440 --> 00:34:34,790


1086
00:34:34,790 --> 00:34:36,389

 

1087
00:34:36,389 --> 00:34:41,270

 

1088
00:34:41,270 --> 00:34:43,109

 

1089
00:34:43,109 --> 00:34:44,790

 

1090
00:34:44,790 --> 00:34:44,800

 

1091
00:34:44,800 --> 00:34:46,869


1092
00:34:46,869 --> 00:34:50,470

 

1093
00:34:50,470 --> 00:34:53,750

 

1094
00:34:53,750 --> 00:34:56,869

 

1095
00:34:56,869 --> 00:34:59,190

 

1096
00:34:59,190 --> 00:35:00,790

 

1097
00:35:00,790 --> 00:35:02,790

 

1098
00:35:02,790 --> 00:35:05,109

 

1099
00:35:05,109 --> 00:35:07,030

 

1100
00:35:07,030 --> 00:35:09,990

 

1101
00:35:09,990 --> 00:35:11,589

 

1102
00:35:11,589 --> 00:35:14,550

 

1103
00:35:14,550 --> 00:35:17,190

 

1104
00:35:17,190 --> 00:35:19,349

 

1105
00:35:19,349 --> 00:35:21,990

 

1106
00:35:21,990 --> 00:35:24,150

 

1107
00:35:24,150 --> 00:35:25,109

 

1108
00:35:25,109 --> 00:35:28,870

 

1109
00:35:28,870 --> 00:35:31,510

 

1110
00:35:31,510 --> 00:35:33,109

 

1111
00:35:33,109 --> 00:35:35,030

 

1112
00:35:35,030 --> 00:35:37,109

 

1113
00:35:37,109 --> 00:35:39,270

 

1114
00:35:39,270 --> 00:35:40,870

 

1115
00:35:40,870 --> 00:35:43,589

 

1116
00:35:43,589 --> 00:35:45,750

 

1117
00:35:45,750 --> 00:35:47,430

 

1118
00:35:47,430 --> 00:35:49,349

 

1119
00:35:49,349 --> 00:35:49,359

 

1120
00:35:49,359 --> 00:35:50,710


1121
00:35:50,710 --> 00:35:53,750

 

1122
00:35:53,750 --> 00:36:01,589

 

1123
00:36:01,589 --> 00:36:05,030

 

1124
00:36:05,030 --> 00:36:07,190

 

1125
00:36:07,190 --> 00:36:08,790

 

1126
00:36:08,790 --> 00:36:15,109

 

1127
00:36:15,109 --> 00:36:15,119

 

1128
00:36:15,119 --> 00:36:16,870


1129
00:36:16,870 --> 00:36:23,910

 

1130
00:36:23,910 --> 00:36:25,829

 

1131
00:36:25,829 --> 00:36:28,310

 

1132
00:36:28,310 --> 00:36:30,630

 

1133
00:36:30,630 --> 00:36:32,230

 

1134
00:36:32,230 --> 00:36:34,790

 

1135
00:36:34,790 --> 00:36:36,550

 

1136
00:36:36,550 --> 00:36:38,150

 

1137
00:36:38,150 --> 00:36:39,990

 

1138
00:36:39,990 --> 00:36:41,750

 

1139
00:36:41,750 --> 00:36:41,760

 

1140
00:36:41,760 --> 00:36:43,349


1141
00:36:43,349 --> 00:36:45,190

 

1142
00:36:45,190 --> 00:36:47,829

 

1143
00:36:47,829 --> 00:36:49,510

 

1144
00:36:49,510 --> 00:36:51,109

 

1145
00:36:51,109 --> 00:36:53,270

 

1146
00:36:53,270 --> 00:36:54,710

 

1147
00:36:54,710 --> 00:36:57,910

 

1148
00:36:57,910 --> 00:37:01,190

 

1149
00:37:01,190 --> 00:37:03,190

 

1150
00:37:03,190 --> 00:37:05,430

 

1151
00:37:05,430 --> 00:37:07,589

 

1152
00:37:07,589 --> 00:37:08,630

 

1153
00:37:08,630 --> 00:37:10,069

 

1154
00:37:10,069 --> 00:37:13,270

 

1155
00:37:13,270 --> 00:37:13,280

 

1156
00:37:13,280 --> 00:37:14,069


1157
00:37:14,069 --> 00:37:15,589

 

1158
00:37:15,589 --> 00:37:17,109

 

1159
00:37:17,109 --> 00:37:18,710

 

1160
00:37:18,710 --> 00:37:22,390

 

1161
00:37:22,390 --> 00:37:25,349

 

1162
00:37:25,349 --> 00:37:27,190

 

1163
00:37:27,190 --> 00:37:28,790

 

1164
00:37:28,790 --> 00:37:28,800

 

1165
00:37:28,800 --> 00:37:30,390


1166
00:37:30,390 --> 00:37:32,550

 

1167
00:37:32,550 --> 00:37:35,270

 

1168
00:37:35,270 --> 00:37:36,630

 

1169
00:37:36,630 --> 00:37:38,790

 

1170
00:37:38,790 --> 00:37:40,150

 

1171
00:37:40,150 --> 00:37:41,670

 

1172
00:37:41,670 --> 00:37:41,680

 

1173
00:37:41,680 --> 00:37:42,870


1174
00:37:42,870 --> 00:37:42,880

 

1175
00:37:42,880 --> 00:37:46,069


1176
00:37:46,069 --> 00:37:47,349

 

1177
00:37:47,349 --> 00:37:49,109

 

1178
00:37:49,109 --> 00:37:51,270

 

1179
00:37:51,270 --> 00:37:53,190

 

1180
00:37:53,190 --> 00:37:55,109

 

1181
00:37:55,109 --> 00:37:55,119

 

1182
00:37:55,119 --> 00:37:56,310


1183
00:37:56,310 --> 00:37:58,390

 

1184
00:37:58,390 --> 00:38:00,230

 

1185
00:38:00,230 --> 00:38:02,230

 

1186
00:38:02,230 --> 00:38:04,470

 

1187
00:38:04,470 --> 00:38:05,829

 

1188
00:38:05,829 --> 00:38:08,230

 

1189
00:38:08,230 --> 00:38:10,630

 

1190
00:38:10,630 --> 00:38:12,390

 

1191
00:38:12,390 --> 00:38:13,990

 

1192
00:38:13,990 --> 00:38:14,000

 

1193
00:38:14,000 --> 00:38:14,950


1194
00:38:14,950 --> 00:38:16,310

 

1195
00:38:16,310 --> 00:38:17,589

 

1196
00:38:17,589 --> 00:38:19,270

 

1197
00:38:19,270 --> 00:38:21,510

 

1198
00:38:21,510 --> 00:38:24,390

 

1199
00:38:24,390 --> 00:38:26,550

 

1200
00:38:26,550 --> 00:38:27,910

 

1201
00:38:27,910 --> 00:38:29,670

 

1202
00:38:29,670 --> 00:38:34,230

 

1203
00:38:34,230 --> 00:38:34,240

 

1204
00:38:34,240 --> 00:38:36,870


1205
00:38:36,870 --> 00:38:38,540

 

1206
00:38:38,540 --> 00:38:38,550

 

1207
00:38:38,550 --> 00:38:41,349


1208
00:38:41,349 --> 00:38:43,670

 

1209
00:38:43,670 --> 00:38:45,910

 

1210
00:38:45,910 --> 00:38:49,829

 

1211
00:38:49,829 --> 00:38:49,839

 

1212
00:38:49,839 --> 00:38:51,109


1213
00:38:51,109 --> 00:38:52,390

 

1214
00:38:52,390 --> 00:38:53,990

 

1215
00:38:53,990 --> 00:38:56,390

 

1216
00:38:56,390 --> 00:38:57,910

 

1217
00:38:57,910 --> 00:39:00,870

 

1218
00:39:00,870 --> 00:39:02,390

 

1219
00:39:02,390 --> 00:39:02,400

 

1220
00:39:02,400 --> 00:39:03,190


1221
00:39:03,190 --> 00:39:03,200

 

1222
00:39:03,200 --> 00:39:04,150


1223
00:39:04,150 --> 00:39:05,430

 

1224
00:39:05,430 --> 00:39:07,510

 

1225
00:39:07,510 --> 00:39:07,520

 

1226
00:39:07,520 --> 00:39:08,790


1227
00:39:08,790 --> 00:39:08,800

 

1228
00:39:08,800 --> 00:39:11,670


1229
00:39:11,670 --> 00:39:16,470

 

1230
00:39:16,470 --> 00:39:18,310

 

1231
00:39:18,310 --> 00:39:19,270

 

1232
00:39:19,270 --> 00:39:21,510

 

1233
00:39:21,510 --> 00:39:22,870

 

1234
00:39:22,870 --> 00:39:25,190

 

1235
00:39:25,190 --> 00:39:26,470

 

1236
00:39:26,470 --> 00:39:28,630

 

1237
00:39:28,630 --> 00:39:33,109

 

1238
00:39:33,109 --> 00:39:33,119

 

1239
00:39:33,119 --> 00:39:34,550


1240
00:39:34,550 --> 00:39:35,990

 

1241
00:39:35,990 --> 00:39:37,990

 

1242
00:39:37,990 --> 00:39:38,000

 

1243
00:39:38,000 --> 00:39:39,109


1244
00:39:39,109 --> 00:39:41,510

 

1245
00:39:41,510 --> 00:39:43,190

 

1246
00:39:43,190 --> 00:39:44,950

 

1247
00:39:44,950 --> 00:39:47,589

 

1248
00:39:47,589 --> 00:39:50,069

 

1249
00:39:50,069 --> 00:39:52,390

 

1250
00:39:52,390 --> 00:39:52,400

 

1251
00:39:52,400 --> 00:39:53,750


1252
00:39:53,750 --> 00:39:57,190

 

1253
00:39:57,190 --> 00:40:02,150

 

1254
00:40:02,150 --> 00:40:05,430

 

1255
00:40:05,430 --> 00:40:09,510

 

1256
00:40:09,510 --> 00:40:09,520

 

1257
00:40:09,520 --> 00:40:12,069


1258
00:40:12,069 --> 00:40:12,079

 

1259
00:40:12,079 --> 00:40:13,589


1260
00:40:13,589 --> 00:40:16,390

 

1261
00:40:16,390 --> 00:40:17,750

 

1262
00:40:17,750 --> 00:40:20,230

 

1263
00:40:20,230 --> 00:40:28,230

 

1264
00:40:28,230 --> 00:40:30,069

 

1265
00:40:30,069 --> 00:40:33,349

 

1266
00:40:33,349 --> 00:40:33,359

 

1267
00:40:33,359 --> 00:40:35,349


1268
00:40:35,349 --> 00:40:37,030

 

1269
00:40:37,030 --> 00:40:38,390

 

1270
00:40:38,390 --> 00:40:40,710

 

1271
00:40:40,710 --> 00:40:42,630

 

1272
00:40:42,630 --> 00:40:44,150

 

1273
00:40:44,150 --> 00:40:44,160

 

1274
00:40:44,160 --> 00:40:48,790


1275
00:40:48,790 --> 00:40:48,800

 

1276
00:40:48,800 --> 00:40:49,829


1277
00:40:49,829 --> 00:40:52,550

 

1278
00:40:52,550 --> 00:40:52,560

 

1279
00:40:52,560 --> 00:40:53,510


1280
00:40:53,510 --> 00:40:53,520

 

1281
00:40:53,520 --> 00:40:55,990


1282
00:40:55,990 --> 00:40:58,710

 

1283
00:40:58,710 --> 00:41:00,150

 

1284
00:41:00,150 --> 00:41:02,550

 

1285
00:41:02,550 --> 00:41:05,190

 

1286
00:41:05,190 --> 00:41:06,870

 

1287
00:41:06,870 --> 00:41:08,790

 

1288
00:41:08,790 --> 00:41:11,190

 

1289
00:41:11,190 --> 00:41:13,990

 

1290
00:41:13,990 --> 00:41:16,950

 

1291
00:41:16,950 --> 00:41:18,470

 

1292
00:41:18,470 --> 00:41:21,430

 

1293
00:41:21,430 --> 00:41:21,440

 

1294
00:41:21,440 --> 00:41:23,030


1295
00:41:23,030 --> 00:41:24,710

 

1296
00:41:24,710 --> 00:41:26,710

 

1297
00:41:26,710 --> 00:41:29,190

 

1298
00:41:29,190 --> 00:41:31,190

 

1299
00:41:31,190 --> 00:41:33,510

 

1300
00:41:33,510 --> 00:41:33,520

 

1301
00:41:33,520 --> 00:41:36,630


1302
00:41:36,630 --> 00:41:38,069

 

1303
00:41:38,069 --> 00:41:39,910

 

1304
00:41:39,910 --> 00:41:41,430

 

1305
00:41:41,430 --> 00:41:43,750

 

1306
00:41:43,750 --> 00:41:43,760

 

1307
00:41:43,760 --> 00:41:44,550


1308
00:41:44,550 --> 00:41:46,470

 

1309
00:41:46,470 --> 00:41:48,710

 

1310
00:41:48,710 --> 00:41:51,829

 

1311
00:41:51,829 --> 00:41:53,349

 

1312
00:41:53,349 --> 00:41:55,349

 

1313
00:41:55,349 --> 00:41:57,030

 

1314
00:41:57,030 --> 00:41:58,710

 

1315
00:41:58,710 --> 00:42:01,109

 

1316
00:42:01,109 --> 00:42:01,119

 

1317
00:42:01,119 --> 00:42:02,390


1318
00:42:02,390 --> 00:42:04,390

 

1319
00:42:04,390 --> 00:42:07,430

 

1320
00:42:07,430 --> 00:42:09,670

 

1321
00:42:09,670 --> 00:42:12,470

 

1322
00:42:12,470 --> 00:42:14,069

 

1323
00:42:14,069 --> 00:42:17,349

 

1324
00:42:17,349 --> 00:42:19,750

 

1325
00:42:19,750 --> 00:42:20,790

 

1326
00:42:20,790 --> 00:42:22,309

 

1327
00:42:22,309 --> 00:42:24,470

 

1328
00:42:24,470 --> 00:42:27,589

 

1329
00:42:27,589 --> 00:42:30,550

 

1330
00:42:30,550 --> 00:42:33,190

 

1331
00:42:33,190 --> 00:42:39,270

 

1332
00:42:39,270 --> 00:42:39,280

 

1333
00:42:39,280 --> 00:42:42,150


1334
00:42:42,150 --> 00:42:43,510

 

1335
00:42:43,510 --> 00:42:43,520

 

1336
00:42:43,520 --> 00:42:49,030


1337
00:42:49,030 --> 00:42:49,040

 

1338
00:42:49,040 --> 00:42:50,630


1339
00:42:50,630 --> 00:42:52,550

 

1340
00:42:52,550 --> 00:42:55,990

 

1341
00:42:55,990 --> 00:42:57,349

 

1342
00:42:57,349 --> 00:42:58,870

 

1343
00:42:58,870 --> 00:43:00,870

 

1344
00:43:00,870 --> 00:43:02,710

 

1345
00:43:02,710 --> 00:43:04,630

 

1346
00:43:04,630 --> 00:43:07,190

 

1347
00:43:07,190 --> 00:43:08,630

 

1348
00:43:08,630 --> 00:43:11,030

 

1349
00:43:11,030 --> 00:43:13,510

 

1350
00:43:13,510 --> 00:43:15,589

 

1351
00:43:15,589 --> 00:43:17,670

 

1352
00:43:17,670 --> 00:43:19,750

 

1353
00:43:19,750 --> 00:43:21,990

 

1354
00:43:21,990 --> 00:43:23,589

 

1355
00:43:23,589 --> 00:43:25,190

 

1356
00:43:25,190 --> 00:43:25,200

 

1357
00:43:25,200 --> 00:43:28,870


1358
00:43:28,870 --> 00:43:28,880

 

1359
00:43:28,880 --> 00:43:30,470


1360
00:43:30,470 --> 00:43:33,910

 

1361
00:43:33,910 --> 00:43:35,589

 

1362
00:43:35,589 --> 00:43:37,190

 

1363
00:43:37,190 --> 00:43:40,710

 

1364
00:43:40,710 --> 00:43:45,190

 

1365
00:43:45,190 --> 00:43:45,200

 

1366
00:43:45,200 --> 00:43:45,570


1367
00:43:45,570 --> 00:43:45,580

 

1368
00:43:45,580 --> 00:43:47,990


1369
00:43:47,990 --> 00:43:48,000

 

1370
00:43:48,000 --> 00:43:48,790


1371
00:43:48,790 --> 00:43:51,030

 

1372
00:43:51,030 --> 00:43:54,790

 

1373
00:43:54,790 --> 00:43:58,150

 

1374
00:43:58,150 --> 00:43:58,160

 

1375
00:43:58,160 --> 00:44:00,230


1376
00:44:00,230 --> 00:44:00,240

 

1377
00:44:00,240 --> 00:44:01,349


1378
00:44:01,349 --> 00:44:03,670

 

1379
00:44:03,670 --> 00:44:05,270

 

1380
00:44:05,270 --> 00:44:09,670

 

1381
00:44:09,670 --> 00:44:11,109

 

1382
00:44:11,109 --> 00:44:13,510

 

1383
00:44:13,510 --> 00:44:16,870

 

1384
00:44:16,870 --> 00:44:18,550

 

1385
00:44:18,550 --> 00:44:21,030

 

1386
00:44:21,030 --> 00:44:23,510

 

1387
00:44:23,510 --> 00:44:25,829

 

1388
00:44:25,829 --> 00:44:27,750

 

1389
00:44:27,750 --> 00:44:27,760

 

1390
00:44:27,760 --> 00:44:28,630


1391
00:44:28,630 --> 00:44:30,630

 

1392
00:44:30,630 --> 00:44:39,349

 

1393
00:44:39,349 --> 00:44:41,270

 

1394
00:44:41,270 --> 00:44:42,710

 

1395
00:44:42,710 --> 00:44:44,630

 

1396
00:44:44,630 --> 00:44:46,950

 

1397
00:44:46,950 --> 00:44:49,349

 

1398
00:44:49,349 --> 00:44:52,150

 

1399
00:44:52,150 --> 00:44:54,390

 

1400
00:44:54,390 --> 00:44:55,670

 

1401
00:44:55,670 --> 00:44:57,829

 

1402
00:44:57,829 --> 00:44:59,430

 

1403
00:44:59,430 --> 00:44:59,440

 

1404
00:44:59,440 --> 00:45:00,470


1405
00:45:00,470 --> 00:45:02,470

 

1406
00:45:02,470 --> 00:45:05,510

 

1407
00:45:05,510 --> 00:45:08,069

 

1408
00:45:08,069 --> 00:45:10,069

 

1409
00:45:10,069 --> 00:45:11,910

 

1410
00:45:11,910 --> 00:45:13,589

 

1411
00:45:13,589 --> 00:45:15,589

 

1412
00:45:15,589 --> 00:45:17,430

 

1413
00:45:17,430 --> 00:45:19,190

 

1414
00:45:19,190 --> 00:45:20,790

 

1415
00:45:20,790 --> 00:45:22,069

 

1416
00:45:22,069 --> 00:45:23,990

 

1417
00:45:23,990 --> 00:45:25,910

 

1418
00:45:25,910 --> 00:45:27,190

 

1419
00:45:27,190 --> 00:45:29,190

 

1420
00:45:29,190 --> 00:45:29,200

 

1421
00:45:29,200 --> 00:45:31,270


1422
00:45:31,270 --> 00:45:33,190

 

1423
00:45:33,190 --> 00:45:33,200

 

1424
00:45:33,200 --> 00:45:34,309


1425
00:45:34,309 --> 00:45:35,910

 

1426
00:45:35,910 --> 00:45:39,510

 

1427
00:45:39,510 --> 00:45:41,190

 

1428
00:45:41,190 --> 00:45:43,829

 

1429
00:45:43,829 --> 00:45:45,829

 

1430
00:45:45,829 --> 00:45:47,349

 

1431
00:45:47,349 --> 00:45:49,510

 

1432
00:45:49,510 --> 00:45:52,950

 

1433
00:45:52,950 --> 00:45:54,790

 

1434
00:45:54,790 --> 00:45:56,309

 

1435
00:45:56,309 --> 00:46:00,550

 

1436
00:46:00,550 --> 00:46:02,870

 

1437
00:46:02,870 --> 00:46:04,950

 

1438
00:46:04,950 --> 00:46:07,670

 

1439
00:46:07,670 --> 00:46:08,710

 

1440
00:46:08,710 --> 00:46:08,720

 

1441
00:46:08,720 --> 00:46:11,349


1442
00:46:11,349 --> 00:46:12,950

 

1443
00:46:12,950 --> 00:46:14,550

 

1444
00:46:14,550 --> 00:46:16,550

 

1445
00:46:16,550 --> 00:46:17,990

 

1446
00:46:17,990 --> 00:46:20,950

 

1447
00:46:20,950 --> 00:46:23,750

 

1448
00:46:23,750 --> 00:46:26,870

 

1449
00:46:26,870 --> 00:46:30,390

 

1450
00:46:30,390 --> 00:46:31,829

 

1451
00:46:31,829 --> 00:46:34,550

 

1452
00:46:34,550 --> 00:46:37,030

 

1453
00:46:37,030 --> 00:46:38,309

 

1454
00:46:38,309 --> 00:46:40,470

 

1455
00:46:40,470 --> 00:46:43,270

 

1456
00:46:43,270 --> 00:46:44,550

 

1457
00:46:44,550 --> 00:46:46,230

 

1458
00:46:46,230 --> 00:46:48,230

 

1459
00:46:48,230 --> 00:46:50,710

 

1460
00:46:50,710 --> 00:46:53,990

 

1461
00:46:53,990 --> 00:46:56,630

 

1462
00:46:56,630 --> 00:46:59,270

 

1463
00:46:59,270 --> 00:47:01,270

 

1464
00:47:01,270 --> 00:47:03,109

 

1465
00:47:03,109 --> 00:47:04,710

 

1466
00:47:04,710 --> 00:47:04,720

 

1467
00:47:04,720 --> 00:47:05,589


1468
00:47:05,589 --> 00:47:07,349

 

1469
00:47:07,349 --> 00:47:09,589

 

1470
00:47:09,589 --> 00:47:11,109

 

1471
00:47:11,109 --> 00:47:13,190

 

1472
00:47:13,190 --> 00:47:14,950

 

1473
00:47:14,950 --> 00:47:16,470

 

1474
00:47:16,470 --> 00:47:18,150

 

1475
00:47:18,150 --> 00:47:20,470

 

1476
00:47:20,470 --> 00:47:20,480

 

1477
00:47:20,480 --> 00:47:21,349


1478
00:47:21,349 --> 00:47:21,359

 

1479
00:47:21,359 --> 00:47:23,430


1480
00:47:23,430 --> 00:47:23,440

 

1481
00:47:23,440 --> 00:47:23,860


1482
00:47:23,860 --> 00:47:23,870

 

1483
00:47:23,870 --> 00:47:25,990


1484
00:47:25,990 --> 00:47:26,000

 

1485
00:47:26,000 --> 00:47:26,870


1486
00:47:26,870 --> 00:47:28,710

 

1487
00:47:28,710 --> 00:47:28,720

 

1488
00:47:28,720 --> 00:47:29,910


1489
00:47:29,910 --> 00:47:32,150

 

1490
00:47:32,150 --> 00:47:34,470

 

1491
00:47:34,470 --> 00:47:36,630

 

1492
00:47:36,630 --> 00:47:39,030

 

1493
00:47:39,030 --> 00:47:42,230

 

1494
00:47:42,230 --> 00:47:43,910

 

1495
00:47:43,910 --> 00:47:47,030

 

1496
00:47:47,030 --> 00:47:49,829

 

1497
00:47:49,829 --> 00:47:52,069

 

1498
00:47:52,069 --> 00:47:56,470

 

1499
00:47:56,470 --> 00:47:58,950

 

1500
00:47:58,950 --> 00:47:58,960

 

1501
00:47:58,960 --> 00:48:01,829


1502
00:48:01,829 --> 00:48:03,750

 

1503
00:48:03,750 --> 00:48:05,430

 

1504
00:48:05,430 --> 00:48:08,309

 

1505
00:48:08,309 --> 00:48:10,710

 

1506
00:48:10,710 --> 00:48:12,710

 

1507
00:48:12,710 --> 00:48:15,190

 

1508
00:48:15,190 --> 00:48:18,390

 

1509
00:48:18,390 --> 00:48:20,309

 

1510
00:48:20,309 --> 00:48:20,319

 

1511
00:48:20,319 --> 00:48:23,670


1512
00:48:23,670 --> 00:48:25,829

 

1513
00:48:25,829 --> 00:48:27,270

 

1514
00:48:27,270 --> 00:48:28,549

 

1515
00:48:28,549 --> 00:48:30,230

 

1516
00:48:30,230 --> 00:48:32,790

 

1517
00:48:32,790 --> 00:48:34,710

 

1518
00:48:34,710 --> 00:48:36,790

 

1519
00:48:36,790 --> 00:48:36,800

 

1520
00:48:36,800 --> 00:48:38,630


1521
00:48:38,630 --> 00:48:41,030

 

1522
00:48:41,030 --> 00:48:43,109

 

1523
00:48:43,109 --> 00:48:44,390

 

1524
00:48:44,390 --> 00:48:45,910

 

1525
00:48:45,910 --> 00:48:48,630

 

1526
00:48:48,630 --> 00:48:51,349

 

1527
00:48:51,349 --> 00:48:53,430

 

1528
00:48:53,430 --> 00:48:55,910

 

1529
00:48:55,910 --> 00:48:58,790

 

1530
00:48:58,790 --> 00:49:02,549

 

1531
00:49:02,549 --> 00:49:08,470

 

1532
00:49:08,470 --> 00:49:08,480

 

1533
00:49:08,480 --> 00:49:10,630


1534
00:49:10,630 --> 00:49:12,470

 

1535
00:49:12,470 --> 00:49:15,109

 

1536
00:49:15,109 --> 00:49:17,109

 

1537
00:49:17,109 --> 00:49:20,069

 

1538
00:49:20,069 --> 00:49:21,750

 

1539
00:49:21,750 --> 00:49:24,309

 

1540
00:49:24,309 --> 00:49:26,870

 

1541
00:49:26,870 --> 00:49:29,990

 

1542
00:49:29,990 --> 00:49:32,069

 

1543
00:49:32,069 --> 00:49:35,349

 

1544
00:49:35,349 --> 00:49:37,510

 

1545
00:49:37,510 --> 00:49:37,520

 

1546
00:49:37,520 --> 00:49:39,190


1547
00:49:39,190 --> 00:49:40,549

 

1548
00:49:40,549 --> 00:49:42,390

 

1549
00:49:42,390 --> 00:49:42,400

 

1550
00:49:42,400 --> 00:49:43,750


1551
00:49:43,750 --> 00:49:46,390

 

1552
00:49:46,390 --> 00:49:47,670

 

1553
00:49:47,670 --> 00:49:47,680

 

1554
00:49:47,680 --> 00:49:49,030


1555
00:49:49,030 --> 00:49:51,829

 

1556
00:49:51,829 --> 00:49:54,630

 

1557
00:49:54,630 --> 00:49:56,390

 

1558
00:49:56,390 --> 00:50:02,470

 

1559
00:50:02,470 --> 00:50:03,829

 

1560
00:50:03,829 --> 00:50:05,349

 

1561
00:50:05,349 --> 00:50:07,829

 

1562
00:50:07,829 --> 00:50:09,910

 

1563
00:50:09,910 --> 00:50:12,230

 

1564
00:50:12,230 --> 00:50:13,270

 

1565
00:50:13,270 --> 00:50:16,069

 

1566
00:50:16,069 --> 00:50:18,390

 

1567
00:50:18,390 --> 00:50:20,069

 

1568
00:50:20,069 --> 00:50:22,790

 

1569
00:50:22,790 --> 00:50:24,950

 

1570
00:50:24,950 --> 00:50:27,190

 

1571
00:50:27,190 --> 00:50:30,630

 

1572
00:50:30,630 --> 00:50:34,230

 

1573
00:50:34,230 --> 00:50:36,710

 

1574
00:50:36,710 --> 00:50:38,470

 

1575
00:50:38,470 --> 00:50:40,870

 

1576
00:50:40,870 --> 00:50:46,710

 

1577
00:50:46,710 --> 00:50:46,720

 

1578
00:50:46,720 --> 00:50:48,549


1579
00:50:48,549 --> 00:50:51,030

 

1580
00:50:51,030 --> 00:50:53,109

 

1581
00:50:53,109 --> 00:50:55,270

 

1582
00:50:55,270 --> 00:50:57,670

 

1583
00:50:57,670 --> 00:50:59,910

 

1584
00:50:59,910 --> 00:51:02,230

 

1585
00:51:02,230 --> 00:51:04,790

 

1586
00:51:04,790 --> 00:51:08,230

 

1587
00:51:08,230 --> 00:51:08,240

 

1588
00:51:08,240 --> 00:51:09,510


1589
00:51:09,510 --> 00:51:10,470

 

1590
00:51:10,470 --> 00:51:13,910

 

1591
00:51:13,910 --> 00:51:13,920

 

1592
00:51:13,920 --> 00:51:15,430


1593
00:51:15,430 --> 00:51:19,030

 

1594
00:51:19,030 --> 00:51:21,190

 

1595
00:51:21,190 --> 00:51:24,710

 

1596
00:51:24,710 --> 00:51:27,109

 

1597
00:51:27,109 --> 00:51:29,829

 

1598
00:51:29,829 --> 00:51:31,349

 

1599
00:51:31,349 --> 00:51:33,670

 

1600
00:51:33,670 --> 00:51:36,069

 

1601
00:51:36,069 --> 00:51:39,670

 

1602
00:51:39,670 --> 00:51:41,510

 

1603
00:51:41,510 --> 00:51:45,349

 

1604
00:51:45,349 --> 00:51:47,349

 

1605
00:51:47,349 --> 00:51:49,109

 

1606
00:51:49,109 --> 00:51:51,109

 

1607
00:51:51,109 --> 00:51:52,790

 

1608
00:51:52,790 --> 00:51:54,390

 

1609
00:51:54,390 --> 00:51:56,470

 

1610
00:51:56,470 --> 00:51:58,230

 

1611
00:51:58,230 --> 00:52:00,630

 

1612
00:52:00,630 --> 00:52:02,950

 

1613
00:52:02,950 --> 00:52:05,670

 

1614
00:52:05,670 --> 00:52:07,430

 

1615
00:52:07,430 --> 00:52:08,950

 

1616
00:52:08,950 --> 00:52:11,030

 

1617
00:52:11,030 --> 00:52:13,109

 

1618
00:52:13,109 --> 00:52:16,150

 

1619
00:52:16,150 --> 00:52:19,190

 

1620
00:52:19,190 --> 00:52:21,270

 

1621
00:52:21,270 --> 00:52:22,870

 

1622
00:52:22,870 --> 00:52:25,670

 

1623
00:52:25,670 --> 00:52:25,680

 

1624
00:52:25,680 --> 00:52:27,109


1625
00:52:27,109 --> 00:52:28,230

 

1626
00:52:28,230 --> 00:52:30,829

 

1627
00:52:30,829 --> 00:52:33,670

 

1628
00:52:33,670 --> 00:52:35,829

 

1629
00:52:35,829 --> 00:52:38,790

 

1630
00:52:38,790 --> 00:52:40,790

 

1631
00:52:40,790 --> 00:52:43,670

 

1632
00:52:43,670 --> 00:52:46,549

 

1633
00:52:46,549 --> 00:52:48,230

 

1634
00:52:48,230 --> 00:52:53,190

 

1635
00:52:53,190 --> 00:52:53,200

 

1636
00:52:53,200 --> 00:52:57,030


1637
00:52:57,030 --> 00:53:02,630

 

1638
00:53:02,630 --> 00:53:04,150

 

1639
00:53:04,150 --> 00:53:07,030

 

1640
00:53:07,030 --> 00:53:09,270

 

1641
00:53:09,270 --> 00:53:12,870

 

1642
00:53:12,870 --> 00:53:14,630

 

1643
00:53:14,630 --> 00:53:16,470

 

1644
00:53:16,470 --> 00:53:17,910

 

1645
00:53:17,910 --> 00:53:19,670

 

1646
00:53:19,670 --> 00:53:21,109

 

1647
00:53:21,109 --> 00:53:22,790

 

1648
00:53:22,790 --> 00:53:25,589

 

1649
00:53:25,589 --> 00:53:27,030

 

1650
00:53:27,030 --> 00:53:29,829

 

1651
00:53:29,829 --> 00:53:32,710

 

1652
00:53:32,710 --> 00:53:35,510

 

1653
00:53:35,510 --> 00:53:37,270

 

1654
00:53:37,270 --> 00:53:39,910

 

1655
00:53:39,910 --> 00:53:42,630

 

1656
00:53:42,630 --> 00:53:44,390

 

1657
00:53:44,390 --> 00:53:47,270

 

1658
00:53:47,270 --> 00:53:51,109

 

1659
00:53:51,109 --> 00:53:54,150

 

1660
00:53:54,150 --> 00:53:57,910

 

1661
00:53:57,910 --> 00:53:57,920

 

1662
00:53:57,920 --> 00:53:58,870


1663
00:53:58,870 --> 00:54:02,230

 

1664
00:54:02,230 --> 00:54:02,240

 

1665
00:54:02,240 --> 00:54:04,710


1666
00:54:04,710 --> 00:54:06,870

 

1667
00:54:06,870 --> 00:54:08,630

 

1668
00:54:08,630 --> 00:54:10,069

 

1669
00:54:10,069 --> 00:54:11,750

 

1670
00:54:11,750 --> 00:54:14,950

 

1671
00:54:14,950 --> 00:54:16,309

 

1672
00:54:16,309 --> 00:54:17,750

 

1673
00:54:17,750 --> 00:54:19,829

 

1674
00:54:19,829 --> 00:54:21,190

 

1675
00:54:21,190 --> 00:54:22,950

 

1676
00:54:22,950 --> 00:54:24,470

 

1677
00:54:24,470 --> 00:54:24,480

 

1678
00:54:24,480 --> 00:54:25,829


1679
00:54:25,829 --> 00:54:28,630

 

1680
00:54:28,630 --> 00:54:33,829

 

1681
00:54:33,829 --> 00:54:33,839

 

1682
00:54:33,839 --> 00:54:36,630


1683
00:54:36,630 --> 00:54:38,950

 

1684
00:54:38,950 --> 00:54:40,230

 

1685
00:54:40,230 --> 00:54:42,950

 

1686
00:54:42,950 --> 00:54:42,960

 

1687
00:54:42,960 --> 00:54:43,990


1688
00:54:43,990 --> 00:54:45,910

 

1689
00:54:45,910 --> 00:54:48,309

 

1690
00:54:48,309 --> 00:54:49,910

 

1691
00:54:49,910 --> 00:54:52,549

 

1692
00:54:52,549 --> 00:54:56,069

 

1693
00:54:56,069 --> 00:54:59,030

 

1694
00:54:59,030 --> 00:55:00,549

 

1695
00:55:00,549 --> 00:55:02,870

 

1696
00:55:02,870 --> 00:55:06,150

 

1697
00:55:06,150 --> 00:55:08,150

 

1698
00:55:08,150 --> 00:55:10,230

 

1699
00:55:10,230 --> 00:55:13,510

 

1700
00:55:13,510 --> 00:55:15,750

 

1701
00:55:15,750 --> 00:55:15,760

 

1702
00:55:15,760 --> 00:55:17,589


1703
00:55:17,589 --> 00:55:19,670

 

1704
00:55:19,670 --> 00:55:21,829

 

1705
00:55:21,829 --> 00:55:23,190

 

1706
00:55:23,190 --> 00:55:25,910

 

1707
00:55:25,910 --> 00:55:27,589

 

1708
00:55:27,589 --> 00:55:29,270

 

1709
00:55:29,270 --> 00:55:29,280

 

1710
00:55:29,280 --> 00:55:30,789


1711
00:55:30,789 --> 00:55:33,270

 

1712
00:55:33,270 --> 00:55:35,510

 

1713
00:55:35,510 --> 00:55:37,589

 

1714
00:55:37,589 --> 00:55:39,750

 

1715
00:55:39,750 --> 00:55:43,190

 

1716
00:55:43,190 --> 00:55:45,589

 

1717
00:55:45,589 --> 00:55:48,069

 

1718
00:55:48,069 --> 00:55:50,470

 

1719
00:55:50,470 --> 00:55:53,589

 

1720
00:55:53,589 --> 00:55:56,069

 

1721
00:55:56,069 --> 00:55:58,309

 

1722
00:55:58,309 --> 00:56:00,230

 

1723
00:56:00,230 --> 00:56:00,240

 

1724
00:56:00,240 --> 00:56:01,190


1725
00:56:01,190 --> 00:56:03,589

 

1726
00:56:03,589 --> 00:56:05,270

 

1727
00:56:05,270 --> 00:56:07,109

 

1728
00:56:07,109 --> 00:56:08,710

 

1729
00:56:08,710 --> 00:56:09,750

 

1730
00:56:09,750 --> 00:56:09,760

 

1731
00:56:09,760 --> 00:56:11,030


1732
00:56:11,030 --> 00:56:11,040

 

1733
00:56:11,040 --> 00:56:11,829


1734
00:56:11,829 --> 00:56:11,839

 

1735
00:56:11,839 --> 00:56:12,950


1736
00:56:12,950 --> 00:56:14,150

 

1737
00:56:14,150 --> 00:56:15,510

 

1738
00:56:15,510 --> 00:56:17,829

 

1739
00:56:17,829 --> 00:56:17,839

 

1740
00:56:17,839 --> 00:56:18,630


1741
00:56:18,630 --> 00:56:20,309

 

1742
00:56:20,309 --> 00:56:22,710

 

1743
00:56:22,710 --> 00:56:25,670

 

1744
00:56:25,670 --> 00:56:27,750

 

1745
00:56:27,750 --> 00:56:30,470

 

1746
00:56:30,470 --> 00:56:30,480

 

1747
00:56:30,480 --> 00:56:31,190


1748
00:56:31,190 --> 00:56:33,030

 

1749
00:56:33,030 --> 00:56:35,430

 

1750
00:56:35,430 --> 00:56:37,829

 

1751
00:56:37,829 --> 00:56:40,069

 

1752
00:56:40,069 --> 00:56:42,230

 

1753
00:56:42,230 --> 00:56:44,870

 

1754
00:56:44,870 --> 00:56:46,630

 

1755
00:56:46,630 --> 00:56:46,640

 

1756
00:56:46,640 --> 00:56:47,750


1757
00:56:47,750 --> 00:56:49,349

 

1758
00:56:49,349 --> 00:56:50,950

 

1759
00:56:50,950 --> 00:56:50,960

 

1760
00:56:50,960 --> 00:56:52,470


1761
00:56:52,470 --> 00:56:52,480

 

1762
00:56:52,480 --> 00:56:54,470


1763
00:56:54,470 --> 00:56:57,190

 

1764
00:56:57,190 --> 00:56:59,510

 

1765
00:56:59,510 --> 00:57:02,069

 

1766
00:57:02,069 --> 00:57:03,910

 

1767
00:57:03,910 --> 00:57:06,309

 

1768
00:57:06,309 --> 00:57:09,109

 

1769
00:57:09,109 --> 00:57:10,950

 

1770
00:57:10,950 --> 00:57:12,549

 

1771
00:57:12,549 --> 00:57:14,390

 

1772
00:57:14,390 --> 00:57:16,549

 

1773
00:57:16,549 --> 00:57:19,190

 

1774
00:57:19,190 --> 00:57:20,870

 

1775
00:57:20,870 --> 00:57:23,349

 

1776
00:57:23,349 --> 00:57:25,030

 

1777
00:57:25,030 --> 00:57:27,750

 

1778
00:57:27,750 --> 00:57:29,349

 

1779
00:57:29,349 --> 00:57:29,359

 

1780
00:57:29,359 --> 00:57:31,589


1781
00:57:31,589 --> 00:57:33,030

 

1782
00:57:33,030 --> 00:57:35,030

 

1783
00:57:35,030 --> 00:57:37,030

 

1784
00:57:37,030 --> 00:57:39,430

 

1785
00:57:39,430 --> 00:57:41,109

 

1786
00:57:41,109 --> 00:57:43,670

 

1787
00:57:43,670 --> 00:57:46,870

 

1788
00:57:46,870 --> 00:57:49,069

 

1789
00:57:49,069 --> 00:57:51,430

 

1790
00:57:51,430 --> 00:57:53,670

 

1791
00:57:53,670 --> 00:57:55,190

 

1792
00:57:55,190 --> 00:57:57,510

 

1793
00:57:57,510 --> 00:57:59,510

 

1794
00:57:59,510 --> 00:58:01,190

 

1795
00:58:01,190 --> 00:58:03,030

 

1796
00:58:03,030 --> 00:58:05,190

 

1797
00:58:05,190 --> 00:58:07,109

 

1798
00:58:07,109 --> 00:58:09,589

 

1799
00:58:09,589 --> 00:58:09,599

 

1800
00:58:09,599 --> 00:58:10,870


1801
00:58:10,870 --> 00:58:14,789

 

1802
00:58:14,789 --> 00:58:17,510

 

1803
00:58:17,510 --> 00:58:19,430

 

1804
00:58:19,430 --> 00:58:19,440

 

1805
00:58:19,440 --> 00:58:20,230


1806
00:58:20,230 --> 00:58:22,549

 

1807
00:58:22,549 --> 00:58:24,390

 

1808
00:58:24,390 --> 00:58:25,910

 

1809
00:58:25,910 --> 00:58:28,150

 

1810
00:58:28,150 --> 00:58:31,670

 

1811
00:58:31,670 --> 00:58:31,680

 

1812
00:58:31,680 --> 00:58:32,549


1813
00:58:32,549 --> 00:58:34,549

 

1814
00:58:34,549 --> 00:58:37,670

 

1815
00:58:37,670 --> 00:58:39,510

 

1816
00:58:39,510 --> 00:58:41,190

 

1817
00:58:41,190 --> 00:58:43,109

 

1818
00:58:43,109 --> 00:58:45,109

 

1819
00:58:45,109 --> 00:58:47,829

 

1820
00:58:47,829 --> 00:58:47,839

 

1821
00:58:47,839 --> 00:58:48,710


1822
00:58:48,710 --> 00:58:51,750

 

1823
00:58:51,750 --> 00:58:54,870

 

1824
00:58:54,870 --> 00:58:57,430

 

1825
00:58:57,430 --> 00:58:59,829

 

1826
00:58:59,829 --> 00:59:01,589

 

1827
00:59:01,589 --> 00:59:03,670

 

1828
00:59:03,670 --> 00:59:07,109

 

1829
00:59:07,109 --> 00:59:09,990

 

1830
00:59:09,990 --> 00:59:11,270

 

1831
00:59:11,270 --> 00:59:13,670

 

1832
00:59:13,670 --> 00:59:15,990

 

1833
00:59:15,990 --> 00:59:16,000

 

1834
00:59:16,000 --> 00:59:19,109


1835
00:59:19,109 --> 00:59:22,069

 

1836
00:59:22,069 --> 00:59:23,990

 

1837
00:59:23,990 --> 00:59:25,750

 

1838
00:59:25,750 --> 00:59:25,760

 

1839
00:59:25,760 --> 00:59:26,950


1840
00:59:26,950 --> 00:59:30,150

 

1841
00:59:30,150 --> 00:59:33,990

 

1842
00:59:33,990 --> 00:59:36,150

 

1843
00:59:36,150 --> 00:59:37,750

 

1844
00:59:37,750 --> 00:59:40,309

 

1845
00:59:40,309 --> 00:59:44,549

 

1846
00:59:44,549 --> 00:59:45,910

 

1847
00:59:45,910 --> 00:59:47,589

 

1848
00:59:47,589 --> 00:59:51,510

 

1849
00:59:51,510 --> 00:59:53,190

 

1850
00:59:53,190 --> 00:59:55,750

 

1851
00:59:55,750 --> 00:59:58,870

 

1852
00:59:58,870 --> 01:00:01,750

 

1853
01:00:01,750 --> 01:00:03,990

 

1854
01:00:03,990 --> 01:00:08,150

 

1855
01:00:08,150 --> 01:00:09,910

 

1856
01:00:09,910 --> 01:00:11,589

 

1857
01:00:11,589 --> 01:00:13,030

 

1858
01:00:13,030 --> 01:00:15,030

 

1859
01:00:15,030 --> 01:00:16,630

 

1860
01:00:16,630 --> 01:00:18,309

 

1861
01:00:18,309 --> 01:00:21,190

 

1862
01:00:21,190 --> 01:00:22,390

 

1863
01:00:22,390 --> 01:00:24,470

 

1864
01:00:24,470 --> 01:00:27,190

 

1865
01:00:27,190 --> 01:00:30,870

 

1866
01:00:30,870 --> 01:00:32,309

 

1867
01:00:32,309 --> 01:00:35,030

 

1868
01:00:35,030 --> 01:00:36,390

 

1869
01:00:36,390 --> 01:00:39,190

 

1870
01:00:39,190 --> 01:00:41,910

 

1871
01:00:41,910 --> 01:00:43,829

 

1872
01:00:43,829 --> 01:00:43,839

 

1873
01:00:43,839 --> 01:00:46,390


1874
01:00:46,390 --> 01:00:48,069

 

1875
01:00:48,069 --> 01:00:50,710

 

1876
01:00:50,710 --> 01:00:53,270

 

1877
01:00:53,270 --> 01:00:55,829

 

1878
01:00:55,829 --> 01:00:58,789

 

1879
01:00:58,789 --> 01:01:01,030

 

1880
01:01:01,030 --> 01:01:02,549

 

1881
01:01:02,549 --> 01:01:04,710

 

1882
01:01:04,710 --> 01:01:06,710

 

1883
01:01:06,710 --> 01:01:08,230

 

1884
01:01:08,230 --> 01:01:08,240

 

1885
01:01:08,240 --> 01:01:09,270


1886
01:01:09,270 --> 01:01:11,030

 

1887
01:01:11,030 --> 01:01:11,040

 

1888
01:01:11,040 --> 01:01:12,230


1889
01:01:12,230 --> 01:01:15,190

 

1890
01:01:15,190 --> 01:01:16,710

 

1891
01:01:16,710 --> 01:01:18,150

 

1892
01:01:18,150 --> 01:01:20,549

 

1893
01:01:20,549 --> 01:01:21,990

 

1894
01:01:21,990 --> 01:01:25,030

 

1895
01:01:25,030 --> 01:01:25,040

 

1896
01:01:25,040 --> 01:01:27,750


1897
01:01:27,750 --> 01:01:30,950

 

1898
01:01:30,950 --> 01:01:34,150

 

1899
01:01:34,150 --> 01:01:34,160

 

1900
01:01:34,160 --> 01:01:35,589


1901
01:01:35,589 --> 01:01:35,599

 

1902
01:01:35,599 --> 01:01:36,710


1903
01:01:36,710 --> 01:01:38,230

 

1904
01:01:38,230 --> 01:01:40,390

 

1905
01:01:40,390 --> 01:01:42,789

 

1906
01:01:42,789 --> 01:01:45,510

 

1907
01:01:45,510 --> 01:01:46,950

 

1908
01:01:46,950 --> 01:01:48,789

 

1909
01:01:48,789 --> 01:01:50,150

 

1910
01:01:50,150 --> 01:01:51,750

 

1911
01:01:51,750 --> 01:01:53,190

 

1912
01:01:53,190 --> 01:01:56,710

 

1913
01:01:56,710 --> 01:01:56,720

 

1914
01:01:56,720 --> 01:01:57,910


1915
01:01:57,910 --> 01:01:59,670

 

1916
01:01:59,670 --> 01:02:02,789

 

1917
01:02:02,789 --> 01:02:04,230

 

1918
01:02:04,230 --> 01:02:06,309

 

1919
01:02:06,309 --> 01:02:08,230

 

1920
01:02:08,230 --> 01:02:09,750

 

1921
01:02:09,750 --> 01:02:11,190

 

1922
01:02:11,190 --> 01:02:13,190

 

1923
01:02:13,190 --> 01:02:15,750

 

1924
01:02:15,750 --> 01:02:18,309

 

1925
01:02:18,309 --> 01:02:20,230

 

1926
01:02:20,230 --> 01:02:22,710

 

1927
01:02:22,710 --> 01:02:24,549

 

1928
01:02:24,549 --> 01:02:27,109

 

1929
01:02:27,109 --> 01:02:29,270

 

1930
01:02:29,270 --> 01:02:31,270

 

1931
01:02:31,270 --> 01:02:33,750

 

1932
01:02:33,750 --> 01:02:35,349

 

1933
01:02:35,349 --> 01:02:37,430

 

1934
01:02:37,430 --> 01:02:40,150

 

1935
01:02:40,150 --> 01:02:42,230

 

1936
01:02:42,230 --> 01:02:44,549

 

1937
01:02:44,549 --> 01:02:46,150

 

1938
01:02:46,150 --> 01:02:48,230

 

1939
01:02:48,230 --> 01:02:49,829

 

1940
01:02:49,829 --> 01:02:49,839

 

1941
01:02:49,839 --> 01:02:52,549


1942
01:02:52,549 --> 01:02:54,710

 

1943
01:02:54,710 --> 01:02:54,720

 

1944
01:02:54,720 --> 01:02:55,510


1945
01:02:55,510 --> 01:02:57,190

 

1946
01:02:57,190 --> 01:02:59,829

 

1947
01:02:59,829 --> 01:03:04,150

 

1948
01:03:04,150 --> 01:03:04,160

 

1949
01:03:04,160 --> 01:03:04,870


1950
01:03:04,870 --> 01:03:06,150

 

1951
01:03:06,150 --> 01:03:07,109

 

1952
01:03:07,109 --> 01:03:09,109

 

1953
01:03:09,109 --> 01:03:09,119

 

1954
01:03:09,119 --> 01:03:16,150


1955
01:03:16,150 --> 01:03:17,349

 

1956
01:03:17,349 --> 01:03:17,359

 

1957
01:03:17,359 --> 01:03:18,630


1958
01:03:18,630 --> 01:03:21,029

 

1959
01:03:21,029 --> 01:03:22,789

 

1960
01:03:22,789 --> 01:03:24,870

 

1961
01:03:24,870 --> 01:03:26,870

 

1962
01:03:26,870 --> 01:03:28,710

 

1963
01:03:28,710 --> 01:03:30,549

 

1964
01:03:30,549 --> 01:03:32,870

 

1965
01:03:32,870 --> 01:03:32,880

 

1966
01:03:32,880 --> 01:03:33,829


1967
01:03:33,829 --> 01:03:35,670

 

1968
01:03:35,670 --> 01:03:37,589

 

1969
01:03:37,589 --> 01:03:39,510

 

1970
01:03:39,510 --> 01:03:40,870

 

1971
01:03:40,870 --> 01:03:43,270

 

1972
01:03:43,270 --> 01:03:44,390

 

1973
01:03:44,390 --> 01:03:46,390

 

1974
01:03:46,390 --> 01:03:47,750

 

1975
01:03:47,750 --> 01:03:51,430

 

1976
01:03:51,430 --> 01:03:53,750

 

1977
01:03:53,750 --> 01:03:55,670

 

1978
01:03:55,670 --> 01:03:57,510

 

1979
01:03:57,510 --> 01:03:58,950

 

1980
01:03:58,950 --> 01:03:58,960

 

1981
01:03:58,960 --> 01:03:59,750


1982
01:03:59,750 --> 01:04:01,670

 

1983
01:04:01,670 --> 01:04:01,680

 

1984
01:04:01,680 --> 01:04:03,190


1985
01:04:03,190 --> 01:04:07,109

 

1986
01:04:07,109 --> 01:04:08,309

 

1987
01:04:08,309 --> 01:04:08,319

 

1988
01:04:08,319 --> 01:04:09,190


1989
01:04:09,190 --> 01:04:13,029

 

1990
01:04:13,029 --> 01:04:15,109

 

1991
01:04:15,109 --> 01:04:17,190

 

1992
01:04:17,190 --> 01:04:19,190

 

1993
01:04:19,190 --> 01:04:21,109

 

1994
01:04:21,109 --> 01:04:22,630

 

1995
01:04:22,630 --> 01:04:24,870

 

1996
01:04:24,870 --> 01:04:26,630

 

1997
01:04:26,630 --> 01:04:26,640

 

1998
01:04:26,640 --> 01:04:28,309


1999
01:04:28,309 --> 01:04:29,990

 

2000
01:04:29,990 --> 01:04:32,230

 

2001
01:04:32,230 --> 01:04:35,109

 

2002
01:04:35,109 --> 01:04:36,710

 

2003
01:04:36,710 --> 01:04:39,829

 

2004
01:04:39,829 --> 01:04:43,430

 

2005
01:04:43,430 --> 01:04:43,440

 

2006
01:04:43,440 --> 01:04:44,309


2007
01:04:44,309 --> 01:04:46,630

 

2008
01:04:46,630 --> 01:04:46,640

 

2009
01:04:46,640 --> 01:04:47,430


2010
01:04:47,430 --> 01:04:47,440

 

2011
01:04:47,440 --> 01:04:49,109


2012
01:04:49,109 --> 01:04:51,029

 

2013
01:04:51,029 --> 01:04:52,829

 

2014
01:04:52,829 --> 01:04:55,190

 

2015
01:04:55,190 --> 01:04:57,109

 

2016
01:04:57,109 --> 01:04:59,750

 

2017
01:04:59,750 --> 01:04:59,760

 

2018
01:04:59,760 --> 01:05:01,109


2019
01:05:01,109 --> 01:05:06,150

 

2020
01:05:06,150 --> 01:05:08,549

 

2021
01:05:08,549 --> 01:05:10,870

 

2022
01:05:10,870 --> 01:05:12,950

 

2023
01:05:12,950 --> 01:05:15,349

 

2024
01:05:15,349 --> 01:05:16,950

 

2025
01:05:16,950 --> 01:05:16,960

 

2026
01:05:16,960 --> 01:05:17,910


2027
01:05:17,910 --> 01:05:17,920

 

2028
01:05:17,920 --> 01:05:21,510


2029
01:05:21,510 --> 01:05:21,520

 

2030
01:05:21,520 --> 01:05:25,589


2031
01:05:25,589 --> 01:05:25,599

 

2032
01:05:25,599 --> 01:05:27,589


2033
01:05:27,589 --> 01:05:32,230

 

2034
01:05:32,230 --> 01:05:34,390

 

2035
01:05:34,390 --> 01:05:36,069

 

2036
01:05:36,069 --> 01:05:37,750

 

2037
01:05:37,750 --> 01:05:39,430

 

2038
01:05:39,430 --> 01:05:42,549

 

2039
01:05:42,549 --> 01:05:45,109

 

2040
01:05:45,109 --> 01:05:46,309

 

2041
01:05:46,309 --> 01:05:47,510

 

2042
01:05:47,510 --> 01:05:49,670

 

2043
01:05:49,670 --> 01:05:51,670

 

2044
01:05:51,670 --> 01:05:53,510

 

2045
01:05:53,510 --> 01:05:55,829

 

2046
01:05:55,829 --> 01:05:58,309

 

2047
01:05:58,309 --> 01:05:59,670

 

2048
01:05:59,670 --> 01:06:02,829

 

2049
01:06:02,829 --> 01:06:09,349

 

2050
01:06:09,349 --> 01:06:11,190

 

2051
01:06:11,190 --> 01:06:12,789

 

2052
01:06:12,789 --> 01:06:14,230

 

2053
01:06:14,230 --> 01:06:16,390

 

2054
01:06:16,390 --> 01:06:19,349

 

2055
01:06:19,349 --> 01:06:21,430

 

2056
01:06:21,430 --> 01:06:22,950

 

2057
01:06:22,950 --> 01:06:24,549

 

2058
01:06:24,549 --> 01:06:26,789

 

2059
01:06:26,789 --> 01:06:29,349

 

2060
01:06:29,349 --> 01:06:31,109

 

2061
01:06:31,109 --> 01:06:32,789

 

2062
01:06:32,789 --> 01:06:32,799

 

2063
01:06:32,799 --> 01:06:36,390


2064
01:06:36,390 --> 01:06:36,400

 

2065
01:06:36,400 --> 01:06:36,870


2066
01:06:36,870 --> 01:06:36,880

 

2067
01:06:36,880 --> 01:06:40,390


2068
01:06:40,390 --> 01:06:40,400

 

2069
01:06:40,400 --> 01:06:41,270


2070
01:06:41,270 --> 01:06:42,870

 

2071
01:06:42,870 --> 01:06:44,630

 

2072
01:06:44,630 --> 01:06:48,069

 

2073
01:06:48,069 --> 01:06:48,079

 

2074
01:06:48,079 --> 01:06:49,349


2075
01:06:49,349 --> 01:06:51,829

 

2076
01:06:51,829 --> 01:06:52,870

 

2077
01:06:52,870 --> 01:06:55,109

 

2078
01:06:55,109 --> 01:06:56,950

 

2079
01:06:56,950 --> 01:06:56,960

 

2080
01:06:56,960 --> 01:06:59,510


2081
01:06:59,510 --> 01:07:01,349

 

2082
01:07:01,349 --> 01:07:01,359

 

2083
01:07:01,359 --> 01:07:02,470


2084
01:07:02,470 --> 01:07:03,430

 

2085
01:07:03,430 --> 01:07:05,670

 

2086
01:07:05,670 --> 01:07:08,390

 

2087
01:07:08,390 --> 01:07:10,470

 

2088
01:07:10,470 --> 01:07:14,630

 

2089
01:07:14,630 --> 01:07:16,630

 

2090
01:07:16,630 --> 01:07:18,069

 

2091
01:07:18,069 --> 01:07:20,069

 

2092
01:07:20,069 --> 01:07:22,549

 

2093
01:07:22,549 --> 01:07:23,990

 

2094
01:07:23,990 --> 01:07:26,069

 

2095
01:07:26,069 --> 01:07:28,309

 

2096
01:07:28,309 --> 01:07:31,109

 

2097
01:07:31,109 --> 01:07:33,349

 

2098
01:07:33,349 --> 01:07:36,069

 

2099
01:07:36,069 --> 01:07:37,510

 

2100
01:07:37,510 --> 01:07:39,349

 

2101
01:07:39,349 --> 01:07:40,789

 

2102
01:07:40,789 --> 01:07:43,510

 

2103
01:07:43,510 --> 01:07:45,990

 

2104
01:07:45,990 --> 01:07:46,000

 

2105
01:07:46,000 --> 01:07:46,950


2106
01:07:46,950 --> 01:07:48,549

 

2107
01:07:48,549 --> 01:07:50,870

 

2108
01:07:50,870 --> 01:07:52,789

 

2109
01:07:52,789 --> 01:07:54,150

 

2110
01:07:54,150 --> 01:07:55,190

 

2111
01:07:55,190 --> 01:07:56,950

 

2112
01:07:56,950 --> 01:07:56,960

 

2113
01:07:56,960 --> 01:07:58,230


2114
01:07:58,230 --> 01:08:00,630

 

2115
01:08:00,630 --> 01:08:03,029

 

2116
01:08:03,029 --> 01:08:07,349

 

2117
01:08:07,349 --> 01:08:09,829

 

2118
01:08:09,829 --> 01:08:12,950

 

2119
01:08:12,950 --> 01:08:15,109

 

2120
01:08:15,109 --> 01:08:17,430

 

2121
01:08:17,430 --> 01:08:19,269

 

2122
01:08:19,269 --> 01:08:24,149

 

2123
01:08:24,149 --> 01:08:25,669

 

2124
01:08:25,669 --> 01:08:27,110

 

2125
01:08:27,110 --> 01:08:29,590

 

2126
01:08:29,590 --> 01:08:32,390

 

2127
01:08:32,390 --> 01:08:34,950

 

2128
01:08:34,950 --> 01:08:37,669

 

2129
01:08:37,669 --> 01:08:37,679

 

2130
01:08:37,679 --> 01:08:39,669


2131
01:08:39,669 --> 01:08:41,430

 

2132
01:08:41,430 --> 01:08:43,749

 

2133
01:08:43,749 --> 01:08:43,759

 

2134
01:08:43,759 --> 01:08:45,669


2135
01:08:45,669 --> 01:08:48,470

 

2136
01:08:48,470 --> 01:08:52,229

 

2137
01:08:52,229 --> 01:08:54,070

 

2138
01:08:54,070 --> 01:08:56,630

 

2139
01:08:56,630 --> 01:08:58,950

 

2140
01:08:58,950 --> 01:09:01,030

 

2141
01:09:01,030 --> 01:09:03,030

 

2142
01:09:03,030 --> 01:09:05,510

 

2143
01:09:05,510 --> 01:09:05,520

 

2144
01:09:05,520 --> 01:09:07,910


2145
01:09:07,910 --> 01:09:08,950

 

2146
01:09:08,950 --> 01:09:12,149

 

2147
01:09:12,149 --> 01:09:14,229

 

2148
01:09:14,229 --> 01:09:16,390

 

2149
01:09:16,390 --> 01:09:17,269

 

2150
01:09:17,269 --> 01:09:18,309

 

2151
01:09:18,309 --> 01:09:20,950

 

2152
01:09:20,950 --> 01:09:22,870

 

2153
01:09:22,870 --> 01:09:24,550

 

2154
01:09:24,550 --> 01:09:25,749

 

2155
01:09:25,749 --> 01:09:28,709

 

2156
01:09:28,709 --> 01:09:35,110

 

2157
01:09:35,110 --> 01:09:39,030

 

2158
01:09:39,030 --> 01:09:40,470

 

2159
01:09:40,470 --> 01:09:43,430

 

2160
01:09:43,430 --> 01:09:45,590

 

2161
01:09:45,590 --> 01:09:49,189

 

2162
01:09:49,189 --> 01:09:51,510

 

2163
01:09:51,510 --> 01:09:53,829

 

2164
01:09:53,829 --> 01:09:55,510

 

2165
01:09:55,510 --> 01:09:57,430

 

2166
01:09:57,430 --> 01:09:58,950

 

2167
01:09:58,950 --> 01:10:01,350

 

2168
01:10:01,350 --> 01:10:03,350

 

2169
01:10:03,350 --> 01:10:05,510

 

2170
01:10:05,510 --> 01:10:08,550

 

2171
01:10:08,550 --> 01:10:09,750

 

2172
01:10:09,750 --> 01:10:11,750

 

2173
01:10:11,750 --> 01:10:14,310

 

2174
01:10:14,310 --> 01:10:16,070

 

2175
01:10:16,070 --> 01:10:18,149

 

2176
01:10:18,149 --> 01:10:19,669

 

2177
01:10:19,669 --> 01:10:21,669

 

2178
01:10:21,669 --> 01:10:23,750

 

2179
01:10:23,750 --> 01:10:25,590

 

2180
01:10:25,590 --> 01:10:27,669

 

2181
01:10:27,669 --> 01:10:31,830

 

2182
01:10:31,830 --> 01:10:35,030

 

2183
01:10:35,030 --> 01:10:37,030

 

2184
01:10:37,030 --> 01:10:38,790

 

2185
01:10:38,790 --> 01:10:40,229

 

2186
01:10:40,229 --> 01:10:42,870

 

2187
01:10:42,870 --> 01:10:44,630

 

2188
01:10:44,630 --> 01:10:44,640

 

2189
01:10:44,640 --> 01:10:45,669


2190
01:10:45,669 --> 01:10:46,790

 

2191
01:10:46,790 --> 01:10:49,270

 

2192
01:10:49,270 --> 01:10:51,270

 

2193
01:10:51,270 --> 01:10:53,430

 

2194
01:10:53,430 --> 01:10:53,440

 

2195
01:10:53,440 --> 01:10:55,270


2196
01:10:55,270 --> 01:10:55,280

 

2197
01:10:55,280 --> 01:10:56,229


2198
01:10:56,229 --> 01:10:58,790

 

2199
01:10:58,790 --> 01:11:00,390

 

2200
01:11:00,390 --> 01:11:00,400

 

2201
01:11:00,400 --> 01:11:01,510


2202
01:11:01,510 --> 01:11:05,270

 

2203
01:11:05,270 --> 01:11:07,430

 

2204
01:11:07,430 --> 01:11:09,270

 

2205
01:11:09,270 --> 01:11:11,110

 

2206
01:11:11,110 --> 01:11:13,270

 

2207
01:11:13,270 --> 01:11:15,030

 

2208
01:11:15,030 --> 01:11:15,040

 

2209
01:11:15,040 --> 01:11:17,430


2210
01:11:17,430 --> 01:11:19,830

 

2211
01:11:19,830 --> 01:11:23,990

 

2212
01:11:23,990 --> 01:11:25,270

 

2213
01:11:25,270 --> 01:11:26,470

 

2214
01:11:26,470 --> 01:11:29,910

 

2215
01:11:29,910 --> 01:11:29,920

 

2216
01:11:29,920 --> 01:11:30,790


2217
01:11:30,790 --> 01:11:32,550

 

2218
01:11:32,550 --> 01:11:32,560

 

2219
01:11:32,560 --> 01:11:37,189


2220
01:11:37,189 --> 01:11:37,199

 

2221
01:11:37,199 --> 01:11:38,830


2222
01:11:38,830 --> 01:11:42,070

 

2223
01:11:42,070 --> 01:11:44,070

 

2224
01:11:44,070 --> 01:11:46,070

 

2225
01:11:46,070 --> 01:11:47,669

 

2226
01:11:47,669 --> 01:11:50,229

 

2227
01:11:50,229 --> 01:11:51,830

 

2228
01:11:51,830 --> 01:11:54,550

 

2229
01:11:54,550 --> 01:11:55,910

 

2230
01:11:55,910 --> 01:11:57,990

 

2231
01:11:57,990 --> 01:11:59,910

 

2232
01:11:59,910 --> 01:12:02,550

 

2233
01:12:02,550 --> 01:12:04,630

 

2234
01:12:04,630 --> 01:12:04,640

 

2235
01:12:04,640 --> 01:12:05,000


2236
01:12:05,000 --> 01:12:05,010

 

2237
01:12:05,010 --> 01:12:06,790


2238
01:12:06,790 --> 01:12:08,550

 

2239
01:12:08,550 --> 01:12:08,560

 

2240
01:12:08,560 --> 01:12:09,510


2241
01:12:09,510 --> 01:12:10,870

 

2242
01:12:10,870 --> 01:12:12,630

 

2243
01:12:12,630 --> 01:12:15,110

 

2244
01:12:15,110 --> 01:12:16,870

 

2245
01:12:16,870 --> 01:12:20,229

 

2246
01:12:20,229 --> 01:12:23,110

 

2247
01:12:23,110 --> 01:12:26,070

 

2248
01:12:26,070 --> 01:12:28,790

 

2249
01:12:28,790 --> 01:12:30,870

 

2250
01:12:30,870 --> 01:12:33,990

 

2251
01:12:33,990 --> 01:12:34,000

 

2252
01:12:34,000 --> 01:12:35,590


2253
01:12:35,590 --> 01:12:38,229

 

2254
01:12:38,229 --> 01:12:38,239

 

2255
01:12:38,239 --> 01:12:39,110


2256
01:12:39,110 --> 01:12:40,870

 

2257
01:12:40,870 --> 01:12:42,790

 

2258
01:12:42,790 --> 01:12:44,709

 

2259
01:12:44,709 --> 01:12:46,550

 

2260
01:12:46,550 --> 01:12:48,470

 

2261
01:12:48,470 --> 01:12:50,390

 

2262
01:12:50,390 --> 01:12:52,070

 

2263
01:12:52,070 --> 01:12:53,910

 

2264
01:12:53,910 --> 01:12:53,920

 

2265
01:12:53,920 --> 01:12:55,030


2266
01:12:55,030 --> 01:12:56,630

 

2267
01:12:56,630 --> 01:13:00,630

 

2268
01:13:00,630 --> 01:13:01,750

 

2269
01:13:01,750 --> 01:13:03,510

 

2270
01:13:03,510 --> 01:13:05,830

 

2271
01:13:05,830 --> 01:13:07,510

 

2272
01:13:07,510 --> 01:13:09,110

 

2273
01:13:09,110 --> 01:13:10,870

 

2274
01:13:10,870 --> 01:13:12,630

 

2275
01:13:12,630 --> 01:13:15,030

 

2276
01:13:15,030 --> 01:13:15,040

 

2277
01:13:15,040 --> 01:13:17,030


2278
01:13:17,030 --> 01:13:19,270

 

2279
01:13:19,270 --> 01:13:22,550

 

2280
01:13:22,550 --> 01:13:23,669

 

2281
01:13:23,669 --> 01:13:25,830

 

2282
01:13:25,830 --> 01:13:27,030

 

2283
01:13:27,030 --> 01:13:28,790

 

2284
01:13:28,790 --> 01:13:31,669

 

2285
01:13:31,669 --> 01:13:33,270

 

2286
01:13:33,270 --> 01:13:35,750

 

2287
01:13:35,750 --> 01:13:37,350

 

2288
01:13:37,350 --> 01:13:39,750

 

2289
01:13:39,750 --> 01:13:39,760

 

2290
01:13:39,760 --> 01:13:40,630


2291
01:13:40,630 --> 01:13:43,189

 

2292
01:13:43,189 --> 01:13:43,199

 

2293
01:13:43,199 --> 01:13:47,430


2294
01:13:47,430 --> 01:13:48,950

 

2295
01:13:48,950 --> 01:13:51,030

 

2296
01:13:51,030 --> 01:13:51,040

 

2297
01:13:51,040 --> 01:13:52,229


2298
01:13:52,229 --> 01:13:54,550

 

2299
01:13:54,550 --> 01:13:56,709

 

2300
01:13:56,709 --> 01:14:00,709

 

2301
01:14:00,709 --> 01:14:02,229

 

2302
01:14:02,229 --> 01:14:03,590

 

2303
01:14:03,590 --> 01:14:05,430

 

2304
01:14:05,430 --> 01:14:07,270

 

2305
01:14:07,270 --> 01:14:09,350

 

2306
01:14:09,350 --> 01:14:11,270

 

2307
01:14:11,270 --> 01:14:13,270

 

2308
01:14:13,270 --> 01:14:14,870

 

2309
01:14:14,870 --> 01:14:16,790

 

2310
01:14:16,790 --> 01:14:18,310

 

2311
01:14:18,310 --> 01:14:20,390

 

2312
01:14:20,390 --> 01:14:22,790

 

2313
01:14:22,790 --> 01:14:25,110

 

2314
01:14:25,110 --> 01:14:26,790

 

2315
01:14:26,790 --> 01:14:28,950

 

2316
01:14:28,950 --> 01:14:31,030

 

2317
01:14:31,030 --> 01:14:32,790

 

2318
01:14:32,790 --> 01:14:32,800

 

2319
01:14:32,800 --> 01:14:33,750


2320
01:14:33,750 --> 01:14:35,750

 

2321
01:14:35,750 --> 01:14:37,590

 

2322
01:14:37,590 --> 01:14:40,229

 

2323
01:14:40,229 --> 01:14:43,030

 

2324
01:14:43,030 --> 01:14:45,350

 

2325
01:14:45,350 --> 01:14:45,360

 

2326
01:14:45,360 --> 01:14:46,470


2327
01:14:46,470 --> 01:14:49,910

 

2328
01:14:49,910 --> 01:14:51,430

 

2329
01:14:51,430 --> 01:14:52,790

 

2330
01:14:52,790 --> 01:14:54,830

 

2331
01:14:54,830 --> 01:14:54,840

 

2332
01:14:54,840 --> 01:14:56,470


2333
01:14:56,470 --> 01:14:58,390

 

2334
01:14:58,390 --> 01:15:00,950

 

2335
01:15:00,950 --> 01:15:03,110

 

2336
01:15:03,110 --> 01:15:04,950

 

2337
01:15:04,950 --> 01:15:06,630

 

2338
01:15:06,630 --> 01:15:09,189

 

2339
01:15:09,189 --> 01:15:11,430

 

2340
01:15:11,430 --> 01:15:15,910

 

2341
01:15:15,910 --> 01:15:15,920

 

2342
01:15:15,920 --> 01:15:17,110


2343
01:15:17,110 --> 01:15:20,630

 

2344
01:15:20,630 --> 01:15:20,640

 

2345
01:15:20,640 --> 01:15:21,590


2346
01:15:21,590 --> 01:15:24,870

 

2347
01:15:24,870 --> 01:15:24,880

 

2348
01:15:24,880 --> 01:15:28,470


2349
01:15:28,470 --> 01:15:31,830

 

2350
01:15:31,830 --> 01:15:33,350

 

2351
01:15:33,350 --> 01:15:36,390

 

2352
01:15:36,390 --> 01:15:36,400

 

2353
01:15:36,400 --> 01:15:37,590


2354
01:15:37,590 --> 01:15:39,510

 

2355
01:15:39,510 --> 01:15:40,870

 

2356
01:15:40,870 --> 01:15:42,550

 

2357
01:15:42,550 --> 01:15:44,630

 

2358
01:15:44,630 --> 01:15:46,310

 

2359
01:15:46,310 --> 01:15:48,630

 

2360
01:15:48,630 --> 01:15:50,709

 

2361
01:15:50,709 --> 01:15:52,790

 

2362
01:15:52,790 --> 01:15:52,800

 

2363
01:15:52,800 --> 01:15:53,669


2364
01:15:53,669 --> 01:15:55,830

 

2365
01:15:55,830 --> 01:15:57,990

 

2366
01:15:57,990 --> 01:16:00,470

 

2367
01:16:00,470 --> 01:16:00,480

 

2368
01:16:00,480 --> 01:16:01,430


2369
01:16:01,430 --> 01:16:03,430

 

2370
01:16:03,430 --> 01:16:05,990

 

2371
01:16:05,990 --> 01:16:08,390

 

2372
01:16:08,390 --> 01:16:10,630

 

2373
01:16:10,630 --> 01:16:12,550

 

2374
01:16:12,550 --> 01:16:14,310

 

2375
01:16:14,310 --> 01:16:15,830

 

2376
01:16:15,830 --> 01:16:18,390

 

2377
01:16:18,390 --> 01:16:18,400

 

2378
01:16:18,400 --> 01:16:20,870


2379
01:16:20,870 --> 01:16:23,350

 

2380
01:16:23,350 --> 01:16:26,470

 

2381
01:16:26,470 --> 01:16:28,470

 

2382
01:16:28,470 --> 01:16:30,630

 

2383
01:16:30,630 --> 01:16:33,830

 

2384
01:16:33,830 --> 01:16:36,229

 

2385
01:16:36,229 --> 01:16:37,669

 

2386
01:16:37,669 --> 01:16:40,229

 

2387
01:16:40,229 --> 01:16:42,550

 

2388
01:16:42,550 --> 01:16:44,229

 

2389
01:16:44,229 --> 01:16:45,910

 

2390
01:16:45,910 --> 01:16:47,590

 

2391
01:16:47,590 --> 01:16:50,550

 

2392
01:16:50,550 --> 01:16:53,430

 

2393
01:16:53,430 --> 01:16:55,270

 

2394
01:16:55,270 --> 01:16:57,510

 

2395
01:16:57,510 --> 01:17:00,630

 

2396
01:17:00,630 --> 01:17:02,149

 

2397
01:17:02,149 --> 01:17:02,159

 

2398
01:17:02,159 --> 01:17:03,110


2399
01:17:03,110 --> 01:17:05,430

 

2400
01:17:05,430 --> 01:17:07,270

 

2401
01:17:07,270 --> 01:17:09,030

 

2402
01:17:09,030 --> 01:17:10,310

 

2403
01:17:10,310 --> 01:17:12,470

 

2404
01:17:12,470 --> 01:17:15,430

 

2405
01:17:15,430 --> 01:17:17,430

 

2406
01:17:17,430 --> 01:17:17,440

 

2407
01:17:17,440 --> 01:17:18,310


2408
01:17:18,310 --> 01:17:20,070

 

2409
01:17:20,070 --> 01:17:21,430

 

2410
01:17:21,430 --> 01:17:24,229

 

2411
01:17:24,229 --> 01:17:24,239

 

2412
01:17:24,239 --> 01:17:25,750


2413
01:17:25,750 --> 01:17:25,760

 

2414
01:17:25,760 --> 01:17:26,630


2415
01:17:26,630 --> 01:17:26,640

 

2416
01:17:26,640 --> 01:17:27,590


2417
01:17:27,590 --> 01:17:28,950

 

2418
01:17:28,950 --> 01:17:30,630

 

2419
01:17:30,630 --> 01:17:32,390

 

2420
01:17:32,390 --> 01:17:33,990

 

2421
01:17:33,990 --> 01:17:36,070

 

2422
01:17:36,070 --> 01:17:38,149

 

2423
01:17:38,149 --> 01:17:39,830

 

2424
01:17:39,830 --> 01:17:41,590

 

2425
01:17:41,590 --> 01:17:43,350

 

2426
01:17:43,350 --> 01:17:45,350

 

2427
01:17:45,350 --> 01:17:46,950

 

2428
01:17:46,950 --> 01:17:46,960

 

2429
01:17:46,960 --> 01:17:47,990


2430
01:17:47,990 --> 01:17:49,830

 

2431
01:17:49,830 --> 01:17:51,430

 

2432
01:17:51,430 --> 01:17:55,030

 

2433
01:17:55,030 --> 01:17:56,950

 

2434
01:17:56,950 --> 01:17:58,310

 

2435
01:17:58,310 --> 01:17:58,320

 

2436
01:17:58,320 --> 01:17:59,270


2437
01:17:59,270 --> 01:18:01,430

 

2438
01:18:01,430 --> 01:18:02,550

 

2439
01:18:02,550 --> 01:18:03,830

 

2440
01:18:03,830 --> 01:18:05,990

 

2441
01:18:05,990 --> 01:18:06,000

 

2442
01:18:06,000 --> 01:18:07,350


2443
01:18:07,350 --> 01:18:08,870

 

2444
01:18:08,870 --> 01:18:10,709

 

2445
01:18:10,709 --> 01:18:12,870

 

2446
01:18:12,870 --> 01:18:13,990

 

2447
01:18:13,990 --> 01:18:17,430

 

2448
01:18:17,430 --> 01:18:19,830

 

2449
01:18:19,830 --> 01:18:20,950

 

2450
01:18:20,950 --> 01:18:23,350

 

2451
01:18:23,350 --> 01:18:25,110

 

2452
01:18:25,110 --> 01:18:27,350

 

2453
01:18:27,350 --> 01:18:29,270

 

2454
01:18:29,270 --> 01:18:31,510

 

2455
01:18:31,510 --> 01:18:33,350

 

2456
01:18:33,350 --> 01:18:35,350

 

2457
01:18:35,350 --> 01:18:37,110

 

2458
01:18:37,110 --> 01:18:38,220

 

2459
01:18:38,220 --> 01:18:38,230

 

2460
01:18:38,230 --> 01:18:39,590


2461
01:18:39,590 --> 01:18:41,189

 

2462
01:18:41,189 --> 01:18:43,669

 

2463
01:18:43,669 --> 01:18:45,510

 

2464
01:18:45,510 --> 01:18:47,030

 

2465
01:18:47,030 --> 01:18:49,590

 

2466
01:18:49,590 --> 01:18:51,189

 

2467
01:18:51,189 --> 01:18:53,750

 

2468
01:18:53,750 --> 01:18:56,229

 

2469
01:18:56,229 --> 01:18:58,870

 

2470
01:18:58,870 --> 01:19:00,470

 

2471
01:19:00,470 --> 01:19:02,390

 

2472
01:19:02,390 --> 01:19:04,790

 

2473
01:19:04,790 --> 01:19:06,310

 

2474
01:19:06,310 --> 01:19:08,310

 

2475
01:19:08,310 --> 01:19:11,590

 

2476
01:19:11,590 --> 01:19:12,790

 

2477
01:19:12,790 --> 01:19:14,310

 

2478
01:19:14,310 --> 01:19:16,550

 

2479
01:19:16,550 --> 01:19:18,950

 

2480
01:19:18,950 --> 01:19:20,550

 

2481
01:19:20,550 --> 01:19:22,149

 

2482
01:19:22,149 --> 01:19:24,709

 

2483
01:19:24,709 --> 01:19:24,719

 

2484
01:19:24,719 --> 01:19:25,669


2485
01:19:25,669 --> 01:19:25,679

 

2486
01:19:25,679 --> 01:19:26,550


2487
01:19:26,550 --> 01:19:27,750

 

2488
01:19:27,750 --> 01:19:29,430

 

2489
01:19:29,430 --> 01:19:31,430

 

2490
01:19:31,430 --> 01:19:33,430

 

2491
01:19:33,430 --> 01:19:35,189

 

2492
01:19:35,189 --> 01:19:36,630

 

2493
01:19:36,630 --> 01:19:39,590

 

2494
01:19:39,590 --> 01:19:40,870

 

2495
01:19:40,870 --> 01:19:42,709

 

2496
01:19:42,709 --> 01:19:46,149

 

2497
01:19:46,149 --> 01:19:48,790

 

2498
01:19:48,790 --> 01:19:50,709

 

2499
01:19:50,709 --> 01:19:53,189

 

2500
01:19:53,189 --> 01:19:55,189

 

2501
01:19:55,189 --> 01:19:57,430

 

2502
01:19:57,430 --> 01:19:57,440

 

2503
01:19:57,440 --> 01:19:58,390


2504
01:19:58,390 --> 01:20:00,149

 

2505
01:20:00,149 --> 01:20:03,270

 

2506
01:20:03,270 --> 01:20:05,030

 

2507
01:20:05,030 --> 01:20:05,040

 

2508
01:20:05,040 --> 01:20:05,910


2509
01:20:05,910 --> 01:20:05,920

 

2510
01:20:05,920 --> 01:20:07,189


2511
01:20:07,189 --> 01:20:09,030

 

2512
01:20:09,030 --> 01:20:11,270

 

2513
01:20:11,270 --> 01:20:14,229

 

2514
01:20:14,229 --> 01:20:16,229

 

2515
01:20:16,229 --> 01:20:17,990

 

2516
01:20:17,990 --> 01:20:19,990

 

2517
01:20:19,990 --> 01:20:24,229

 

2518
01:20:24,229 --> 01:20:26,149

 

2519
01:20:26,149 --> 01:20:27,910

 

2520
01:20:27,910 --> 01:20:29,910

 

2521
01:20:29,910 --> 01:20:32,390

 

2522
01:20:32,390 --> 01:20:33,750

 

2523
01:20:33,750 --> 01:20:33,760

 

2524
01:20:33,760 --> 01:20:34,709


2525
01:20:34,709 --> 01:20:37,350

 

2526
01:20:37,350 --> 01:20:39,750

 

2527
01:20:39,750 --> 01:20:39,760

 

2528
01:20:39,760 --> 01:20:41,270


2529
01:20:41,270 --> 01:20:43,830

 

2530
01:20:43,830 --> 01:20:47,669

 

2531
01:20:47,669 --> 01:20:47,679

 

2532
01:20:47,679 --> 01:20:48,550


2533
01:20:48,550 --> 01:20:50,149

 

2534
01:20:50,149 --> 01:20:50,159

 

2535
01:20:50,159 --> 01:20:51,189


2536
01:20:51,189 --> 01:20:54,790

 

2537
01:20:54,790 --> 01:20:58,070

 

2538
01:20:58,070 --> 01:21:03,030

 

2539
01:21:03,030 --> 01:21:05,350

 

2540
01:21:05,350 --> 01:21:05,360

 

2541
01:21:05,360 --> 01:21:06,550


2542
01:21:06,550 --> 01:21:09,510

 

2543
01:21:09,510 --> 01:21:11,510

 

2544
01:21:11,510 --> 01:21:13,510

 

2545
01:21:13,510 --> 01:21:16,149

 

2546
01:21:16,149 --> 01:21:19,750

 

2547
01:21:19,750 --> 01:21:22,629

 

2548
01:21:22,629 --> 01:21:25,030

 

2549
01:21:25,030 --> 01:21:27,430

 

2550
01:21:27,430 --> 01:21:29,510

 

2551
01:21:29,510 --> 01:21:31,430

 

2552
01:21:31,430 --> 01:21:32,950

 

2553
01:21:32,950 --> 01:21:35,030

 

2554
01:21:35,030 --> 01:21:37,830

 

2555
01:21:37,830 --> 01:21:39,669

 

2556
01:21:39,669 --> 01:21:39,679

 

2557
01:21:39,679 --> 01:21:40,830


2558
01:21:40,830 --> 01:21:42,629

 

2559
01:21:42,629 --> 01:21:44,709

 

2560
01:21:44,709 --> 01:21:47,189

 

2561
01:21:47,189 --> 01:21:48,470

 

2562
01:21:48,470 --> 01:21:48,480

 

2563
01:21:48,480 --> 01:21:49,590


2564
01:21:49,590 --> 01:21:51,669

 

2565
01:21:51,669 --> 01:21:53,510

 

2566
01:21:53,510 --> 01:21:56,709

 

2567
01:21:56,709 --> 01:21:56,719

 

2568
01:21:56,719 --> 01:21:58,790


2569
01:21:58,790 --> 01:22:00,550

 

2570
01:22:00,550 --> 01:22:02,950

 

2571
01:22:02,950 --> 01:22:03,910

 

2572
01:22:03,910 --> 01:22:03,920

 

2573
01:22:03,920 --> 01:22:08,310


2574
01:22:08,310 --> 01:22:10,310

 

2575
01:22:10,310 --> 01:22:11,750

 

2576
01:22:11,750 --> 01:22:14,149

 

2577
01:22:14,149 --> 01:22:16,550

 

2578
01:22:16,550 --> 01:22:18,390

 

2579
01:22:18,390 --> 01:22:20,709

 

2580
01:22:20,709 --> 01:22:23,350

 

2581
01:22:23,350 --> 01:22:24,629

 

2582
01:22:24,629 --> 01:22:26,229

 

2583
01:22:26,229 --> 01:22:28,709

 

2584
01:22:28,709 --> 01:22:30,310

 

2585
01:22:30,310 --> 01:22:31,830

 

2586
01:22:31,830 --> 01:22:33,910

 

2587
01:22:33,910 --> 01:22:36,149

 

2588
01:22:36,149 --> 01:22:38,709

 

2589
01:22:38,709 --> 01:22:40,950

 

2590
01:22:40,950 --> 01:22:40,960

 

2591
01:22:40,960 --> 01:22:41,990


2592
01:22:41,990 --> 01:22:44,229

 

2593
01:22:44,229 --> 01:22:45,990

 

2594
01:22:45,990 --> 01:22:48,229

 

2595
01:22:48,229 --> 01:22:49,669

 

2596
01:22:49,669 --> 01:22:49,679

 

2597
01:22:49,679 --> 01:22:51,189


2598
01:22:51,189 --> 01:22:52,229

 

2599
01:22:52,229 --> 01:22:53,910

 

2600
01:22:53,910 --> 01:22:54,950

 

2601
01:22:54,950 --> 01:22:54,960

 

2602
01:22:54,960 --> 01:22:55,910


2603
01:22:55,910 --> 01:22:57,830

 

2604
01:22:57,830 --> 01:22:59,669

 

2605
01:22:59,669 --> 01:23:02,229

 

2606
01:23:02,229 --> 01:23:03,510

 

2607
01:23:03,510 --> 01:23:06,790

 

2608
01:23:06,790 --> 01:23:09,350

 

2609
01:23:09,350 --> 01:23:10,709

 

2610
01:23:10,709 --> 01:23:12,310

 

2611
01:23:12,310 --> 01:23:14,390

 

2612
01:23:14,390 --> 01:23:16,390

 

2613
01:23:16,390 --> 01:23:18,550

 

2614
01:23:18,550 --> 01:23:18,560

 

2615
01:23:18,560 --> 01:23:19,669


2616
01:23:19,669 --> 01:23:20,790

 

2617
01:23:20,790 --> 01:23:22,950

 

2618
01:23:22,950 --> 01:23:24,709

 

2619
01:23:24,709 --> 01:23:26,629

 

2620
01:23:26,629 --> 01:23:27,590

 

2621
01:23:27,590 --> 01:23:29,750

 

2622
01:23:29,750 --> 01:23:29,760

 

2623
01:23:29,760 --> 01:23:31,270


2624
01:23:31,270 --> 01:23:33,030

 

2625
01:23:33,030 --> 01:23:35,990

 

2626
01:23:35,990 --> 01:23:36,000

 

2627
01:23:36,000 --> 01:23:36,870


2628
01:23:36,870 --> 01:23:40,229

 

2629
01:23:40,229 --> 01:23:40,239

 

2630
01:23:40,239 --> 01:23:41,510


2631
01:23:41,510 --> 01:23:43,030

 

2632
01:23:43,030 --> 01:23:45,990

 

2633
01:23:45,990 --> 01:23:47,510

 

2634
01:23:47,510 --> 01:23:49,270

 

2635
01:23:49,270 --> 01:23:51,430

 

2636
01:23:51,430 --> 01:23:53,350

 

2637
01:23:53,350 --> 01:23:55,990

 

2638
01:23:55,990 --> 01:23:58,229

 

2639
01:23:58,229 --> 01:23:59,750

 

2640
01:23:59,750 --> 01:24:01,750

 

2641
01:24:01,750 --> 01:24:01,760

 

2642
01:24:01,760 --> 01:24:04,470


2643
01:24:04,470 --> 01:24:07,430

 

2644
01:24:07,430 --> 01:24:09,590

 

2645
01:24:09,590 --> 01:24:12,310

 

2646
01:24:12,310 --> 01:24:15,189

 

2647
01:24:15,189 --> 01:24:16,629

 

2648
01:24:16,629 --> 01:24:19,350

 

2649
01:24:19,350 --> 01:24:21,590

 

2650
01:24:21,590 --> 01:24:23,669

 

2651
01:24:23,669 --> 01:24:25,910

 

2652
01:24:25,910 --> 01:24:28,310

 

2653
01:24:28,310 --> 01:24:31,110

 

2654
01:24:31,110 --> 01:24:31,120

 

2655
01:24:31,120 --> 01:24:31,990


2656
01:24:31,990 --> 01:24:36,870

 

2657
01:24:36,870 --> 01:24:36,880

 

2658
01:24:36,880 --> 01:24:38,709


2659
01:24:38,709 --> 01:24:40,470

 

2660
01:24:40,470 --> 01:24:42,390

 

2661
01:24:42,390 --> 01:24:43,910

 

2662
01:24:43,910 --> 01:24:46,950

 

2663
01:24:46,950 --> 01:24:48,870

 

2664
01:24:48,870 --> 01:24:48,880

 

2665
01:24:48,880 --> 01:24:50,390


2666
01:24:50,390 --> 01:24:52,310

 

2667
01:24:52,310 --> 01:24:54,709

 

2668
01:24:54,709 --> 01:24:57,350

 

2669
01:24:57,350 --> 01:24:59,350

 

2670
01:24:59,350 --> 01:25:01,430

 

2671
01:25:01,430 --> 01:25:02,790

 

2672
01:25:02,790 --> 01:25:04,310

 

2673
01:25:04,310 --> 01:25:06,790

 

2674
01:25:06,790 --> 01:25:09,830

 

2675
01:25:09,830 --> 01:25:09,840

 

2676
01:25:09,840 --> 01:25:11,030


2677
01:25:11,030 --> 01:25:13,590

 

2678
01:25:13,590 --> 01:25:15,189

 

2679
01:25:15,189 --> 01:25:18,070

 

2680
01:25:18,070 --> 01:25:18,080

 

2681
01:25:18,080 --> 01:25:19,030


2682
01:25:19,030 --> 01:25:20,790

 

2683
01:25:20,790 --> 01:25:23,270

 

2684
01:25:23,270 --> 01:25:27,239