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welcome everyone um  back to um theory of computation  so  we are now at lecture number 15 and  if this is an important  lecture um well all of our lectures are  important but  this is i'm going to introduce  one of the major topics  that we're going to see  going on in various forms through the  rest of the semester which is the notion  of a complete problem in this case it's  going to be an np complete problem  and  i think some of you have probably heard  of that concept already maybe you've  seen it in some courses or other but  we're going to do that in a  in a rather careful and formal way  over the next couple of lectures  so we are following up on our  previous discussions about complexity  time complexity  we defined  the time and non-deterministic time  complexity classes as you may remember  the classes p and np  talked about the p versus np problem  looked at some interesting algorithm for  showing problems in p called dynamic  programming  and we started to move toward our  discussion of np completeness with the  introduction of polynomial time  reducibility  which is  related to some of the earlier  reducibility notions that we discussed  in the computability section  so  let's  [Music]  quick review um  what it means for one problem to be  polynomial time reducible to another  this follows our pattern of  reducibility concepts where  um if a problem is reducible to another  problem and that other problem is  solvable then the first problem is  solvable so if a problem is reducible to  an easy problem that problem becomes  easy too  um and  the kind of reduction that we're going  to be looking at here are the mapping  reductions but now where the reductions  are computable in polynomial time  and so we had this um  result we mentioned last time that if a  is polynomial time reducible to b and b  is in p then a is also in p  um  [Music]  so that's going to be um  critically important for the  you know for the  for the whole um  of discussion that's coming  um  our intuition about pnnp um  repeating that here um the np problems  are the ones where you can verify  membership easily  as opposed to  being able to test membership easily  those are the problems that are in p the  verification typically requires  some kind of a  certificate  that establishes kind of a proof  that the input is a member of that  language  um  and the  big question of the field which remains  an unsolved problem  is whether p equals np it's called the p  versus np question  and we don't know the answer so whether  there are problems that are  in np um solvable and nondeterministic  polynomial time typically these are  problems that involve searching whether  they can be solved in polynomial time  typically without the searching um  and  uh if p  were equal to np then you could always  eliminate searching and a p were  different from np then there were cases  where you need to search um and we don't  know the answer to that uh  so  in  the direction of exploring uh this  this question and its ramifications  we introduced this problem called sat  these are the boolean formulas  which have an assignment  that  enable that event makes them evaluate to  true  um  so we call those satisfiable formulas  and uh we um mentioned but have not yet  proven and we will not prove until the  next lecture on tuesday  that there is this theorem  uh kind of a very remarkable theorem  that says that if you have uh you take  the satisfiability problem and if it is  solvable quickly then all of the np  problems are solvable quickly so if sat  is in p then p equals np so in a sense  sat is kind of you know the  uh  it's sort of the super  np problem in the sense that all of the  difficulty of any np problem is  is embedded within sat so if sat becomes  easy then all of the other np problems  become easy um and we'll eventually  prove that but right now we're kind of  setting up  the um terminology to allow us to do  that  um  so uh anyway we'll we'll we'll get there  um  so  the the the key ingredient for proving  this kook 11 theorem is polynomial time  reducibility  what we're going to show  is that every problem in np  can be polynomial time reduced to set  so every np problem can be converted  into a set problem  and so  working toward that because when you  think about it you know there are  infinitely many problems in np and  um  being able to show that all of them  are reducible to sat as in a sense kind  of um you have to prove a kind of a  higher level theorem it's not just a  single reduction we're kind of  exhibiting a schema of reductions which  shows that all of these problems can be  reduced to set  um and kind of  developing  our  um  you know our intuition toward that  direction we're going to look at some  specific polynomial time reductions  today um  and  we'll start out with a reduction between  two problems we have not yet seen one of  them called threesat and the other one  called clique so on this slide we're  going to introduce those two problems  so remembering again about boolean  formulas  we're going to consider a special  class  of boolean formulas a restricted form of  boolean form formulas called conjunctive  normal form  and so this formula here in particular  is in that conjunctive normal form  um so just remember um i was explaining  some of this to someone else uh  earlier this morning showing some of  these slides and it was pointed out that  not all of you may have be familiar with  these um the boolean operations of and  and or  uh so this is the or symbol here  this is the and symbol here  you know they operate you know hopefully  you've seen the just the concept of  boolean and or  or is in a sense a little bit like a  union um and it's like an intersection  and the the symbols here are kind of a  little bit similar to the union and  intersections you know the intersection  the union intersection symbols  themselves like or the v-shaped is a  little bit like a  pointy union symbol and the in the um  upside-down v-shape is a little like a  pointy intersection symbol so maybe  that's if you haven't seen those seen  that connection before maybe it's  interesting to observe that  but anyway um  what makes this  uh formula be in conjunctive normal form  well the conjunctive normal form  formulas  are organized in a certain way they have  um  these groups called clauses within the  parentheses  that are added together that are  connected by these and  and operations um  and  within those groups which are called  clauses  the um  elements are  ored together  those elements  are going to be either variables  or  variables with negation negated  variables um  so uh just to repeat that  uh these well i'll just just state those  those variables or negated variables are  going to be called literals um  and  the  ors of a bunch of literals are going to  be called clauses  okay this is just the standard  terminology in the field  okay so  um a literal is a variable or a negated  variable  and a clause is an or  of literals  okay just a definition  all right so we have a bunch of these  clauses here each of the clauses has an  or of of liter of literals in it  now  a chomsky  conjunctive normal form  uh formula i guess they're both c and  f's but uh  it's conjunctive normal form formula is  one that's written as an and of these  clauses  okay so we take these clauses which  themselves are ors we and them together  we get a formula that's conjunctive  normal form conjunctive stands for and i  think uh conjunction is an and um  and then a three  cnf  is a cnf where each clause has exactly  three literals  so for example this particular  formula is a cnf but it's not a three  cnf because it has at least two clauses  which violate the condition of three  literals per clause this one is okay  obviously  this first clause  and the three sat problem  so this is the first of the two problems  we're going to be talking about  the three sad problem  is  the satisfiability problem restricted to  these three cnf formulas  so this is the collection of  setup of three cnf formulas which are  satisfiable  so you can think of it as a kind of a  special case  of the sat problem where we only care  about formulas of the special kind  um  this being a special case you might  imagine that this might be an easier  problem to solve but in general it turns  out not to be as we will see this is  just as solving this special case is  just as har as hard as solving the  general case for satisfiability  um  now let's turn to the second  of these two languages up in the  headline  uh  the clique problem  um  so for that we gonna turn to graph  theory  so we're going to consider graphs  you know  points and lines connecting them  and we will say that a clique  in a graph  is a collection of nodes collection of  the points that are all pairwise  connected by lines  and a k click is one where you have case  arch nodes  so  here we have a three click four click  and five click  um and the clique problem  is to try to find cliques of a certain  specified size  embedded within a given graph  so you'd like to know whether you're  going to be given a graph and it targets  click-sized k  and you want to know is there a subset  of the nodes in the graph  of size k  that are all connected to one another  that's the cleat problem  um  so obviously  um  the clique problem just as with the  satisfiability problem is a decidable  problem you can just try every possible  subset of k nodes  and see whether it constitutes a clique  but that's in general going to be an  exponential algorithm if you have you  know  you know k is a large value like it  might be half the size of the graph you  might be looking for a very large clique  then you have to try  many many subsets in order to see  whether uh any one of them is a clique  it's also going to be and i hope you get  this intuition this is a problem that's  in np  the cleat problem is an np problem  because you can easily verify  that a graph uh has a k click just by  exhibiting the click  okay um  now uh  uh  so the clique language here is you're  given a graph and a k and you want to  know does the graph contain a cake  league um and what we're going to show  is that these two problems are connected  that the three set problem and the cleat  problem are related in that you can  reduce in polynomial time three sat to  click  okay so this is um  you know  at  you know just standing back at it  back for a minute you know that seems  kind of surprising there's no real  reason  you know uh obviously anyway no obvious  reason why there should be a connection  between three sat and click click they  look very different from one another  but we do give that uh we will give that  reduction  and um it then that implies that if you  can find a way of solving the clique  problem quickly that'll give you a way  of solving the the three set problem  quickly  um and that's going to be the whole  point of this um  so we're going to show this over the  next  slide or two um this this polynomial  time reduction and this is something i'm  going to walk through it slowly  but this is one where you it's really  important to try to get a sense of how  it's working um  uh because this is the kind of thing  that we're going to be doing a lot of  and you're going to be asked to do it  also on the homework and and possibly on  the final exam i hate to use that as the  motivating force here but at least it  might be one motivation for some of you  that you have to understand how to do uh  this kind of a reduction  okay so  you know i i think at a high level  what's going on is we're showing how to  recode  a  boolean formula satisfiability problem  into the problem of testing whether a  graph has a clique  um  so  let's see if we have any questions here  that are coming up in the in the chat um  okay so i this is an interesting  question here can we always convert  a boolean  formula  into conjunctive into conjunctive normal  form  first of all yes the answer is you can  always convert a boolean formula into an  equivalent one in ins into in cnf  but  in general that might make the formula  exponentially larger  [Music]  so  just the mere fact that you can convert  formulas into into conjunctive normal  form doesn't mean  that solving um  that you know that solving uh  um  conjunctive normal form uh formulas for  testing satisfiability of the of the cnf  formulas is going to be as hard as  testing the general case because just  the conversion might be exponential um  there's something more uh more  a little bit more complicated going on  um  than that  so let's uh  um  let me just see here  uh if he is a satisfiable formula which  is not in cnf can be  so a similar  um you know similar question so the  questions that i'm getting are about  converting formulas uh to cnf um  so yeah you can do it but not in  polynomial time in general because the  resulting formula you get might be much  larger  um if you if you're looking for an  equivalent formula if you're not looking  for an equivalent formula then you know  of course then depending upon what  you're looking for you might be able to  find something smaller  um  ah  this is a this is a i guess a good basic  question  why is clique in np  um  because isn't doesn't verifying that you  have a clique require going through all  the possible cliques  once i  you have to understand what verifying  means verifying means you can verify  something if you're given a certificate  in the case of a clique  the click problem the certificate is the  clique  um so once you have the certificate you  can do the verification in polynomial  time finding the certificate of course  might be difficult  so you only think of np in the context  of having that certificate  so in a case for like  compositeness the certificate might be  the factor  um  you know  anyway i hope that's you know it's there  there are problems sometimes with what  the certificate is is not necessarily  obvious um but there can be a  certificate um for  um showing that the  that the input is in the language but in  the ones that we've done so far maybe  you can argue that this certificate is  sort of an obvious thing but it's not  always an obvious thing  um  okay  why don't we move on then  um  so let's see how do we do  this polynomial time reduction from  three sat to click  okay  here we go  um  so i'm just going to give a reduction  that's that's that's what the definition  means i'm going to give a way of  converting  formulas  to  pairs  a g and a k  where the formula is going to be solid  satisfiable if and only if the graph has  a k click  okay so let's kind of a little bit do it  by example  and  in order to do that  so here's going to be a formula now it's  in three cnf that's what i need you know  in order to be doing this reduction i'm  converting three cnf formulas into  clique problems  we have to a little bit understand what  it means  when we say  uh we talk about the satisfiability of a  formula like this because it's going to  be helpful in doing the reduction you  know obviously it's satisfiable means  that the form you can find an assignment  of the variables so you're going to set  each of these variables a b and c and so  on to true or false  and you want to make the whole formula  evaluate to true but what does that  actually mean  [Music]  it means that because of the structure  of the formula  that making this formula true  corresponds to making each clause true  because  the clauses are all added together so  the only way for the formula to be true  is to make each clause true  and to make a clause true  you have to make at least one of the  literals true  so it's another way of thinking about  uh satisfying this formula  you need satisfying these and a  satisfying assignment  makes at least one true literal in every  clause  it's really important to think about it  that way  because that's what's going to be um  [Music]  kind of the basis  for doing  this reduction and all of the reductions  it's kind of what makes three set easy  to think about in terms of its  satisfiability  if you had a sort of a general um  satisfiability problem and you had a  satisfiable formula there's no sort of  obvious way of sort of seeing what the  satisfying assignment looks like but  here we understand what it looks like it  has that very special form  making one true literal in every clause  at least one true literal in every  clause  okay so now we're going to do the  reduction so i'm going to take from this  formula you and  you have to understand you know i know  for some of you you know you're going to  be you know chafing why am i going so  slowly but i want to make sure that  we're all together in understanding what  the rules of the game are and what we're  trying to do  um  we're trying to convert  this formula into a graph and a number  so right now my job is to do this  reduction is to exhibit that graph  okay so i'm going to do that in kind of  two steps i'll first i'm going to tell  you what the nodes of the graph are  then then i'll tell you what the edges  of that graph  finally i'll tell you what the number k  is  that's the way this polynomial time  reduction is going to work  and we have to also  observe at the very end  that the reduction that i'm given giving  you this  procedure for building this graph can be  done in polynomial time but that i think  you'll see once i'm done that that's  pretty obvious  okay so first as i promised the nodes um  so the nodes of this graph are going to  correspond to the literals of the  formula every literal is going to become  a node in the graph and it's going to be  labeled  with  the um name of that literal so it's  every node is going to be labeled an a  or b or or c bar  and so on so here it goes  okay so those are the nodes  of the graph g  one for each literal in the formula  okay  labeled as promised  now i have to tell you what the edges  look like  the edges are going to be  um i'm going to tell you what the edges  are by first telling you what they are  not  i'm going to first explain to you what  the forbidden edges are  what what edges i'm going to  promise not to include  and then the ones that i  will include are going to be all the  others  okay so what are going to be the  forbidden edges first of all um  the the forbidden  edges are going to be of two types one  is the edges between nodes  that come from  literals in the same clause so i would  just call that  no edges within a clause  okay so for example these three nodes  will not be connected to one another and  i'm going to indicate that by writing  red dashed lines which means there's not  an edge there  okay those are those are forbidden from  having an edge  okay so these three have no edge and the  same thing for every other uh triple of  three nodes that come from uh clauses  okay those are  no edges there  and there's one other category of edge  that i'm forbidding  and that are edges that can  that um go between  inconsistent labels and the nodes with  inconsistent labels so for example  between a and a bar those are  inconsistent  you know nothing's wrong with a going a  to d those are not inconsistent those  are just different labels or going from  a to a that's okay  but from a to a bar that's that's not  allowed um so that's going to be another  forbidden edge  or for example a to a bar here  for b a bar to a or for example from  this c bar to c  forbidden  okay so imagine you're going to write  down all of those forbidden edges and  then once you've done that  uh  those are all of the the forbidden edges  and then after  [Music]  taking those away  you're going to be putting in all the  other edges possible  so for example as a let me just uh  gray those out so they don't interfere  with the picture  b from a to a to d  um there's going to be an edge um  because those are not forbid you know a  to a not forbidden because  they they're in different  um  they are in different uh  clauses and they're not inconsistent  they're they're consistent with one  another so here are a whole bunch of  other i'm not showing them all becomes  very messy but uh here are all the other  edges between nodes uh which are  where there isn't there's  no pro  within they're not forbidden  okay and that's  that's the graph  that's g  g is all  those nodes and those edges um which are  not forbidden  now i just have to tell you what k a is  k is going to be the number of clauses  that's going to be the size of the  clique i'm looking for in this graph g  that i just spelled out for you  and i'm going to claim  that  this graph here that i just described  will have a k clique  k is the number of clauses  exactly when  fee was satisfiable  it's kind of kind of cool you know um if  he is satisfiable then there will be a  cake leak here  and if he was not satisfiable okay clean  so we have to we're gonna prove that as  a claim on the next slide  okay so k is the number of clauses  all right  um  any questions on that on that  construction so i've done with the  construction what's left is to argue why  the construction works  so far so good let's move on  okay  um  so um  so here is that very same construction i  eliminated the red uh  forbidden edges these are the ones that  were remaining  plus you know anything else that was not  forbidden  that was the same formula that i had  from the previous slide  now i'm going to claim that that formula  is satisfiable exactly when g has a k  click now why in the world is that  so this is an if and only if it's proved  in both directions  and this is going to be the typical kind  of thing that you're going to want to do  when you're exhibiting a  uh  you know one of these reductions  which is you're going to have an  opportunity to do that um and we'll do  do that too  in our examples  uh  okay  so now  uh what i'm what i want to show  is that  uh  if fee is satisfiable so let's prove the  forward direction if p is satisfiable  then g has a k click  okay so first of all if fee is  satisfiable that means it has a  satisfying assignment  okay  now here's a common confusion i'm not  sure whether it's helpful to  sprinkle the confusions in along with  the discussion but  in case you're worried  you might ask well how do i find that  satisfying assignment i thought that was  you know  exponential  exponentially hard to do  this is a proof this is not an algorithm  okay we're just trying to argue the con  correctness of this construction  so there's no longer any concern about  how to find an assignment i'm just  saying if there isn't a sign if there is  a satisfying assignment then something  will happen something that then then um  something will happen in particular then  we will show that g has a cake league  so  let's say  fee is satisfiable  so it has some assignment let's take  that satisfying assignment  and remember  that in any satisfying assignment  uh to a formula in cnf  that makes at least one literal true in  every clause  so  let's just pick one of those there might  be several clauses which have multiple  true literals in that case just pick one  of them  arbitrarily  okay so i don't know what the i don't  have this sort of indicated on this  diagram here but imagine you know maybe  in the very first clause a was true in  the second clause b was true in the  third clause  e compliment ebar was true and you know  not not  you know  e bar was true which means e itself was  false but e bar was true and that's the  way each of those clauses in turn uh got  to be true you know in this particular  satisfying assignment  because you're going to pick one true  literal in every clause and now  from  that choice  of literals one per clause  i'm going to look at the corresponding  nodes in g  and i'm going to claim  that those nodes taken together  form a k click  okay so first of all  do they have the right number of nodes  well sure because  i'm picking one node per click i already  said k is the number of cliques so i'm  getting exactly ka nodes so i have at  least the right number  of nodes but how do i know they're all  connected to one another those nodes uh  that i just picked  well they're all connected to each other  because i'm going to say well because  there were no forbidden edges among them  and remember that we put in all possible  edges except for the forbidden ones  so how do i know there are no forbidden  edges  well what were the rules for being  forbidden means they two nodes in the  same clique in the same clause  well i'm picking one node per clause  so i can never be having two nodes from  the same clause so i'm never going to  run into trouble with uh the first  condition of being forbidden so what's  the second possibility for being  forbidden is that i'm picking two nodes  which are inconsistent  well how do i know that i didn't end up  with two nodes with inconsistent labels  that would be bad because then they  would have a forbidden edge and what my  result would not be a clique  how do i know i didn't end up picking  um  uh  in this group i pick a and in this group  i pick a a bar  well because they all came from the same  assignment  those are all of the true letters they  were all true literals  in clauses it's not possible that a was  true in this clause and a bar is true in  that clause because if a is true here a  bar has got to be false it can't be the  true literal in this clause  so i cannot have any inconsistent any  inconsistent nodes  appearing  among the nodes of my clique  and so they're not in the same clause  they're not inconsistent so the edge has  to be there and that's going to be true  for every pair of nodes in that clique  in that in that group of nodes and  that's why it's a click  um  so that proves one direction now we stop  to prove the other direction because we  have to say well if g has a k clique how  do we know that ph is satisfied so  that's the reverse  so let's just take any k clique that's  in g  and how do i know i can get from that a  satisfying assignment to the formula  um  uh  good um  getting some good questions here in the  chat but let me just move on  um take any uh so  proving the reverse  this the reverse direction but we're  assuming we have a cake leak to take any  such k-click  now first of all you observe that it's  got to have  one node in every clause  it can't have two nodes in the same  clause or zero nodes in in a clause  first of all it can't have two nodes in  the same clause why well because those  nodes are never connected to one another  so that they cannot be both in the same  clique because all the nodes in a clique  are connected to one another  um by construction by the way we  construct a g  nodes from the same clause are not  connected so they cannot appear in a  clique together  so um  there's going to be at most one node  from every clause appearing in this  clique but and but but  appearing in in this clip but we also  have we know we have k notes  so that means if there is at most one  per um per clause  and we only have k clauses that means  every clause has got to have one  you know if some clause is missing or  note then there's going to be some other  clause that has two  so we know that every um  that this clique that would that g  has has exactly one node in every clause  so how do we get from that a satisfying  assignment  so what we're going to do is  take the current  corresponding literals  that correspond to that one node in the  that was in the uh clique  take those literals and set those  literals to be true  which means setting the variable to be  true  um or setting um you know  if the literal was a bar then setting a  to be false because you want a bar to be  true you want to set the literals to be  true  well now we're setting um there's one  note in each clause we're setting one  literal uh the corresponding literal  true so that's going to set one true  literal in every clause  so that means it's going to be  satisfying there's one thing that one  has to really to be  careful here to double check to make  sure that we're doing this consistently  that we're not being asked to set you  know a  true and also a bar true because then  that would not be  possible  but a and a bar are not connected  so it's we're never going to be trying  to set both a and a bar true if so we're  going to be trying to set a true we're  never going to try to set a bar true  because those are not those a and a bar  cannot be in the same clique they're not  connected to each other  okay so  that  is the argument um and lastly  i claim we're not going to look at this  in detail that it's kind of obvious that  this reduction can be done in polynomial  time namely if i give you that formula  you can write out that graph in  pretty easily  there's no hard work to be done in  writing out that that graph or counting  up the number of clauses  so that's that's the proof  that i can reduce three statically  um  and it tells you also that if i could  solve click quickly  then i can stop threesat quickly  and this is the whole point  because i can convert now clique  problems to three sat problems  no  no i couldn't i'm sorry i got backwards  convert three set problems the clique  problem so if i can solve quick click  easily then i can solve three set easily  right i just showed a way of converting  these formulas to graphs so if that's so  that says if i can solve the graphs  easily then i can solve the formulas  okay  why don't we uh turn to some oh question  checking but you know we can also  um why don't i i'll launch a check-in  but we'll  we  let's just see  i'm seeing many many questions here  which are actually what my check-in is  about  i'll turn it back to you guys uh  okay so  where did we use the fact  um that we have  three  uh literals per clause what does this  thing just work even if we had any  number of literals per clause  um  what do you think  we're having a  we've got a tight race here  hmm uh  truth is losing out unfortunately  all right oh it's really close but still  okay  uh  one more okay it's neck and neck  okay almost done  10  seconds  are we done here okay that's it  uh ending polling  uh  yes  truth is  thank god  um  yeah it works for any size clause we  didn't use the fact that it's a three c  and f it could have been any number here  uh  so it's well we've got a i guess it's a  plural plurality here though that not a  majority um  no we didn't where did we use three in  any of these argument i didn't mention  it maybe you were imagining that that is  going to be part of it but three does  not come into this at this discussion at  all if you think about how it's why it's  working  um even if we had you know one of these  clauses had 10 literals in it  as long as k is the number of clauses  and we don't connect  any variables any literals internal to a  clause  this whole argument is going to still  work so please check that make sure you  understand what's going on because i can  see that a good chunk of you  have not got this right  um  uh so i got a good question here uh what  if it's only just one big clause so it's  like the whole formula is just one big  war so what does happen in that case  good question so in that case um  uh  so the you know suppose has one big  clause with a hundred literals  that's the whole formula it's just a big  war so we know the formula is going to  be satisfiable by the way obviously um  so if you look at the corresponding g  it's going to have 100 nodes because  there's one for every literal um  none of them are going to be connected  to each other because they're all in the  same clause  um so it looks like we're going to be in  tough shape trying to find a clique  there because there's no edges at all  everything's forbidden but what is k  k is going to be 1 in that case because  there's only one clause  and  uh  kind of a degenerate case but a clique  with just one  you know just a single node  is it counts as a one clique  because it's just one node there's no  need for any edges at all it still  counts as a clique so it'll still it'll  still work out um  uh let's see what else here uh that was  kind of a fun question that you asked me  thank you um  uh there are a lot of questions um  i think we're actually pretty near the  break also we'll just see  yeah um  so why don't i um  oh  many many questions here uh  so i i think i'm gonna start start the  clock going down for our um our coffee  break and um  and i will take some of these questions  i'll try to answer some of these  questions oops  um try to answer some of these questions  um  afterward  uh all right  good  all right  um  so i got a question about  making sure why nodes don't have  inconsistent labels if i understand the  question correctly  um so these would be nodes  um  so i never put edges between nodes with  inconsistent labels that's the rule  that's my construction i get to i get to  say how the construction works so those  two nodes with inconsistent labels can  never be in the same clique  because  there's no edge between them  so i'm not sure that answered the  question there but that was uh  um  let's see  so getting another question here  since any boolean formula can be  converted to cnf  does that mean um is sat polynomial time  reducible to clique as well  monomial time reducible to clique but  not for the reason that any boolean  formula can be converted to c and f  because  that conversion  um  is going to be an exponential conversion  in general the conversion to c and f so  you have to be careful about what we  mean by converting a formula to cnf  um  uh so also getting questions what  happens in the case of one set well we  didn't really talk about one set  uh suppose or i guess so the question is  suppose there's only a single  literal per clause  that's an interesting another  sort of edge case here what happens  under those circumstances  so um there's only a single literal per  clause um  uh so you have to work out what happens  but in that case  um  uh  all the clauses just have one literal in  them  and now they have um  uh  um  the only way  uh  this  if you think about what is a when you  have a  one cnf so there's only um  one  literal per clause the only way that can  be satisfiable is you have to make each  literal true because there's no warring  anymore so every literal has to be true  in the assignment so that means you can  never have inconsistent literals um and  that's the only case when you would not  have an edge because you're not the  forbidden is the forbidden condition  won't happen because you don't have more  than one uh node per per clause every  clause is going to have just a single  node in it uh so  it really comes down to whether or not  you have an inconsistently labeled um  inconsistent clauses  i i didn't explain that super well but  it's the  the argument still works there you  should check it for yourself  um  so this is a basic question how do we  see that f is polynomial time  i i'm not sure i want to expend a lot of  you know a lot of discussion on that  because  the the conversion from the formula  to the graph  is kind of a one-to-one  you know every no every uh literal  becomes a node  um the rule for when edges are present  it's a very simple rule so i guess i  would argue  i'm not sure what what more to say it  seems pretty clear that the  conversion that reduction is polynomial  time computable i mean if you wrote a  program to do that it would operate very  quickly  um  is it possible for g to have a k plus n  clique where n is greater than zero and  does that matter  now if you think about it  the biggest possible clique that this  graph g could have is k  it could never have more than  uh uh you know a bigger clique because  two nodes in the same uh clause are  never connected  you only have k clauses  so the answer is no you cannot have a  bigger  a bigger than a k click it's just not  going to happen  does each clause need to have the same  number of literals i don't see why  um  uh  so question is why are we worrying about  threesat um  since it didn't seem to matter here  there are other examples where it does  matter i i i'm actually not sure if  we'll see one or not but um there there  are cases where it does matter  um let me just see if there's any quick  questions here  uh  somebody's saying doesn't does  do we have to worry about there being a  polynomial number of literals  so you have to think about what you what  that question means the size of the  input which includes all of the literals  is n  so there's going to be at most n  literals  um because that's the size of the input  uh the i mean  at most uh and literals appearing so  you don't have to worry about that being  polynomial because by the way we're  we're defining the size of the input is  going to be at most n literals  i hope that's clear  uh  okay  when we talk about power  okay so this is a good question when we  talk about polynomial time  it's polynomial in the representation of  of the input which is if you want to  think about it in terms of bits that's  fine or whatever it's not going to  matter if you use some larger alphabet  but the number of symbols you need you  know in your fixed size alphabet the  number of symbols you need to write down  that input in whatever your reasonable  encoding is is going to be n and it's  going to be polynomial in n polynomial  in that  length of the input  um  so another question is does sat reduce  the three set yes we that we will see um  sat actually does polynomial time reduce  the three set not by converting to an  equivalent formula but by some more uh  somewhat more involved argument uh than  that okay why don't we move on some of  these are going to come out anyway in  the lecture  okay so now let's talk about mp  completeness because we've kind of set  things up we're not going to prove that  you know the basic theorem the cook 11  theorem about mp completeness but at  least we'll be able to make the  definition  okay here's our definition you know what  it means for a problem to be in be  complete  um  uh  so a language  b is called np complete  if it has two properties  number one is that has to be a member of  np  and number two  every language in np has the polynomial  time reduced to that  to that language to that np complete  language in order for it to be np  complete  okay so  simple picture  has to be in nnp  and everything in np reduces to it  um  and so that's kind of the magical  property that we claimed that that sat  has sat for for one thing is obviously  in np  and  as we  the cook 11 theorem shows  or will show  everything in npe is reducible to sat so  that's going to be our first example of  an np complete problem  um  uh so now  uh and we're going to get what we  claimed also for sat that if sat or any  other np-complete problem  turns out to be solvable in polynomial  time  then every np problem is solvable in  polynomial time and and that's immediate  because everything is reducible in  polynomial time to the np complete  problem  so if you can do it easily you can do  everything easily just by going through  the reduction  okay  so the quick level theorem as i  mentioned is that sad is np-complete  and we're going to actually prove it  next lecture but let's assume for the  remainder of this lecture that we know  it to be true because so i'll use the  terminology of problems being np  complete uh assuming that we know um  that we have uh sat as empty complete  okay  so we're gonna we're gonna be  using some of the things that were  proving next lecture uh just  in the terminology that we're going to  be talking today  okay so here's the picture  here's the class np  and everything in np  is  polynomial time reducible to set  sat itself is a member of np but i  didn't want to show it that way because  it makes the picture kind of hard to uh  hard to uh  hard to display  so  just from the perspective of the  reduction everything in np is polynomial  time reducible to sad we'll show that  next lecture  another thing that we'll show next  lecture  is that sat in turn is polynomial time  reducible to three sat so  three stat as you remember are just  those problems that are in conjunction  three  are in three c and f  um  and then  what we showed today is that three set  is polynomial time reducible to clique  so now  taking the assumption that sat is np  complete  so everything is polynomial time  reducible to sat which is in turn  polynomial time reducible to three sat  in in turn reducible to click  these reductions as we've seen before  composed  um  you can just reply apply one reduction  function after the next  if each one each one individually is  polynomial the whole thing as a  combination is going to be polynomial so  now we know that three sat is going to  be also np complete because  we can reduce anything in np to stat and  then to three set and then we get a  reduction directly to three sat by  composing those two reductions and then  furthermore at the clique so now we  uh have several np complete problems and  moving beyond that  we have the hand path problem which we  are going to talk about next and we'll  show another  reduction um  in addition to the one we just showed to  click now one going from three set to  hand  um  okay uh so in general  i think the the the the take away  message is that to show some language as  np complete you want to show that three  sat is polynomial time reducible to it  um  uh  okay  some good questions coming in i'll try  to answer those so you're going to  retake  3sat and reduced to c that's the sort of  the most typical case  or you might have there's going to be  other examples too we might start with  another problem that you've already  shown to be np complete and read and  reduce it to your language so it doesn't  have to start with reset  though  often it does  um  okay why do we why is this concept  important  so i would say there are two reasons um  and this is going to get a little bit at  the uh some of the chat questions um  first of all  if you're faced with some copy new  computational problem you've got some  robotics problem  um that you want to solve in your thesis  and you need to some algorithm about  whether the robot arm can do such and  move in a certain such a way and involve  searching you know possibly searching  through a space of different kinds of  motions and you want to know you know i  need a either i'd like to have a i like  to find a polynomial algorithm to solve  that problem  i'm using this as an example because  this actually did happen to one of the  former students from this class who was  working in robotics and they actually  end up proving that the problem that  they were trying to solve is np complete  um  so  um  uh  that's useful information because even  though knowing a problem is np complete  doesn't guarantee that it's not in p  because conceivably p equals np  what it does tell you that if you show  if you have a problem that's np complete  and it does turn out to be in p  then all then p equals np so the tremend  would be tremendous surprising  consequences  of  of your problem if it was known to be  np-complete ending up in p so generally  people take a proof of np completeness  as powerful evidence that the problem is  not in p even though it's not quite a  proof  it's powerful evidence that it's not in  p and so you might as well give up  working on trying to find a polynomial  algorithm for it because if you do  you know you don't have to worry about  robotics anymore you're going to you  know become famous as a complexity  theory person um  so i wouldn't worry about so if you show  problems np complete you can pretty much  assume it's almost certainly not in p um  now there's another reason related to  the to the for the theorists uh to be  care about mp completeness and that is  if you're trying to prove p is different  from np or p equals np as one of the  chat questions is raising um  you know the problem you know  in order to prove p different from np  you know the most likely approach is you  pick some p problem in np and show it's  not in p  right that's what it would mean for them  to be different you're going to pick  some np problem and show it's not not a  p problem  well  um  one thing would be terrible  is possibly p is different from np and  you pick the wrong problem  like suppose i'd spent all my time  trying to show  you know back  20 years ago i'm working really hard  trying to show composites  is not in pain  which is would have been perfectly  reasonable to do  because composites is an np and was not  known to be in p 20 years ago  um and then i invested tons of effort to  try to prove because you know i like  number theory or god knows what to prove  that composites is not in p  and then it turns out composites wasn't  in p it was the wrong problem to pick  even though p might be different from np  but what np completeness guarantees is  that if you work on a problem which is  np-complete you could you can't pick the  wrong problem because if any problem is  in np and not in p an np complete  problem is going to be an example of  that  right because if the np complete problem  is in p everything in np is in p  so if np is different from p you know  the complete problems are not in p so  you might as well work on one of those  so those are two  ways in which np completeness and has  turned out to be is an important concept  okay so here's a check-in um  uh  again this  you guys are kind of getting maybe  you're getting  started to think the way i think you get  is some at least one of you would ask  this question in the chat  um  which language that we've probably seen  is most analogous to sat  okay atm etm  or zero to the k1 to the k  uh  obviously this is maybe subjective you  may have your own interpretation of what  that means  this in a sense perhaps no right answer  but what do you think  uh  okay that's it  um i don't know i don't know how in the  world that you could see this problem is  analogous to to zero to the k1 to the k  but okay um  i'm sure you have your reason  um  yes this is a lot like eight  atm why  well because for one thing we showed in  a homework problem  that all  um touring recognizable languages are  reducible to atm mapping reducible to  atm so that's a little bit like the  notion of completeness  that we have for satisfiability  um because all np problems are going to  be reducible to sad  um  and the other thing too is that once we  start once we  uh we started we want to show other  problems are undecidable we reduced atm  to them and that's also very similar  we're going to be reducing sat or  threesat  so it's indirectly from sat to other  problems in order to show that they're  np complete so that they're hard they're  they're that they're hard in a sense  and so uh in kind of both those ways  atm and sat are kind of playing similar  roles one key difference however  between atm and sat is in the for atm we  can prove  that it's undecidable  but for sat we don't know how to prove  it's outside of p  that would those would be the analogous  situations  um  and  uh so  the the story for sat which is kind of  easily solved by a diagonalization  argument for atm  there's reasons to believe that we will  see later that diagonalization is not  going to work to prove sat  outside of p  and  besides that we don't really have any  good methods  um  so  uh anyway  uh let's move on  why is etm less analogous  uh  i know because atm was the first problem  we showed undecidable and sad is kind of  the first problem that we're going to be  showing np complete  i guess that would be my answer  um  okay let's continue so let's show now  that  handpath is np-complete assuming that we  know sat  or threesat is np complete  okay so we're going to give a reduction  from 3sat to handpath that that  that that's what that's what this is  about it's just like what we did for  clique but now for him path um  and and this is going to be very typical  in in these reductions typically what  happens  is that you're trying to simulate  a formula  use you know a boolean formula for  satisfiability  from the satisfiability perspective  you're trying to simulate that formula  with some sort of structures  inside the target language which would  be hand path  and so what the lingo that  people use is that you're going to build  gadgets to simulate the structures in  the formula namely  the variables the literals and the and  the clauses  okay these are going to be substructures  of  the um  either you know the graph in this case  that you're that you're building um  we'll see what that means so let's take  a formula here  and  uh let's again try to imagine how we  would reduce that  to the hand path problem so the  reduction would produce  a graph no it would produce a hand path  instance so a g s and t want to know is  there a hamiltonian path from s to t in  the graph  and this is going to be not the whole  graph but this is going to be a  substructure in that graph the next  slide is going to have the global  structure of the graph graph  but here this is going to be a key  element and it's we're going to call  that  the variable gadget  okay what does it look like i don't know  if you can see it on your um  clearly enough but these  uh  edges so there are four outside notes  here the edges connecting them are all  kind of pointed downward  and then there are these horizontal  nodes here  and there are edges connecting them both  left to right and right to left  okay there's a row of these horizontal  nodes okay so you get the picture of  what this looks like  um  you have to look carefully to see the  arrowheads i  maybe should have made those a little  bigger  okay so now um  uh  whoops  let me let me just see so now if we're  trying to get from this node to that  node imagine now you're trying to build  a hamiltonian path  because this is going to be a part  of that graph g  that i'm constructing  now remember for them there to be a  hamiltonian graph  that means you have to go through every  node  in the graph so if i want to get from s  to t  the only way i'm going to be able to to  go to hit these  horizontal nodes here  is  um  by picking them up  uh  as i go from s to t  so the only possibility is if you think  about it is it would be for the  sequence to go like this  if you could if that comes through for  you so the the path would go from s to  this node and then through  these hops of  along these horizontal nodes and then  down to the bottom node here  okay so that's one way  that you can get from here down to there  and pick up all the other nodes along  the way which is they're not going to  have any other possibility possible ways  of getting to them  so i'm going to call that a zigzag  okay but there's another way to get from  s to the bottom node  okay which is going to be by doing sort  of the  dual going to the right then going out  to the left and then down to the bottom  i'm going to call that  a zigzig  okay  and uh with a little kind of little  diagram here just to summarize what it  means  and this is the this is the classic  thing for a variable gadget  because it's a structure  that when you're trying to think about  how the object that you're  asking whether it exists or not the  hamiltonian path how it relates to that  object it's going to have two  possibilities which are going to  correspond to the variable variable  being set true or false in the formula  so we're showing how to  set  you know the simulate the variable  in this hand path instance  so setting the variable is going to  correspond to  to to  to constructing the path  okay  now we also have to make sure not only  that the variable gets set to true or  false but that it gets a set to true and  false in a way that makes this a  satisfying assignment namely that we get  one true literal in every clause  so i'm going to add another gadget here  called a clause gadget  which is just a single node  and visiting that node here  is going to correspond to  um  satisfying that clause to having a true  literal in that clause  um  and now how am i going to get i'm going  to  have it enable  a detour  from these horizontal nodes out to this  clause gadget  so here it is here's that detour  as i'm going from left to right  i can instead of jump doing a single  jump here i could branch out and visit  that clause node  and then come back and pick up my left  to right  uh path as as i was doing before  you realize what i'm trying what's  i hope you're seeing the big picture  because this has to be a hamiltonian  path it has to hit every node that's one  of the requirements so i've got a this  is going to be one of the nodes in my  graph it's the path has got to hit that  node the only way it's going to be able  to hit that node  is by taking a detour off of this  uh horizontal path here  but notice and this is the this is the  key  if i'm doing a zigzag  then i can make the detour and visit  that clause gadget that clause node  but if i'm doing a zag zig  the way this detour is structured does  not allow me  to visit that node  because if i'm going from right to left  here by the time i get to  to this outgoing edge  now i come want to come back  that note has already been taken  it only works if i'm going zag zig from  going from right to left  the left to right if i'm going from left  to right then i can do it for going  right to left no  so this is going to correspond if you  think of left the left to right  direction as true that's going to  correspond to that variable appearing in  the positive way in that clause um but  not in the negative way the negative way  i would reverse  the in and out  of the dot of the detour  flip that around  so now i could only do the detail detour  when i'm going right to left instead of  left to right i hope you get the picture  this is how the structure is working now  what's only left me to do is put it all  together  but  this slide contains the guts um of  what's happening  um so you wanna is is there any question  that i can answer for you  um on this  let's move on to the next slide and then  as questions come up you can be typing  them in we can maybe answer it there  okay so here is the big picture  so imagine we have  that  formula  that we started we're reducing that  formula  um uh  and um  uh  and let's say it has m variables and k  clauses  i'm going to call them clause c1 c2 up  to ck here the m variables appearing  either  positively or  negated in that formula  and this is the way the global structure  of g is going to look like now i'm  getting a question here what do those  horizontal nodes what role do they play  there those nodes are there to allow me  to visit  those  clause  nodes  the the nodes which represent the clause  gadgets which i'm going to place over  here  so imag so here here is that  um here is i mean this is almost a hole  of g that i'm just missing uh some a few  edges but these are all the nodes of g  so remember i'm trying to find out is  there a hamiltonian path from s to t  now if i didn't have to worry about the  these c nodes  the answer would be just yes  in fact there would be many hamiltonian  pairs from s to t because i can do a  zigzag or zx zig through each of these  uh variable gadgets and i would take me  from s to t and i'd be good  it's just these c nodes these nodes  these ci nodes  you know i have to hit them too  so  they're going to be  um  [Music]  visiting them is going to be enabled by  detours from here so  let me just try to show you what that  looks like so i'm going to magnify  little pieces here  uh from from these gadgets here and show  you how these guys are connected up  so here is the x1 gadget  so x1 appears positively in c1  right here's x1 and c1 and so that means  when i'm going left to right i'm going  to be a possibility of visiting c1  um  but now let's look what happens with um  uh  uh  which was the next one i met i i had  here  uh  okay um  so  uh  right  so the next one is oh yeah so that's so  x1 appears positively in c1 and that's  why i have the connection like this uh  now  x1 appears negated in c2  so i only want to do enable the detour  to visit c2 when i'm going right to left  as opposed to left to right  so  this  uh  set of  horizontal nodes is only going to allow  me to take a detour either out to c1  or out to c2 but not to both  because the hamiltonian path is either  going to go left to right or right to  left it can't do both  when it's going through x the x one uh  gadget  okay you need to  didn't kind of you have to i'll try to  help you through it but you have to try  to think about why it's working  so what let's think again together  so x2  also appears in c1  but now it appears negated so i'm going  to have edges from this x2  gadget  whoops  x the x2 gadget  but now look at the look at how i've  i've arranged that detour you know i can  leave on the left leftward side and  return on the rightward side  um which means  i can only  do that detour when i'm going right to  left  okay um  and that's because x2 is negated in c1  okay so there's  maybe it's a lot here if you're if  you're not quite getting it but  the point is let's say try to quickly  prove so that's the whole construction  you just do that for every single  appearance of a literal  in a clause you're going to add these  detours which allow you possibly to go  visit  the uh the clause  um  so the forward direction is  um  why is this true so you take any  satisfying assignment as i suggested  you make the corresponding zigzags or  zag zigs through the variable gadgets  from s to t and then take the detours to  visit the cloud the clause nodes  um  the reverse direction actually is  slightly trickier we're not gonna have  time to go through the subtlety of it  um  but uh  what you want to make sure here  is that you don't have a kind of a a  weird path  um  occurring because i'm going to start  with a hamiltonian path now and build an  assignment  uh and what we want to make sure that  the the path that i'm constructing  doesn't  go from one from this  horizontal nodes  to a  clause node and then back to somebody  else's horizontal nodes um and  is kind of a hodgepodge of things which  don't make any sense in trying to  reconstruct a satisfying assignment what  you what you really want to have happen  is the hamiltonian path should have  clear zigzags and zag zigs that allow  you to decide  how to set the variables  and so that's the role of these little  nodes here these are spacers  that separate the detours from one  another  and that force  um  a visit to the  clause node to come back  to the same  uh place from which it left  otherwise you would never be able to  visit those uh  spacer notes  you have to look in the book for that uh  but  there is a little bit of a detail that  you have to go through you have to show  it must be zig zags and zig zag zigs  and then you get the corresponding truth  assignment and the most satisfied fee uh  for all paths  okay  let's uh again the reduction is  polynomial time computable i'm not going  to say more about that we're kind of a  little bit low on time  um last check in  would this construction still work if g  was undirected  suppose  i just eliminated all the directions  from the from the edges  uh made them lines instead of arrows  would that now show that the undirected  hamiltonian path problem is np-complete  let me see  what do this  okay what do the c nodes represent here  there's one c node for every clause  so there are k clauses  named one c one to ck  and there are k  c nodes these are the so called clause  gadgets  which are going to force  there to be one true literal  um in every clause  for the satisfying assignment  you know it's uh  you know  uh  you have to look at it or or maybe we  can spend a little bit more time  explaining it but uh that's what that's  what the purpose is  um does that mean we need only  does that mean we need only two  inside nodes  uh do we know that so the  the horizontal nodes do we only need two  of them  you won't be able to reuse the reuse  these nodes for multiple detours for one  thing  you know once you've gone to a detour  you come back to the node next over and  so you better not overlay multiple  detours  and  and  also you need to keep them separated  from each other don't forget and this  node x1 can appear in many many  different clauses  um  so  you would need to have um  possibly many  of these horizontal nodes  um  so some now says 2k inside nodes would  suffice  probably 2k i would say 3k just to be  safe for the spacer nodes um you need to  look at carefully at the argument which  is laid out in the textbook um you may  not actually need the spacer nodes but  then it makes the argument just more  more ugly so the way the construction is  done is you have three k  inside notes um that  okay uh again several questions like  that  the graph would start looking messy if  x9 was in c c1  yeah if x9 down here was in c1 yeah i  would  it'd be messy it's okay my c is allowed  uh  all right i think we're let's end the  polling you all are you all in  all right  share results  yes  the answer is no  uh the construction depends on this  being directed i mean you can see that  all over the place but for one thing  uh  you know the whole point of these  detours is the directions of the of the  um edges and so  without that  the the this construction me is is going  to be just a bunch of  you know not going to mean anything it's  not going to prove anything it's  probably always going to be have a have  a hamiltonian path without the  directions um  so i think we're kind of out of time um  uh  a quick review um these are the topics  we've covered i think we're out of time  so i should let you go  um  but i'll stick around for a few minutes  um  in case any of you have any questions  but i need to run off it for myself for  another meeting so i don't have much  that much time  um  clarify my comment about picking the  wrong problem to tackle p versus np  um  when i use composites as an example  you know if i worked hard to prove that  composites is not in p as a way of  proving that there is some np language  which is not in p  um  that would that would have been a  mistake because composites is in p  that would be just i would would have  been working hard to prove something  which we now know was false  so we don't want to spend time  working on the wrong language  um  you know but the nice thing about np  complete languages is that we have a  guarantee  that if p is different from np that that  language is not in p  are there problems that are not in p  that are in np but are not np complete  oh that's a good question  are there  problems in between  uh p and np complete so np complete is  sort of like the hardest problems in np  and the p problems are obviously the  easy problems that are in np and how  about is everything either np complete  or mp  so for one thing there are problems  that are not known to be in either  category  [Music]  uh so we'll discuss some of those in due  course but one of them is like the graph  isomorphism problem testing two graphs  if they're really just permutations of  one another it's clearly a problem in np  but not known to bnp um so there are  problems that are  not known to be either np complete or in  p so there are problems that might be in  between um  but then there's another theorem out  there which says that if you if you  assume that p is different from np then  you can construct  problems which are in between which are  neither np complete nor in p they're np  problems but they're not empty complete  not in p um so  uh those problems themselves are perhaps  somewhat artificial but they at least  prove the point that it is possible to  have these intermediate problems  um  so somebody's asking isn't factorization  one not no you know we  for the case of factorization  um or you have to make sure to make a  language out of that by the way um  uh  but  um  it's a um  uh  stuff because factorization is a  function so we won't really want to be  talking about languages as kind of as  the homework suggests  np and p are classes of languages but  okay that's a separate separate note  there factorization um  is  uh not no  it could factorization could be in p and  it could be np-complete  you know both of those are not ruled out  so uh you know i think most people would  probably venture to guess that it's it's  a problem that's in the in between state  that's neither p nor mp complete but not  known  um  uh do we  who who first thought of this reduction  from three set to hand path it's so  clever  well it wasn't me uh  i believe that uh reduction from  uh  i think that is due to dick carp  who was one of my professors at berkeley  where i was a graduate student he that  was done before i got there it was  around 1971 um when so there were two  famous papers there's the cook paper  there's also the lemon paper that was in  russian that took a while for people to  discover out here in the west but  the cook paper was 1971 and very quickly  followed after he just showed sat as np  complete  but uh after that uh carp  was  he  he had a paper called reducibility among  combinatorial problems and he had a list  of about 20 problems that he showed were  np complete  by reduction from sad um  include clique include handpath and a  bunch of other things um  so  and that's was also a very famous uh  very famous paper um both of those were  people who often talk about cook carp as  sort of together they  uh they they they really sort of  show the importance of np completeness  and exist and no whole notion of np  completeness  um  yeah 21 problems so  yeah so car proof 21 problems wouldn't  be complete in 1972  um so that was shortly after coke showed  that sat was mp complete  um incidentally i think the terminology  np complete wasn't around uh  uh until a little later  and that might have been might be due to  knuth  i'm not sure i remember he did a big  poll  of people about what should be the right  language to use for the for that term  and i think he came up with it  um  all right i'm gonna head off guys nice  uh seeing you all uh so until uh  thursday oh until tuesday uh bye-bye

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welcome everyone um  back to um theory of computation  so  we are now at lecture number 15 and  if this is an important  lecture um well all of our lectures are  important but  this is i'm going to introduce  one of the major topics  that we're going to see  going on in various forms through the  rest of the semester which is the notion  of a complete problem in this case it's  going to be an np complete problem  and  i think some of you have probably heard  of that concept already maybe you've  seen it in some courses or other but  we're going to do that in a  in a rather careful and formal way  over the next couple of lectures  so we are following up on our  previous discussions about complexity  time complexity  we defined  the time and non-deterministic time  complexity classes as you may remember  the classes p and np  talked about the p versus np problem  looked at some interesting algorithm for  showing problems in p called dynamic  programming  and we started to move toward our  discussion of np completeness with the  introduction of polynomial time  reducibility  which is  related to some of the earlier  reducibility notions that we discussed  in the computability section  so  let's  [Music]  quick review um  what it means for one problem to be  polynomial time reducible to another  this follows our pattern of  reducibility concepts where  um if a problem is reducible to another  problem and that other problem is  solvable then the first problem is  solvable so if a problem is reducible to  an easy problem that problem becomes  easy too  um and  the kind of reduction that we're going  to be looking at here are the mapping  reductions but now where the reductions  are computable in polynomial time  and so we had this um  result we mentioned last time that if a  is polynomial time reducible to b and b  is in p then a is also in p  um  [Music]  so that's going to be um  critically important for the  you know for the  for the whole um  of discussion that's coming  um  our intuition about pnnp um  repeating that here um the np problems  are the ones where you can verify  membership easily  as opposed to  being able to test membership easily  those are the problems that are in p the  verification typically requires  some kind of a  certificate  that establishes kind of a proof  that the input is a member of that  language  um  and the  big question of the field which remains  an unsolved problem  is whether p equals np it's called the p  versus np question  and we don't know the answer so whether  there are problems that are  in np um solvable and nondeterministic  polynomial time typically these are  problems that involve searching whether  they can be solved in polynomial time  typically without the searching um  and  uh if p  were equal to np then you could always  eliminate searching and a p were  different from np then there were cases  where you need to search um and we don't  know the answer to that uh  so  in  the direction of exploring uh this  this question and its ramifications  we introduced this problem called sat  these are the boolean formulas  which have an assignment  that  enable that event makes them evaluate to  true  um  so we call those satisfiable formulas  and uh we um mentioned but have not yet  proven and we will not prove until the  next lecture on tuesday  that there is this theorem  uh kind of a very remarkable theorem  that says that if you have uh you take  the satisfiability problem and if it is  solvable quickly then all of the np  problems are solvable quickly so if sat  is in p then p equals np so in a sense  sat is kind of you know the  uh  it's sort of the super  np problem in the sense that all of the  difficulty of any np problem is  is embedded within sat so if sat becomes  easy then all of the other np problems  become easy um and we'll eventually  prove that but right now we're kind of  setting up  the um terminology to allow us to do  that  um  so uh anyway we'll we'll we'll get there  um  so  the the the key ingredient for proving  this kook 11 theorem is polynomial time  reducibility  what we're going to show  is that every problem in np  can be polynomial time reduced to set  so every np problem can be converted  into a set problem  and so  working toward that because when you  think about it you know there are  infinitely many problems in np and  um  being able to show that all of them  are reducible to sat as in a sense kind  of um you have to prove a kind of a  higher level theorem it's not just a  single reduction we're kind of  exhibiting a schema of reductions which  shows that all of these problems can be  reduced to set  um and kind of  developing  our  um  you know our intuition toward that  direction we're going to look at some  specific polynomial time reductions  today um  and  we'll start out with a reduction between  two problems we have not yet seen one of  them called threesat and the other one  called clique so on this slide we're  going to introduce those two problems  so remembering again about boolean  formulas  we're going to consider a special  class  of boolean formulas a restricted form of  boolean form formulas called conjunctive  normal form  and so this formula here in particular  is in that conjunctive normal form  um so just remember um i was explaining  some of this to someone else uh  earlier this morning showing some of  these slides and it was pointed out that  not all of you may have be familiar with  these um the boolean operations of and  and or  uh so this is the or symbol here  this is the and symbol here  you know they operate you know hopefully  you've seen the just the concept of  boolean and or  or is in a sense a little bit like a  union um and it's like an intersection  and the the symbols here are kind of a  little bit similar to the union and  intersections you know the intersection  the union intersection symbols  themselves like or the v-shaped is a  little bit like a  pointy union symbol and the in the um  upside-down v-shape is a little like a  pointy intersection symbol so maybe  that's if you haven't seen those seen  that connection before maybe it's  interesting to observe that  but anyway um  what makes this  uh formula be in conjunctive normal form  well the conjunctive normal form  formulas  are organized in a certain way they have  um  these groups called clauses within the  parentheses  that are added together that are  connected by these and  and operations um  and  within those groups which are called  clauses  the um  elements are  ored together  those elements  are going to be either variables  or  variables with negation negated  variables um  so uh just to repeat that  uh these well i'll just just state those  those variables or negated variables are  going to be called literals um  and  the  ors of a bunch of literals are going to  be called clauses  okay this is just the standard  terminology in the field  okay so  um a literal is a variable or a negated  variable  and a clause is an or  of literals  okay just a definition  all right so we have a bunch of these  clauses here each of the clauses has an  or of of liter of literals in it  now  a chomsky  conjunctive normal form  uh formula i guess they're both c and  f's but uh  it's conjunctive normal form formula is  one that's written as an and of these  clauses  okay so we take these clauses which  themselves are ors we and them together  we get a formula that's conjunctive  normal form conjunctive stands for and i  think uh conjunction is an and um  and then a three  cnf  is a cnf where each clause has exactly  three literals  so for example this particular  formula is a cnf but it's not a three  cnf because it has at least two clauses  which violate the condition of three  literals per clause this one is okay  obviously  this first clause  and the three sat problem  so this is the first of the two problems  we're going to be talking about  the three sad problem  is  the satisfiability problem restricted to  these three cnf formulas  so this is the collection of  setup of three cnf formulas which are  satisfiable  so you can think of it as a kind of a  special case  of the sat problem where we only care  about formulas of the special kind  um  this being a special case you might  imagine that this might be an easier  problem to solve but in general it turns  out not to be as we will see this is  just as solving this special case is  just as har as hard as solving the  general case for satisfiability  um  now let's turn to the second  of these two languages up in the  headline  uh  the clique problem  um  so for that we gonna turn to graph  theory  so we're going to consider graphs  you know  points and lines connecting them  and we will say that a clique  in a graph  is a collection of nodes collection of  the points that are all pairwise  connected by lines  and a k click is one where you have case  arch nodes  so  here we have a three click four click  and five click  um and the clique problem  is to try to find cliques of a certain  specified size  embedded within a given graph  so you'd like to know whether you're  going to be given a graph and it targets  click-sized k  and you want to know is there a subset  of the nodes in the graph  of size k  that are all connected to one another  that's the cleat problem  um  so obviously  um  the clique problem just as with the  satisfiability problem is a decidable  problem you can just try every possible  subset of k nodes  and see whether it constitutes a clique  but that's in general going to be an  exponential algorithm if you have you  know  you know k is a large value like it  might be half the size of the graph you  might be looking for a very large clique  then you have to try  many many subsets in order to see  whether uh any one of them is a clique  it's also going to be and i hope you get  this intuition this is a problem that's  in np  the cleat problem is an np problem  because you can easily verify  that a graph uh has a k click just by  exhibiting the click  okay um  now uh  uh  so the clique language here is you're  given a graph and a k and you want to  know does the graph contain a cake  league um and what we're going to show  is that these two problems are connected  that the three set problem and the cleat  problem are related in that you can  reduce in polynomial time three sat to  click  okay so this is um  you know  at  you know just standing back at it  back for a minute you know that seems  kind of surprising there's no real  reason  you know uh obviously anyway no obvious  reason why there should be a connection  between three sat and click click they  look very different from one another  but we do give that uh we will give that  reduction  and um it then that implies that if you  can find a way of solving the clique  problem quickly that'll give you a way  of solving the the three set problem  quickly  um and that's going to be the whole  point of this um  so we're going to show this over the  next  slide or two um this this polynomial  time reduction and this is something i'm  going to walk through it slowly  but this is one where you it's really  important to try to get a sense of how  it's working um  uh because this is the kind of thing  that we're going to be doing a lot of  and you're going to be asked to do it  also on the homework and and possibly on  the final exam i hate to use that as the  motivating force here but at least it  might be one motivation for some of you  that you have to understand how to do uh  this kind of a reduction  okay so  you know i i think at a high level  what's going on is we're showing how to  recode  a  boolean formula satisfiability problem  into the problem of testing whether a  graph has a clique  um  so  let's see if we have any questions here  that are coming up in the in the chat um  okay so i this is an interesting  question here can we always convert  a boolean  formula  into conjunctive into conjunctive normal  form  first of all yes the answer is you can  always convert a boolean formula into an  equivalent one in ins into in cnf  but  in general that might make the formula  exponentially larger  [Music]  so  just the mere fact that you can convert  formulas into into conjunctive normal  form doesn't mean  that solving um  that you know that solving uh  um  conjunctive normal form uh formulas for  testing satisfiability of the of the cnf  formulas is going to be as hard as  testing the general case because just  the conversion might be exponential um  there's something more uh more  a little bit more complicated going on  um  than that  so let's uh  um  let me just see here  uh if he is a satisfiable formula which  is not in cnf can be  so a similar  um you know similar question so the  questions that i'm getting are about  converting formulas uh to cnf um  so yeah you can do it but not in  polynomial time in general because the  resulting formula you get might be much  larger  um if you if you're looking for an  equivalent formula if you're not looking  for an equivalent formula then you know  of course then depending upon what  you're looking for you might be able to  find something smaller  um  ah  this is a this is a i guess a good basic  question  why is clique in np  um  because isn't doesn't verifying that you  have a clique require going through all  the possible cliques  once i  you have to understand what verifying  means verifying means you can verify  something if you're given a certificate  in the case of a clique  the click problem the certificate is the  clique  um so once you have the certificate you  can do the verification in polynomial  time finding the certificate of course  might be difficult  so you only think of np in the context  of having that certificate  so in a case for like  compositeness the certificate might be  the factor  um  you know  anyway i hope that's you know it's there  there are problems sometimes with what  the certificate is is not necessarily  obvious um but there can be a  certificate um for  um showing that the  that the input is in the language but in  the ones that we've done so far maybe  you can argue that this certificate is  sort of an obvious thing but it's not  always an obvious thing  um  okay  why don't we move on then  um  so let's see how do we do  this polynomial time reduction from  three sat to click  okay  here we go  um  so i'm just going to give a reduction  that's that's that's what the definition  means i'm going to give a way of  converting  formulas  to  pairs  a g and a k  where the formula is going to be solid  satisfiable if and only if the graph has  a k click  okay so let's kind of a little bit do it  by example  and  in order to do that  so here's going to be a formula now it's  in three cnf that's what i need you know  in order to be doing this reduction i'm  converting three cnf formulas into  clique problems  we have to a little bit understand what  it means  when we say  uh we talk about the satisfiability of a  formula like this because it's going to  be helpful in doing the reduction you  know obviously it's satisfiable means  that the form you can find an assignment  of the variables so you're going to set  each of these variables a b and c and so  on to true or false  and you want to make the whole formula  evaluate to true but what does that  actually mean  [Music]  it means that because of the structure  of the formula  that making this formula true  corresponds to making each clause true  because  the clauses are all added together so  the only way for the formula to be true  is to make each clause true  and to make a clause true  you have to make at least one of the  literals true  so it's another way of thinking about  uh satisfying this formula  you need satisfying these and a  satisfying assignment  makes at least one true literal in every  clause  it's really important to think about it  that way  because that's what's going to be um  [Music]  kind of the basis  for doing  this reduction and all of the reductions  it's kind of what makes three set easy  to think about in terms of its  satisfiability  if you had a sort of a general um  satisfiability problem and you had a  satisfiable formula there's no sort of  obvious way of sort of seeing what the  satisfying assignment looks like but  here we understand what it looks like it  has that very special form  making one true literal in every clause  at least one true literal in every  clause  okay so now we're going to do the  reduction so i'm going to take from this  formula you and  you have to understand you know i know  for some of you you know you're going to  be you know chafing why am i going so  slowly but i want to make sure that  we're all together in understanding what  the rules of the game are and what we're  trying to do  um  we're trying to convert  this formula into a graph and a number  so right now my job is to do this  reduction is to exhibit that graph  okay so i'm going to do that in kind of  two steps i'll first i'm going to tell  you what the nodes of the graph are  then then i'll tell you what the edges  of that graph  finally i'll tell you what the number k  is  that's the way this polynomial time  reduction is going to work  and we have to also  observe at the very end  that the reduction that i'm given giving  you this  procedure for building this graph can be  done in polynomial time but that i think  you'll see once i'm done that that's  pretty obvious  okay so first as i promised the nodes um  so the nodes of this graph are going to  correspond to the literals of the  formula every literal is going to become  a node in the graph and it's going to be  labeled  with  the um name of that literal so it's  every node is going to be labeled an a  or b or or c bar  and so on so here it goes  okay so those are the nodes  of the graph g  one for each literal in the formula  okay  labeled as promised  now i have to tell you what the edges  look like  the edges are going to be  um i'm going to tell you what the edges  are by first telling you what they are  not  i'm going to first explain to you what  the forbidden edges are  what what edges i'm going to  promise not to include  and then the ones that i  will include are going to be all the  others  okay so what are going to be the  forbidden edges first of all um  the the forbidden  edges are going to be of two types one  is the edges between nodes  that come from  literals in the same clause so i would  just call that  no edges within a clause  okay so for example these three nodes  will not be connected to one another and  i'm going to indicate that by writing  red dashed lines which means there's not  an edge there  okay those are those are forbidden from  having an edge  okay so these three have no edge and the  same thing for every other uh triple of  three nodes that come from uh clauses  okay those are  no edges there  and there's one other category of edge  that i'm forbidding  and that are edges that can  that um go between  inconsistent labels and the nodes with  inconsistent labels so for example  between a and a bar those are  inconsistent  you know nothing's wrong with a going a  to d those are not inconsistent those  are just different labels or going from  a to a that's okay  but from a to a bar that's that's not  allowed um so that's going to be another  forbidden edge  or for example a to a bar here  for b a bar to a or for example from  this c bar to c  forbidden  okay so imagine you're going to write  down all of those forbidden edges and  then once you've done that  uh  those are all of the the forbidden edges  and then after  [Music]  taking those away  you're going to be putting in all the  other edges possible  so for example as a let me just uh  gray those out so they don't interfere  with the picture  b from a to a to d  um there's going to be an edge um  because those are not forbid you know a  to a not forbidden because  they they're in different  um  they are in different uh  clauses and they're not inconsistent  they're they're consistent with one  another so here are a whole bunch of  other i'm not showing them all becomes  very messy but uh here are all the other  edges between nodes uh which are  where there isn't there's  no pro  within they're not forbidden  okay and that's  that's the graph  that's g  g is all  those nodes and those edges um which are  not forbidden  now i just have to tell you what k a is  k is going to be the number of clauses  that's going to be the size of the  clique i'm looking for in this graph g  that i just spelled out for you  and i'm going to claim  that  this graph here that i just described  will have a k clique  k is the number of clauses  exactly when  fee was satisfiable  it's kind of kind of cool you know um if  he is satisfiable then there will be a  cake leak here  and if he was not satisfiable okay clean  so we have to we're gonna prove that as  a claim on the next slide  okay so k is the number of clauses  all right  um  any questions on that on that  construction so i've done with the  construction what's left is to argue why  the construction works  so far so good let's move on  okay  um  so um  so here is that very same construction i  eliminated the red uh  forbidden edges these are the ones that  were remaining  plus you know anything else that was not  forbidden  that was the same formula that i had  from the previous slide  now i'm going to claim that that formula  is satisfiable exactly when g has a k  click now why in the world is that  so this is an if and only if it's proved  in both directions  and this is going to be the typical kind  of thing that you're going to want to do  when you're exhibiting a  uh  you know one of these reductions  which is you're going to have an  opportunity to do that um and we'll do  do that too  in our examples  uh  okay  so now  uh what i'm what i want to show  is that  uh  if fee is satisfiable so let's prove the  forward direction if p is satisfiable  then g has a k click  okay so first of all if fee is  satisfiable that means it has a  satisfying assignment  okay  now here's a common confusion i'm not  sure whether it's helpful to  sprinkle the confusions in along with  the discussion but  in case you're worried  you might ask well how do i find that  satisfying assignment i thought that was  you know  exponential  exponentially hard to do  this is a proof this is not an algorithm  okay we're just trying to argue the con  correctness of this construction  so there's no longer any concern about  how to find an assignment i'm just  saying if there isn't a sign if there is  a satisfying assignment then something  will happen something that then then um  something will happen in particular then  we will show that g has a cake league  so  let's say  fee is satisfiable  so it has some assignment let's take  that satisfying assignment  and remember  that in any satisfying assignment  uh to a formula in cnf  that makes at least one literal true in  every clause  so  let's just pick one of those there might  be several clauses which have multiple  true literals in that case just pick one  of them  arbitrarily  okay so i don't know what the i don't  have this sort of indicated on this  diagram here but imagine you know maybe  in the very first clause a was true in  the second clause b was true in the  third clause  e compliment ebar was true and you know  not not  you know  e bar was true which means e itself was  false but e bar was true and that's the  way each of those clauses in turn uh got  to be true you know in this particular  satisfying assignment  because you're going to pick one true  literal in every clause and now  from  that choice  of literals one per clause  i'm going to look at the corresponding  nodes in g  and i'm going to claim  that those nodes taken together  form a k click  okay so first of all  do they have the right number of nodes  well sure because  i'm picking one node per click i already  said k is the number of cliques so i'm  getting exactly ka nodes so i have at  least the right number  of nodes but how do i know they're all  connected to one another those nodes uh  that i just picked  well they're all connected to each other  because i'm going to say well because  there were no forbidden edges among them  and remember that we put in all possible  edges except for the forbidden ones  so how do i know there are no forbidden  edges  well what were the rules for being  forbidden means they two nodes in the  same clique in the same clause  well i'm picking one node per clause  so i can never be having two nodes from  the same clause so i'm never going to  run into trouble with uh the first  condition of being forbidden so what's  the second possibility for being  forbidden is that i'm picking two nodes  which are inconsistent  well how do i know that i didn't end up  with two nodes with inconsistent labels  that would be bad because then they  would have a forbidden edge and what my  result would not be a clique  how do i know i didn't end up picking  um  uh  in this group i pick a and in this group  i pick a a bar  well because they all came from the same  assignment  those are all of the true letters they  were all true literals  in clauses it's not possible that a was  true in this clause and a bar is true in  that clause because if a is true here a  bar has got to be false it can't be the  true literal in this clause  so i cannot have any inconsistent any  inconsistent nodes  appearing  among the nodes of my clique  and so they're not in the same clause  they're not inconsistent so the edge has  to be there and that's going to be true  for every pair of nodes in that clique  in that in that group of nodes and  that's why it's a click  um  so that proves one direction now we stop  to prove the other direction because we  have to say well if g has a k clique how  do we know that ph is satisfied so  that's the reverse  so let's just take any k clique that's  in g  and how do i know i can get from that a  satisfying assignment to the formula  um  uh  good um  getting some good questions here in the  chat but let me just move on  um take any uh so  proving the reverse  this the reverse direction but we're  assuming we have a cake leak to take any  such k-click  now first of all you observe that it's  got to have  one node in every clause  it can't have two nodes in the same  clause or zero nodes in in a clause  first of all it can't have two nodes in  the same clause why well because those  nodes are never connected to one another  so that they cannot be both in the same  clique because all the nodes in a clique  are connected to one another  um by construction by the way we  construct a g  nodes from the same clause are not  connected so they cannot appear in a  clique together  so um  there's going to be at most one node  from every clause appearing in this  clique but and but but  appearing in in this clip but we also  have we know we have k notes  so that means if there is at most one  per um per clause  and we only have k clauses that means  every clause has got to have one  you know if some clause is missing or  note then there's going to be some other  clause that has two  so we know that every um  that this clique that would that g  has has exactly one node in every clause  so how do we get from that a satisfying  assignment  so what we're going to do is  take the current  corresponding literals  that correspond to that one node in the  that was in the uh clique  take those literals and set those  literals to be true  which means setting the variable to be  true  um or setting um you know  if the literal was a bar then setting a  to be false because you want a bar to be  true you want to set the literals to be  true  well now we're setting um there's one  note in each clause we're setting one  literal uh the corresponding literal  true so that's going to set one true  literal in every clause  so that means it's going to be  satisfying there's one thing that one  has to really to be  careful here to double check to make  sure that we're doing this consistently  that we're not being asked to set you  know a  true and also a bar true because then  that would not be  possible  but a and a bar are not connected  so it's we're never going to be trying  to set both a and a bar true if so we're  going to be trying to set a true we're  never going to try to set a bar true  because those are not those a and a bar  cannot be in the same clique they're not  connected to each other  okay so  that  is the argument um and lastly  i claim we're not going to look at this  in detail that it's kind of obvious that  this reduction can be done in polynomial  time namely if i give you that formula  you can write out that graph in  pretty easily  there's no hard work to be done in  writing out that that graph or counting  up the number of clauses  so that's that's the proof  that i can reduce three statically  um  and it tells you also that if i could  solve click quickly  then i can stop threesat quickly  and this is the whole point  because i can convert now clique  problems to three sat problems  no  no i couldn't i'm sorry i got backwards  convert three set problems the clique  problem so if i can solve quick click  easily then i can solve three set easily  right i just showed a way of converting  these formulas to graphs so if that's so  that says if i can solve the graphs  easily then i can solve the formulas  okay  why don't we uh turn to some oh question  checking but you know we can also  um why don't i i'll launch a check-in  but we'll  we  let's just see  i'm seeing many many questions here  which are actually what my check-in is  about  i'll turn it back to you guys uh  okay so  where did we use the fact  um that we have  three  uh literals per clause what does this  thing just work even if we had any  number of literals per clause  um  what do you think  we're having a  we've got a tight race here  hmm uh  truth is losing out unfortunately  all right oh it's really close but still  okay  uh  one more okay it's neck and neck  okay almost done  10  seconds  are we done here okay that's it  uh ending polling  uh  yes  truth is  thank god  um  yeah it works for any size clause we  didn't use the fact that it's a three c  and f it could have been any number here  uh  so it's well we've got a i guess it's a  plural plurality here though that not a  majority um  no we didn't where did we use three in  any of these argument i didn't mention  it maybe you were imagining that that is  going to be part of it but three does  not come into this at this discussion at  all if you think about how it's why it's  working  um even if we had you know one of these  clauses had 10 literals in it  as long as k is the number of clauses  and we don't connect  any variables any literals internal to a  clause  this whole argument is going to still  work so please check that make sure you  understand what's going on because i can  see that a good chunk of you  have not got this right  um  uh so i got a good question here uh what  if it's only just one big clause so it's  like the whole formula is just one big  war so what does happen in that case  good question so in that case um  uh  so the you know suppose has one big  clause with a hundred literals  that's the whole formula it's just a big  war so we know the formula is going to  be satisfiable by the way obviously um  so if you look at the corresponding g  it's going to have 100 nodes because  there's one for every literal um  none of them are going to be connected  to each other because they're all in the  same clause  um so it looks like we're going to be in  tough shape trying to find a clique  there because there's no edges at all  everything's forbidden but what is k  k is going to be 1 in that case because  there's only one clause  and  uh  kind of a degenerate case but a clique  with just one  you know just a single node  is it counts as a one clique  because it's just one node there's no  need for any edges at all it still  counts as a clique so it'll still it'll  still work out um  uh let's see what else here uh that was  kind of a fun question that you asked me  thank you um  uh there are a lot of questions um  i think we're actually pretty near the  break also we'll just see  yeah um  so why don't i um  oh  many many questions here uh  so i i think i'm gonna start start the  clock going down for our um our coffee  break and um  and i will take some of these questions  i'll try to answer some of these  questions oops  um try to answer some of these questions  um  afterward  uh all right  good  all right  um  so i got a question about  making sure why nodes don't have  inconsistent labels if i understand the  question correctly  um so these would be nodes  um  so i never put edges between nodes with  inconsistent labels that's the rule  that's my construction i get to i get to  say how the construction works so those  two nodes with inconsistent labels can  never be in the same clique  because  there's no edge between them  so i'm not sure that answered the  question there but that was uh  um  let's see  so getting another question here  since any boolean formula can be  converted to cnf  does that mean um is sat polynomial time  reducible to clique as well  monomial time reducible to clique but  not for the reason that any boolean  formula can be converted to c and f  because  that conversion  um  is going to be an exponential conversion  in general the conversion to c and f so  you have to be careful about what we  mean by converting a formula to cnf  um  uh so also getting questions what  happens in the case of one set well we  didn't really talk about one set  uh suppose or i guess so the question is  suppose there's only a single  literal per clause  that's an interesting another  sort of edge case here what happens  under those circumstances  so um there's only a single literal per  clause um  uh so you have to work out what happens  but in that case  um  uh  all the clauses just have one literal in  them  and now they have um  uh  um  the only way  uh  this  if you think about what is a when you  have a  one cnf so there's only um  one  literal per clause the only way that can  be satisfiable is you have to make each  literal true because there's no warring  anymore so every literal has to be true  in the assignment so that means you can  never have inconsistent literals um and  that's the only case when you would not  have an edge because you're not the  forbidden is the forbidden condition  won't happen because you don't have more  than one uh node per per clause every  clause is going to have just a single  node in it uh so  it really comes down to whether or not  you have an inconsistently labeled um  inconsistent clauses  i i didn't explain that super well but  it's the  the argument still works there you  should check it for yourself  um  so this is a basic question how do we  see that f is polynomial time  i i'm not sure i want to expend a lot of  you know a lot of discussion on that  because  the the conversion from the formula  to the graph  is kind of a one-to-one  you know every no every uh literal  becomes a node  um the rule for when edges are present  it's a very simple rule so i guess i  would argue  i'm not sure what what more to say it  seems pretty clear that the  conversion that reduction is polynomial  time computable i mean if you wrote a  program to do that it would operate very  quickly  um  is it possible for g to have a k plus n  clique where n is greater than zero and  does that matter  now if you think about it  the biggest possible clique that this  graph g could have is k  it could never have more than  uh uh you know a bigger clique because  two nodes in the same uh clause are  never connected  you only have k clauses  so the answer is no you cannot have a  bigger  a bigger than a k click it's just not  going to happen  does each clause need to have the same  number of literals i don't see why  um  uh  so question is why are we worrying about  threesat um  since it didn't seem to matter here  there are other examples where it does  matter i i i'm actually not sure if  we'll see one or not but um there there  are cases where it does matter  um let me just see if there's any quick  questions here  uh  somebody's saying doesn't does  do we have to worry about there being a  polynomial number of literals  so you have to think about what you what  that question means the size of the  input which includes all of the literals  is n  so there's going to be at most n  literals  um because that's the size of the input  uh the i mean  at most uh and literals appearing so  you don't have to worry about that being  polynomial because by the way we're  we're defining the size of the input is  going to be at most n literals  i hope that's clear  uh  okay  when we talk about power  okay so this is a good question when we  talk about polynomial time  it's polynomial in the representation of  of the input which is if you want to  think about it in terms of bits that's  fine or whatever it's not going to  matter if you use some larger alphabet  but the number of symbols you need you  know in your fixed size alphabet the  number of symbols you need to write down  that input in whatever your reasonable  encoding is is going to be n and it's  going to be polynomial in n polynomial  in that  length of the input  um  so another question is does sat reduce  the three set yes we that we will see um  sat actually does polynomial time reduce  the three set not by converting to an  equivalent formula but by some more uh  somewhat more involved argument uh than  that okay why don't we move on some of  these are going to come out anyway in  the lecture  okay so now let's talk about mp  completeness because we've kind of set  things up we're not going to prove that  you know the basic theorem the cook 11  theorem about mp completeness but at  least we'll be able to make the  definition  okay here's our definition you know what  it means for a problem to be in be  complete  um  uh  so a language  b is called np complete  if it has two properties  number one is that has to be a member of  np  and number two  every language in np has the polynomial  time reduced to that  to that language to that np complete  language in order for it to be np  complete  okay so  simple picture  has to be in nnp  and everything in np reduces to it  um  and so that's kind of the magical  property that we claimed that that sat  has sat for for one thing is obviously  in np  and  as we  the cook 11 theorem shows  or will show  everything in npe is reducible to sat so  that's going to be our first example of  an np complete problem  um  uh so now  uh and we're going to get what we  claimed also for sat that if sat or any  other np-complete problem  turns out to be solvable in polynomial  time  then every np problem is solvable in  polynomial time and and that's immediate  because everything is reducible in  polynomial time to the np complete  problem  so if you can do it easily you can do  everything easily just by going through  the reduction  okay  so the quick level theorem as i  mentioned is that sad is np-complete  and we're going to actually prove it  next lecture but let's assume for the  remainder of this lecture that we know  it to be true because so i'll use the  terminology of problems being np  complete uh assuming that we know um  that we have uh sat as empty complete  okay  so we're gonna we're gonna be  using some of the things that were  proving next lecture uh just  in the terminology that we're going to  be talking today  okay so here's the picture  here's the class np  and everything in np  is  polynomial time reducible to set  sat itself is a member of np but i  didn't want to show it that way because  it makes the picture kind of hard to uh  hard to uh  hard to display  so  just from the perspective of the  reduction everything in np is polynomial  time reducible to sad we'll show that  next lecture  another thing that we'll show next  lecture  is that sat in turn is polynomial time  reducible to three sat so  three stat as you remember are just  those problems that are in conjunction  three  are in three c and f  um  and then  what we showed today is that three set  is polynomial time reducible to clique  so now  taking the assumption that sat is np  complete  so everything is polynomial time  reducible to sat which is in turn  polynomial time reducible to three sat  in in turn reducible to click  these reductions as we've seen before  composed  um  you can just reply apply one reduction  function after the next  if each one each one individually is  polynomial the whole thing as a  combination is going to be polynomial so  now we know that three sat is going to  be also np complete because  we can reduce anything in np to stat and  then to three set and then we get a  reduction directly to three sat by  composing those two reductions and then  furthermore at the clique so now we  uh have several np complete problems and  moving beyond that  we have the hand path problem which we  are going to talk about next and we'll  show another  reduction um  in addition to the one we just showed to  click now one going from three set to  hand  um  okay uh so in general  i think the the the the take away  message is that to show some language as  np complete you want to show that three  sat is polynomial time reducible to it  um  uh  okay  some good questions coming in i'll try  to answer those so you're going to  retake  3sat and reduced to c that's the sort of  the most typical case  or you might have there's going to be  other examples too we might start with  another problem that you've already  shown to be np complete and read and  reduce it to your language so it doesn't  have to start with reset  though  often it does  um  okay why do we why is this concept  important  so i would say there are two reasons um  and this is going to get a little bit at  the uh some of the chat questions um  first of all  if you're faced with some copy new  computational problem you've got some  robotics problem  um that you want to solve in your thesis  and you need to some algorithm about  whether the robot arm can do such and  move in a certain such a way and involve  searching you know possibly searching  through a space of different kinds of  motions and you want to know you know i  need a either i'd like to have a i like  to find a polynomial algorithm to solve  that problem  i'm using this as an example because  this actually did happen to one of the  former students from this class who was  working in robotics and they actually  end up proving that the problem that  they were trying to solve is np complete  um  so  um  uh  that's useful information because even  though knowing a problem is np complete  doesn't guarantee that it's not in p  because conceivably p equals np  what it does tell you that if you show  if you have a problem that's np complete  and it does turn out to be in p  then all then p equals np so the tremend  would be tremendous surprising  consequences  of  of your problem if it was known to be  np-complete ending up in p so generally  people take a proof of np completeness  as powerful evidence that the problem is  not in p even though it's not quite a  proof  it's powerful evidence that it's not in  p and so you might as well give up  working on trying to find a polynomial  algorithm for it because if you do  you know you don't have to worry about  robotics anymore you're going to you  know become famous as a complexity  theory person um  so i wouldn't worry about so if you show  problems np complete you can pretty much  assume it's almost certainly not in p um  now there's another reason related to  the to the for the theorists uh to be  care about mp completeness and that is  if you're trying to prove p is different  from np or p equals np as one of the  chat questions is raising um  you know the problem you know  in order to prove p different from np  you know the most likely approach is you  pick some p problem in np and show it's  not in p  right that's what it would mean for them  to be different you're going to pick  some np problem and show it's not not a  p problem  well  um  one thing would be terrible  is possibly p is different from np and  you pick the wrong problem  like suppose i'd spent all my time  trying to show  you know back  20 years ago i'm working really hard  trying to show composites  is not in pain  which is would have been perfectly  reasonable to do  because composites is an np and was not  known to be in p 20 years ago  um and then i invested tons of effort to  try to prove because you know i like  number theory or god knows what to prove  that composites is not in p  and then it turns out composites wasn't  in p it was the wrong problem to pick  even though p might be different from np  but what np completeness guarantees is  that if you work on a problem which is  np-complete you could you can't pick the  wrong problem because if any problem is  in np and not in p an np complete  problem is going to be an example of  that  right because if the np complete problem  is in p everything in np is in p  so if np is different from p you know  the complete problems are not in p so  you might as well work on one of those  so those are two  ways in which np completeness and has  turned out to be is an important concept  okay so here's a check-in um  uh  again this  you guys are kind of getting maybe  you're getting  started to think the way i think you get  is some at least one of you would ask  this question in the chat  um  which language that we've probably seen  is most analogous to sat  okay atm etm  or zero to the k1 to the k  uh  obviously this is maybe subjective you  may have your own interpretation of what  that means  this in a sense perhaps no right answer  but what do you think  uh  okay that's it  um i don't know i don't know how in the  world that you could see this problem is  analogous to to zero to the k1 to the k  but okay um  i'm sure you have your reason  um  yes this is a lot like eight  atm why  well because for one thing we showed in  a homework problem  that all  um touring recognizable languages are  reducible to atm mapping reducible to  atm so that's a little bit like the  notion of completeness  that we have for satisfiability  um because all np problems are going to  be reducible to sad  um  and the other thing too is that once we  start once we  uh we started we want to show other  problems are undecidable we reduced atm  to them and that's also very similar  we're going to be reducing sat or  threesat  so it's indirectly from sat to other  problems in order to show that they're  np complete so that they're hard they're  they're that they're hard in a sense  and so uh in kind of both those ways  atm and sat are kind of playing similar  roles one key difference however  between atm and sat is in the for atm we  can prove  that it's undecidable  but for sat we don't know how to prove  it's outside of p  that would those would be the analogous  situations  um  and  uh so  the the story for sat which is kind of  easily solved by a diagonalization  argument for atm  there's reasons to believe that we will  see later that diagonalization is not  going to work to prove sat  outside of p  and  besides that we don't really have any  good methods  um  so  uh anyway  uh let's move on  why is etm less analogous  uh  i know because atm was the first problem  we showed undecidable and sad is kind of  the first problem that we're going to be  showing np complete  i guess that would be my answer  um  okay let's continue so let's show now  that  handpath is np-complete assuming that we  know sat  or threesat is np complete  okay so we're going to give a reduction  from 3sat to handpath that that  that that's what that's what this is  about it's just like what we did for  clique but now for him path um  and and this is going to be very typical  in in these reductions typically what  happens  is that you're trying to simulate  a formula  use you know a boolean formula for  satisfiability  from the satisfiability perspective  you're trying to simulate that formula  with some sort of structures  inside the target language which would  be hand path  and so what the lingo that  people use is that you're going to build  gadgets to simulate the structures in  the formula namely  the variables the literals and the and  the clauses  okay these are going to be substructures  of  the um  either you know the graph in this case  that you're that you're building um  we'll see what that means so let's take  a formula here  and  uh let's again try to imagine how we  would reduce that  to the hand path problem so the  reduction would produce  a graph no it would produce a hand path  instance so a g s and t want to know is  there a hamiltonian path from s to t in  the graph  and this is going to be not the whole  graph but this is going to be a  substructure in that graph the next  slide is going to have the global  structure of the graph graph  but here this is going to be a key  element and it's we're going to call  that  the variable gadget  okay what does it look like i don't know  if you can see it on your um  clearly enough but these  uh  edges so there are four outside notes  here the edges connecting them are all  kind of pointed downward  and then there are these horizontal  nodes here  and there are edges connecting them both  left to right and right to left  okay there's a row of these horizontal  nodes okay so you get the picture of  what this looks like  um  you have to look carefully to see the  arrowheads i  maybe should have made those a little  bigger  okay so now um  uh  whoops  let me let me just see so now if we're  trying to get from this node to that  node imagine now you're trying to build  a hamiltonian path  because this is going to be a part  of that graph g  that i'm constructing  now remember for them there to be a  hamiltonian graph  that means you have to go through every  node  in the graph so if i want to get from s  to t  the only way i'm going to be able to to  go to hit these  horizontal nodes here  is  um  by picking them up  uh  as i go from s to t  so the only possibility is if you think  about it is it would be for the  sequence to go like this  if you could if that comes through for  you so the the path would go from s to  this node and then through  these hops of  along these horizontal nodes and then  down to the bottom node here  okay so that's one way  that you can get from here down to there  and pick up all the other nodes along  the way which is they're not going to  have any other possibility possible ways  of getting to them  so i'm going to call that a zigzag  okay but there's another way to get from  s to the bottom node  okay which is going to be by doing sort  of the  dual going to the right then going out  to the left and then down to the bottom  i'm going to call that  a zigzig  okay  and uh with a little kind of little  diagram here just to summarize what it  means  and this is the this is the classic  thing for a variable gadget  because it's a structure  that when you're trying to think about  how the object that you're  asking whether it exists or not the  hamiltonian path how it relates to that  object it's going to have two  possibilities which are going to  correspond to the variable variable  being set true or false in the formula  so we're showing how to  set  you know the simulate the variable  in this hand path instance  so setting the variable is going to  correspond to  to to  to constructing the path  okay  now we also have to make sure not only  that the variable gets set to true or  false but that it gets a set to true and  false in a way that makes this a  satisfying assignment namely that we get  one true literal in every clause  so i'm going to add another gadget here  called a clause gadget  which is just a single node  and visiting that node here  is going to correspond to  um  satisfying that clause to having a true  literal in that clause  um  and now how am i going to get i'm going  to  have it enable  a detour  from these horizontal nodes out to this  clause gadget  so here it is here's that detour  as i'm going from left to right  i can instead of jump doing a single  jump here i could branch out and visit  that clause node  and then come back and pick up my left  to right  uh path as as i was doing before  you realize what i'm trying what's  i hope you're seeing the big picture  because this has to be a hamiltonian  path it has to hit every node that's one  of the requirements so i've got a this  is going to be one of the nodes in my  graph it's the path has got to hit that  node the only way it's going to be able  to hit that node  is by taking a detour off of this  uh horizontal path here  but notice and this is the this is the  key  if i'm doing a zigzag  then i can make the detour and visit  that clause gadget that clause node  but if i'm doing a zag zig  the way this detour is structured does  not allow me  to visit that node  because if i'm going from right to left  here by the time i get to  to this outgoing edge  now i come want to come back  that note has already been taken  it only works if i'm going zag zig from  going from right to left  the left to right if i'm going from left  to right then i can do it for going  right to left no  so this is going to correspond if you  think of left the left to right  direction as true that's going to  correspond to that variable appearing in  the positive way in that clause um but  not in the negative way the negative way  i would reverse  the in and out  of the dot of the detour  flip that around  so now i could only do the detail detour  when i'm going right to left instead of  left to right i hope you get the picture  this is how the structure is working now  what's only left me to do is put it all  together  but  this slide contains the guts um of  what's happening  um so you wanna is is there any question  that i can answer for you  um on this  let's move on to the next slide and then  as questions come up you can be typing  them in we can maybe answer it there  okay so here is the big picture  so imagine we have  that  formula  that we started we're reducing that  formula  um uh  and um  uh  and let's say it has m variables and k  clauses  i'm going to call them clause c1 c2 up  to ck here the m variables appearing  either  positively or  negated in that formula  and this is the way the global structure  of g is going to look like now i'm  getting a question here what do those  horizontal nodes what role do they play  there those nodes are there to allow me  to visit  those  clause  nodes  the the nodes which represent the clause  gadgets which i'm going to place over  here  so imag so here here is that  um here is i mean this is almost a hole  of g that i'm just missing uh some a few  edges but these are all the nodes of g  so remember i'm trying to find out is  there a hamiltonian path from s to t  now if i didn't have to worry about the  these c nodes  the answer would be just yes  in fact there would be many hamiltonian  pairs from s to t because i can do a  zigzag or zx zig through each of these  uh variable gadgets and i would take me  from s to t and i'd be good  it's just these c nodes these nodes  these ci nodes  you know i have to hit them too  so  they're going to be  um  [Music]  visiting them is going to be enabled by  detours from here so  let me just try to show you what that  looks like so i'm going to magnify  little pieces here  uh from from these gadgets here and show  you how these guys are connected up  so here is the x1 gadget  so x1 appears positively in c1  right here's x1 and c1 and so that means  when i'm going left to right i'm going  to be a possibility of visiting c1  um  but now let's look what happens with um  uh  uh  which was the next one i met i i had  here  uh  okay um  so  uh  right  so the next one is oh yeah so that's so  x1 appears positively in c1 and that's  why i have the connection like this uh  now  x1 appears negated in c2  so i only want to do enable the detour  to visit c2 when i'm going right to left  as opposed to left to right  so  this  uh  set of  horizontal nodes is only going to allow  me to take a detour either out to c1  or out to c2 but not to both  because the hamiltonian path is either  going to go left to right or right to  left it can't do both  when it's going through x the x one uh  gadget  okay you need to  didn't kind of you have to i'll try to  help you through it but you have to try  to think about why it's working  so what let's think again together  so x2  also appears in c1  but now it appears negated so i'm going  to have edges from this x2  gadget  whoops  x the x2 gadget  but now look at the look at how i've  i've arranged that detour you know i can  leave on the left leftward side and  return on the rightward side  um which means  i can only  do that detour when i'm going right to  left  okay um  and that's because x2 is negated in c1  okay so there's  maybe it's a lot here if you're if  you're not quite getting it but  the point is let's say try to quickly  prove so that's the whole construction  you just do that for every single  appearance of a literal  in a clause you're going to add these  detours which allow you possibly to go  visit  the uh the clause  um  so the forward direction is  um  why is this true so you take any  satisfying assignment as i suggested  you make the corresponding zigzags or  zag zigs through the variable gadgets  from s to t and then take the detours to  visit the cloud the clause nodes  um  the reverse direction actually is  slightly trickier we're not gonna have  time to go through the subtlety of it  um  but uh  what you want to make sure here  is that you don't have a kind of a a  weird path  um  occurring because i'm going to start  with a hamiltonian path now and build an  assignment  uh and what we want to make sure that  the the path that i'm constructing  doesn't  go from one from this  horizontal nodes  to a  clause node and then back to somebody  else's horizontal nodes um and  is kind of a hodgepodge of things which  don't make any sense in trying to  reconstruct a satisfying assignment what  you what you really want to have happen  is the hamiltonian path should have  clear zigzags and zag zigs that allow  you to decide  how to set the variables  and so that's the role of these little  nodes here these are spacers  that separate the detours from one  another  and that force  um  a visit to the  clause node to come back  to the same  uh place from which it left  otherwise you would never be able to  visit those uh  spacer notes  you have to look in the book for that uh  but  there is a little bit of a detail that  you have to go through you have to show  it must be zig zags and zig zag zigs  and then you get the corresponding truth  assignment and the most satisfied fee uh  for all paths  okay  let's uh again the reduction is  polynomial time computable i'm not going  to say more about that we're kind of a  little bit low on time  um last check in  would this construction still work if g  was undirected  suppose  i just eliminated all the directions  from the from the edges  uh made them lines instead of arrows  would that now show that the undirected  hamiltonian path problem is np-complete  let me see  what do this  okay what do the c nodes represent here  there's one c node for every clause  so there are k clauses  named one c one to ck  and there are k  c nodes these are the so called clause  gadgets  which are going to force  there to be one true literal  um in every clause  for the satisfying assignment  you know it's uh  you know  uh  you have to look at it or or maybe we  can spend a little bit more time  explaining it but uh that's what that's  what the purpose is  um does that mean we need only  does that mean we need only two  inside nodes  uh do we know that so the  the horizontal nodes do we only need two  of them  you won't be able to reuse the reuse  these nodes for multiple detours for one  thing  you know once you've gone to a detour  you come back to the node next over and  so you better not overlay multiple  detours  and  and  also you need to keep them separated  from each other don't forget and this  node x1 can appear in many many  different clauses  um  so  you would need to have um  possibly many  of these horizontal nodes  um  so some now says 2k inside nodes would  suffice  probably 2k i would say 3k just to be  safe for the spacer nodes um you need to  look at carefully at the argument which  is laid out in the textbook um you may  not actually need the spacer nodes but  then it makes the argument just more  more ugly so the way the construction is  done is you have three k  inside notes um that  okay uh again several questions like  that  the graph would start looking messy if  x9 was in c c1  yeah if x9 down here was in c1 yeah i  would  it'd be messy it's okay my c is allowed  uh  all right i think we're let's end the  polling you all are you all in  all right  share results  yes  the answer is no  uh the construction depends on this  being directed i mean you can see that  all over the place but for one thing  uh  you know the whole point of these  detours is the directions of the of the  um edges and so  without that  the the this construction me is is going  to be just a bunch of  you know not going to mean anything it's  not going to prove anything it's  probably always going to be have a have  a hamiltonian path without the  directions um  so i think we're kind of out of time um  uh  a quick review um these are the topics  we've covered i think we're out of time  so i should let you go  um  but i'll stick around for a few minutes  um  in case any of you have any questions  but i need to run off it for myself for  another meeting so i don't have much  that much time  um  clarify my comment about picking the  wrong problem to tackle p versus np  um  when i use composites as an example  you know if i worked hard to prove that  composites is not in p as a way of  proving that there is some np language  which is not in p  um  that would that would have been a  mistake because composites is in p  that would be just i would would have  been working hard to prove something  which we now know was false  so we don't want to spend time  working on the wrong language  um  you know but the nice thing about np  complete languages is that we have a  guarantee  that if p is different from np that that  language is not in p  are there problems that are not in p  that are in np but are not np complete  oh that's a good question  are there  problems in between  uh p and np complete so np complete is  sort of like the hardest problems in np  and the p problems are obviously the  easy problems that are in np and how  about is everything either np complete  or mp  so for one thing there are problems  that are not known to be in either  category  [Music]  uh so we'll discuss some of those in due  course but one of them is like the graph  isomorphism problem testing two graphs  if they're really just permutations of  one another it's clearly a problem in np  but not known to bnp um so there are  problems that are  not known to be either np complete or in  p so there are problems that might be in  between um  but then there's another theorem out  there which says that if you if you  assume that p is different from np then  you can construct  problems which are in between which are  neither np complete nor in p they're np  problems but they're not empty complete  not in p um so  uh those problems themselves are perhaps  somewhat artificial but they at least  prove the point that it is possible to  have these intermediate problems  um  so somebody's asking isn't factorization  one not no you know we  for the case of factorization  um or you have to make sure to make a  language out of that by the way um  uh  but  um  it's a um  uh  stuff because factorization is a  function so we won't really want to be  talking about languages as kind of as  the homework suggests  np and p are classes of languages but  okay that's a separate separate note  there factorization um  is  uh not no  it could factorization could be in p and  it could be np-complete  you know both of those are not ruled out  so uh you know i think most people would  probably venture to guess that it's it's  a problem that's in the in between state  that's neither p nor mp complete but not  known  um  uh do we  who who first thought of this reduction  from three set to hand path it's so  clever  well it wasn't me uh  i believe that uh reduction from  uh  i think that is due to dick carp  who was one of my professors at berkeley  where i was a graduate student he that  was done before i got there it was  around 1971 um when so there were two  famous papers there's the cook paper  there's also the lemon paper that was in  russian that took a while for people to  discover out here in the west but  the cook paper was 1971 and very quickly  followed after he just showed sat as np  complete  but uh after that uh carp  was  he  he had a paper called reducibility among  combinatorial problems and he had a list  of about 20 problems that he showed were  np complete  by reduction from sad um  include clique include handpath and a  bunch of other things um  so  and that's was also a very famous uh  very famous paper um both of those were  people who often talk about cook carp as  sort of together they  uh they they they really sort of  show the importance of np completeness  and exist and no whole notion of np  completeness  um  yeah 21 problems so  yeah so car proof 21 problems wouldn't  be complete in 1972  um so that was shortly after coke showed  that sat was mp complete  um incidentally i think the terminology  np complete wasn't around uh  uh until a little later  and that might have been might be due to  knuth  i'm not sure i remember he did a big  poll  of people about what should be the right  language to use for the for that term  and i think he came up with it  um  all right i'm gonna head off guys nice  uh seeing you all uh so until uh  thursday oh until tuesday uh bye-bye
 

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00:08:26,639 --> 00:08:27,749


251
00:08:27,749 --> 00:08:31,189

 

252
00:08:31,189 --> 00:08:31,199

 

253
00:08:31,199 --> 00:08:32,230


254
00:08:32,230 --> 00:08:34,870

 

255
00:08:34,870 --> 00:08:34,880

 

256
00:08:34,880 --> 00:08:36,070


257
00:08:36,070 --> 00:08:38,149

 

258
00:08:38,149 --> 00:08:39,990

 

259
00:08:39,990 --> 00:08:42,389

 

260
00:08:42,389 --> 00:08:42,399

 

261
00:08:42,399 --> 00:08:43,829


262
00:08:43,829 --> 00:08:45,910

 

263
00:08:45,910 --> 00:08:45,920

 

264
00:08:45,920 --> 00:08:47,110


265
00:08:47,110 --> 00:08:48,630

 

266
00:08:48,630 --> 00:08:50,150

 

267
00:08:50,150 --> 00:08:52,949

 

268
00:08:52,949 --> 00:08:54,470

 

269
00:08:54,470 --> 00:08:56,630

 

270
00:08:56,630 --> 00:08:56,640

 

271
00:08:56,640 --> 00:08:57,430


272
00:08:57,430 --> 00:08:59,190

 

273
00:08:59,190 --> 00:09:01,910

 

274
00:09:01,910 --> 00:09:04,550

 

275
00:09:04,550 --> 00:09:08,230

 

276
00:09:08,230 --> 00:09:10,310

 

277
00:09:10,310 --> 00:09:13,509

 

278
00:09:13,509 --> 00:09:13,519

 

279
00:09:13,519 --> 00:09:14,389


280
00:09:14,389 --> 00:09:14,399

 

281
00:09:14,399 --> 00:09:15,590


282
00:09:15,590 --> 00:09:18,070

 

283
00:09:18,070 --> 00:09:20,310

 

284
00:09:20,310 --> 00:09:21,750

 

285
00:09:21,750 --> 00:09:23,509

 

286
00:09:23,509 --> 00:09:24,949

 

287
00:09:24,949 --> 00:09:28,230

 

288
00:09:28,230 --> 00:09:28,240

 

289
00:09:28,240 --> 00:09:30,070


290
00:09:30,070 --> 00:09:32,150

 

291
00:09:32,150 --> 00:09:34,310

 

292
00:09:34,310 --> 00:09:36,470

 

293
00:09:36,470 --> 00:09:37,750

 

294
00:09:37,750 --> 00:09:40,230

 

295
00:09:40,230 --> 00:09:43,430

 

296
00:09:43,430 --> 00:09:43,440

 

297
00:09:43,440 --> 00:09:44,790


298
00:09:44,790 --> 00:09:47,269

 

299
00:09:47,269 --> 00:09:49,430

 

300
00:09:49,430 --> 00:09:51,430

 

301
00:09:51,430 --> 00:09:53,110

 

302
00:09:53,110 --> 00:09:55,590

 

303
00:09:55,590 --> 00:09:57,910

 

304
00:09:57,910 --> 00:09:57,920

 

305
00:09:57,920 --> 00:10:00,150


306
00:10:00,150 --> 00:10:02,470

 

307
00:10:02,470 --> 00:10:04,870

 

308
00:10:04,870 --> 00:10:07,110

 

309
00:10:07,110 --> 00:10:09,509

 

310
00:10:09,509 --> 00:10:14,150

 

311
00:10:14,150 --> 00:10:15,990

 

312
00:10:15,990 --> 00:10:16,000

 

313
00:10:16,000 --> 00:10:17,430


314
00:10:17,430 --> 00:10:20,630

 

315
00:10:20,630 --> 00:10:22,389

 

316
00:10:22,389 --> 00:10:25,750

 

317
00:10:25,750 --> 00:10:28,230

 

318
00:10:28,230 --> 00:10:31,750

 

319
00:10:31,750 --> 00:10:33,590

 

320
00:10:33,590 --> 00:10:35,430

 

321
00:10:35,430 --> 00:10:35,440

 

322
00:10:35,440 --> 00:10:36,550


323
00:10:36,550 --> 00:10:39,269

 

324
00:10:39,269 --> 00:10:41,829

 

325
00:10:41,829 --> 00:10:43,430

 

326
00:10:43,430 --> 00:10:45,750

 

327
00:10:45,750 --> 00:10:47,670

 

328
00:10:47,670 --> 00:10:47,680

 

329
00:10:47,680 --> 00:10:48,710


330
00:10:48,710 --> 00:10:52,230

 

331
00:10:52,230 --> 00:10:55,110

 

332
00:10:55,110 --> 00:10:57,190

 

333
00:10:57,190 --> 00:10:59,670

 

334
00:10:59,670 --> 00:10:59,680

 

335
00:10:59,680 --> 00:11:01,590


336
00:11:01,590 --> 00:11:03,030

 

337
00:11:03,030 --> 00:11:04,470

 

338
00:11:04,470 --> 00:11:06,790

 

339
00:11:06,790 --> 00:11:10,630

 

340
00:11:10,630 --> 00:11:10,640

 

341
00:11:10,640 --> 00:11:11,509


342
00:11:11,509 --> 00:11:13,030

 

343
00:11:13,030 --> 00:11:14,470

 

344
00:11:14,470 --> 00:11:17,030

 

345
00:11:17,030 --> 00:11:19,509

 

346
00:11:19,509 --> 00:11:21,750

 

347
00:11:21,750 --> 00:11:23,750

 

348
00:11:23,750 --> 00:11:26,470

 

349
00:11:26,470 --> 00:11:26,480

 

350
00:11:26,480 --> 00:11:29,030


351
00:11:29,030 --> 00:11:30,949

 

352
00:11:30,949 --> 00:11:32,389

 

353
00:11:32,389 --> 00:11:32,399

 

354
00:11:32,399 --> 00:11:33,670


355
00:11:33,670 --> 00:11:33,680

 

356
00:11:33,680 --> 00:11:34,470


357
00:11:34,470 --> 00:11:36,389

 

358
00:11:36,389 --> 00:11:36,399

 

359
00:11:36,399 --> 00:11:37,350


360
00:11:37,350 --> 00:11:39,990

 

361
00:11:39,990 --> 00:11:40,000

 

362
00:11:40,000 --> 00:11:40,949


363
00:11:40,949 --> 00:11:43,670

 

364
00:11:43,670 --> 00:11:44,870

 

365
00:11:44,870 --> 00:11:47,350

 

366
00:11:47,350 --> 00:11:50,790

 

367
00:11:50,790 --> 00:11:52,150

 

368
00:11:52,150 --> 00:11:54,150

 

369
00:11:54,150 --> 00:11:56,870

 

370
00:11:56,870 --> 00:11:58,389

 

371
00:11:58,389 --> 00:12:00,629

 

372
00:12:00,629 --> 00:12:01,990

 

373
00:12:01,990 --> 00:12:02,000

 

374
00:12:02,000 --> 00:12:03,670


375
00:12:03,670 --> 00:12:06,069

 

376
00:12:06,069 --> 00:12:07,590

 

377
00:12:07,590 --> 00:12:09,910

 

378
00:12:09,910 --> 00:12:13,269

 

379
00:12:13,269 --> 00:12:15,430

 

380
00:12:15,430 --> 00:12:19,829

 

381
00:12:19,829 --> 00:12:21,990

 

382
00:12:21,990 --> 00:12:24,230

 

383
00:12:24,230 --> 00:12:25,829

 

384
00:12:25,829 --> 00:12:28,470

 

385
00:12:28,470 --> 00:12:30,470

 

386
00:12:30,470 --> 00:12:31,910

 

387
00:12:31,910 --> 00:12:34,550

 

388
00:12:34,550 --> 00:12:36,470

 

389
00:12:36,470 --> 00:12:36,480

 

390
00:12:36,480 --> 00:12:37,590


391
00:12:37,590 --> 00:12:39,350

 

392
00:12:39,350 --> 00:12:39,360

 

393
00:12:39,360 --> 00:12:40,870


394
00:12:40,870 --> 00:12:42,629

 

395
00:12:42,629 --> 00:12:44,629

 

396
00:12:44,629 --> 00:12:47,590

 

397
00:12:47,590 --> 00:12:49,509

 

398
00:12:49,509 --> 00:12:52,550

 

399
00:12:52,550 --> 00:12:54,069

 

400
00:12:54,069 --> 00:12:55,910

 

401
00:12:55,910 --> 00:12:55,920

 

402
00:12:55,920 --> 00:12:57,670


403
00:12:57,670 --> 00:12:59,990

 

404
00:12:59,990 --> 00:13:01,910

 

405
00:13:01,910 --> 00:13:03,990

 

406
00:13:03,990 --> 00:13:05,350

 

407
00:13:05,350 --> 00:13:07,190

 

408
00:13:07,190 --> 00:13:10,069

 

409
00:13:10,069 --> 00:13:11,509

 

410
00:13:11,509 --> 00:13:13,670

 

411
00:13:13,670 --> 00:13:15,030

 

412
00:13:15,030 --> 00:13:16,790

 

413
00:13:16,790 --> 00:13:20,389

 

414
00:13:20,389 --> 00:13:23,990

 

415
00:13:23,990 --> 00:13:26,629

 

416
00:13:26,629 --> 00:13:28,310

 

417
00:13:28,310 --> 00:13:30,389

 

418
00:13:30,389 --> 00:13:30,399

 

419
00:13:30,399 --> 00:13:31,910


420
00:13:31,910 --> 00:13:33,829

 

421
00:13:33,829 --> 00:13:35,670

 

422
00:13:35,670 --> 00:13:37,190

 

423
00:13:37,190 --> 00:13:39,670

 

424
00:13:39,670 --> 00:13:42,710

 

425
00:13:42,710 --> 00:13:44,710

 

426
00:13:44,710 --> 00:13:46,870

 

427
00:13:46,870 --> 00:13:49,910

 

428
00:13:49,910 --> 00:13:49,920

 

429
00:13:49,920 --> 00:13:51,910


430
00:13:51,910 --> 00:13:54,150

 

431
00:13:54,150 --> 00:13:55,990

 

432
00:13:55,990 --> 00:13:56,000

 

433
00:13:56,000 --> 00:13:56,949


434
00:13:56,949 --> 00:14:00,310

 

435
00:14:00,310 --> 00:14:01,829

 

436
00:14:01,829 --> 00:14:03,910

 

437
00:14:03,910 --> 00:14:03,920

 

438
00:14:03,920 --> 00:14:05,670


439
00:14:05,670 --> 00:14:08,389

 

440
00:14:08,389 --> 00:14:10,310

 

441
00:14:10,310 --> 00:14:12,230

 

442
00:14:12,230 --> 00:14:14,389

 

443
00:14:14,389 --> 00:14:16,629

 

444
00:14:16,629 --> 00:14:16,639

 

445
00:14:16,639 --> 00:14:17,750


446
00:14:17,750 --> 00:14:21,590

 

447
00:14:21,590 --> 00:14:23,110

 

448
00:14:23,110 --> 00:14:24,949

 

449
00:14:24,949 --> 00:14:26,870

 

450
00:14:26,870 --> 00:14:26,880

 

451
00:14:26,880 --> 00:14:27,750


452
00:14:27,750 --> 00:14:29,030

 

453
00:14:29,030 --> 00:14:30,710

 

454
00:14:30,710 --> 00:14:32,870

 

455
00:14:32,870 --> 00:14:32,880

 

456
00:14:32,880 --> 00:14:33,829


457
00:14:33,829 --> 00:14:36,790

 

458
00:14:36,790 --> 00:14:38,790

 

459
00:14:38,790 --> 00:14:40,710

 

460
00:14:40,710 --> 00:14:43,590

 

461
00:14:43,590 --> 00:14:46,150

 

462
00:14:46,150 --> 00:14:48,150

 

463
00:14:48,150 --> 00:14:49,829

 

464
00:14:49,829 --> 00:14:51,110

 

465
00:14:51,110 --> 00:14:52,230

 

466
00:14:52,230 --> 00:14:54,150

 

467
00:14:54,150 --> 00:14:56,230

 

468
00:14:56,230 --> 00:14:59,030

 

469
00:14:59,030 --> 00:15:00,790

 

470
00:15:00,790 --> 00:15:03,670

 

471
00:15:03,670 --> 00:15:05,750

 

472
00:15:05,750 --> 00:15:07,030

 

473
00:15:07,030 --> 00:15:08,949

 

474
00:15:08,949 --> 00:15:11,430

 

475
00:15:11,430 --> 00:15:11,440

 

476
00:15:11,440 --> 00:15:13,030


477
00:15:13,030 --> 00:15:13,040

 

478
00:15:13,040 --> 00:15:13,829


479
00:15:13,829 --> 00:15:16,790

 

480
00:15:16,790 --> 00:15:18,310

 

481
00:15:18,310 --> 00:15:19,829

 

482
00:15:19,829 --> 00:15:19,839

 

483
00:15:19,839 --> 00:15:21,030


484
00:15:21,030 --> 00:15:21,040

 

485
00:15:21,040 --> 00:15:23,269


486
00:15:23,269 --> 00:15:24,870

 

487
00:15:24,870 --> 00:15:36,550

 

488
00:15:36,550 --> 00:15:39,350

 

489
00:15:39,350 --> 00:15:43,670

 

490
00:15:43,670 --> 00:15:45,110

 

491
00:15:45,110 --> 00:15:45,120

 

492
00:15:45,120 --> 00:15:46,550


493
00:15:46,550 --> 00:15:48,790

 

494
00:15:48,790 --> 00:15:48,800

 

495
00:15:48,800 --> 00:15:49,670


496
00:15:49,670 --> 00:15:51,749

 

497
00:15:51,749 --> 00:15:54,550

 

498
00:15:54,550 --> 00:15:58,949

 

499
00:15:58,949 --> 00:15:58,959

 

500
00:15:58,959 --> 00:16:00,470


501
00:16:00,470 --> 00:16:02,910

 

502
00:16:02,910 --> 00:16:04,970

 

503
00:16:04,970 --> 00:16:04,980

 

504
00:16:04,980 --> 00:16:06,310


505
00:16:06,310 --> 00:16:06,320

 

506
00:16:06,320 --> 00:16:07,189


507
00:16:07,189 --> 00:16:09,350

 

508
00:16:09,350 --> 00:16:11,350

 

509
00:16:11,350 --> 00:16:13,269

 

510
00:16:13,269 --> 00:16:15,430

 

511
00:16:15,430 --> 00:16:18,069

 

512
00:16:18,069 --> 00:16:18,079

 

513
00:16:18,079 --> 00:16:19,110


514
00:16:19,110 --> 00:16:22,389

 

515
00:16:22,389 --> 00:16:25,030

 

516
00:16:25,030 --> 00:16:26,949

 

517
00:16:26,949 --> 00:16:29,030

 

518
00:16:29,030 --> 00:16:31,910

 

519
00:16:31,910 --> 00:16:34,949

 

520
00:16:34,949 --> 00:16:36,790

 

521
00:16:36,790 --> 00:16:36,800

 

522
00:16:36,800 --> 00:16:38,389


523
00:16:38,389 --> 00:16:39,590

 

524
00:16:39,590 --> 00:16:42,870

 

525
00:16:42,870 --> 00:16:42,880

 

526
00:16:42,880 --> 00:16:44,710


527
00:16:44,710 --> 00:16:48,829

 

528
00:16:48,829 --> 00:16:52,389

 

529
00:16:52,389 --> 00:16:56,150

 

530
00:16:56,150 --> 00:16:58,069

 

531
00:16:58,069 --> 00:17:00,150

 

532
00:17:00,150 --> 00:17:01,990

 

533
00:17:01,990 --> 00:17:05,829

 

534
00:17:05,829 --> 00:17:08,630

 

535
00:17:08,630 --> 00:17:10,789

 

536
00:17:10,789 --> 00:17:12,470

 

537
00:17:12,470 --> 00:17:12,480

 

538
00:17:12,480 --> 00:17:16,829


539
00:17:16,829 --> 00:17:19,510

 

540
00:17:19,510 --> 00:17:21,990

 

541
00:17:21,990 --> 00:17:24,390

 

542
00:17:24,390 --> 00:17:26,309

 

543
00:17:26,309 --> 00:17:27,510

 

544
00:17:27,510 --> 00:17:29,590

 

545
00:17:29,590 --> 00:17:29,600

 

546
00:17:29,600 --> 00:17:33,029


547
00:17:33,029 --> 00:17:33,039

 

548
00:17:33,039 --> 00:17:33,909


549
00:17:33,909 --> 00:17:36,470

 

550
00:17:36,470 --> 00:17:36,480

 

551
00:17:36,480 --> 00:17:37,430


552
00:17:37,430 --> 00:17:39,990

 

553
00:17:39,990 --> 00:17:40,000

 

554
00:17:40,000 --> 00:17:41,110


555
00:17:41,110 --> 00:17:43,190

 

556
00:17:43,190 --> 00:17:44,870

 

557
00:17:44,870 --> 00:17:47,669

 

558
00:17:47,669 --> 00:17:48,789

 

559
00:17:48,789 --> 00:17:50,310

 

560
00:17:50,310 --> 00:17:53,270

 

561
00:17:53,270 --> 00:17:56,470

 

562
00:17:56,470 --> 00:17:58,470

 

563
00:17:58,470 --> 00:18:00,789

 

564
00:18:00,789 --> 00:18:00,799

 

565
00:18:00,799 --> 00:18:02,470


566
00:18:02,470 --> 00:18:04,950

 

567
00:18:04,950 --> 00:18:06,950

 

568
00:18:06,950 --> 00:18:08,950

 

569
00:18:08,950 --> 00:18:11,110

 

570
00:18:11,110 --> 00:18:13,110

 

571
00:18:13,110 --> 00:18:14,950

 

572
00:18:14,950 --> 00:18:16,710

 

573
00:18:16,710 --> 00:18:18,549

 

574
00:18:18,549 --> 00:18:19,669

 

575
00:18:19,669 --> 00:18:19,679

 

576
00:18:19,679 --> 00:18:20,789


577
00:18:20,789 --> 00:18:23,669

 

578
00:18:23,669 --> 00:18:25,590

 

579
00:18:25,590 --> 00:18:27,029

 

580
00:18:27,029 --> 00:18:29,350

 

581
00:18:29,350 --> 00:18:32,070

 

582
00:18:32,070 --> 00:18:33,990

 

583
00:18:33,990 --> 00:18:36,630

 

584
00:18:36,630 --> 00:18:38,710

 

585
00:18:38,710 --> 00:18:40,470

 

586
00:18:40,470 --> 00:18:42,230

 

587
00:18:42,230 --> 00:18:43,830

 

588
00:18:43,830 --> 00:18:45,350

 

589
00:18:45,350 --> 00:18:45,360

 

590
00:18:45,360 --> 00:18:46,390


591
00:18:46,390 --> 00:18:46,400

 

592
00:18:46,400 --> 00:18:47,830


593
00:18:47,830 --> 00:18:49,510

 

594
00:18:49,510 --> 00:18:49,520

 

595
00:18:49,520 --> 00:18:50,630


596
00:18:50,630 --> 00:18:52,549

 

597
00:18:52,549 --> 00:18:55,190

 

598
00:18:55,190 --> 00:18:57,110

 

599
00:18:57,110 --> 00:18:57,120

 

600
00:18:57,120 --> 00:18:57,909


601
00:18:57,909 --> 00:18:59,029

 

602
00:18:59,029 --> 00:18:59,039

 

603
00:18:59,039 --> 00:19:03,190


604
00:19:03,190 --> 00:19:04,470

 

605
00:19:04,470 --> 00:19:06,230

 

606
00:19:06,230 --> 00:19:07,350

 

607
00:19:07,350 --> 00:19:07,360

 

608
00:19:07,360 --> 00:19:09,110


609
00:19:09,110 --> 00:19:09,120

 

610
00:19:09,120 --> 00:19:10,549


611
00:19:10,549 --> 00:19:10,559

 

612
00:19:10,559 --> 00:19:11,430


613
00:19:11,430 --> 00:19:11,440

 

614
00:19:11,440 --> 00:19:12,549


615
00:19:12,549 --> 00:19:15,029

 

616
00:19:15,029 --> 00:19:16,470

 

617
00:19:16,470 --> 00:19:19,430

 

618
00:19:19,430 --> 00:19:22,390

 

619
00:19:22,390 --> 00:19:24,630

 

620
00:19:24,630 --> 00:19:26,789

 

621
00:19:26,789 --> 00:19:26,799

 

622
00:19:26,799 --> 00:19:27,590


623
00:19:27,590 --> 00:19:29,430

 

624
00:19:29,430 --> 00:19:31,590

 

625
00:19:31,590 --> 00:19:34,230

 

626
00:19:34,230 --> 00:19:35,510

 

627
00:19:35,510 --> 00:19:37,990

 

628
00:19:37,990 --> 00:19:42,470

 

629
00:19:42,470 --> 00:19:44,230

 

630
00:19:44,230 --> 00:19:45,430

 

631
00:19:45,430 --> 00:19:47,350

 

632
00:19:47,350 --> 00:19:49,909

 

633
00:19:49,909 --> 00:19:51,590

 

634
00:19:51,590 --> 00:19:53,669

 

635
00:19:53,669 --> 00:19:55,510

 

636
00:19:55,510 --> 00:19:57,350

 

637
00:19:57,350 --> 00:19:58,710

 

638
00:19:58,710 --> 00:20:01,430

 

639
00:20:01,430 --> 00:20:03,190

 

640
00:20:03,190 --> 00:20:04,789

 

641
00:20:04,789 --> 00:20:06,789

 

642
00:20:06,789 --> 00:20:08,080

 

643
00:20:08,080 --> 00:20:08,090

 

644
00:20:08,090 --> 00:20:10,390


645
00:20:10,390 --> 00:20:12,149

 

646
00:20:12,149 --> 00:20:13,750

 

647
00:20:13,750 --> 00:20:16,310

 

648
00:20:16,310 --> 00:20:21,990

 

649
00:20:21,990 --> 00:20:22,000

 

650
00:20:22,000 --> 00:20:23,590


651
00:20:23,590 --> 00:20:26,310

 

652
00:20:26,310 --> 00:20:28,230

 

653
00:20:28,230 --> 00:20:30,470

 

654
00:20:30,470 --> 00:20:32,470

 

655
00:20:32,470 --> 00:20:34,230

 

656
00:20:34,230 --> 00:20:37,190

 

657
00:20:37,190 --> 00:20:39,590

 

658
00:20:39,590 --> 00:20:43,190

 

659
00:20:43,190 --> 00:20:44,789

 

660
00:20:44,789 --> 00:20:46,470

 

661
00:20:46,470 --> 00:20:49,909

 

662
00:20:49,909 --> 00:20:49,919

 

663
00:20:49,919 --> 00:20:51,190


664
00:20:51,190 --> 00:20:53,830

 

665
00:20:53,830 --> 00:20:56,549

 

666
00:20:56,549 --> 00:20:58,580

 

667
00:20:58,580 --> 00:20:58,590

 

668
00:20:58,590 --> 00:20:59,830


669
00:20:59,830 --> 00:21:01,590

 

670
00:21:01,590 --> 00:21:02,789

 

671
00:21:02,789 --> 00:21:04,710

 

672
00:21:04,710 --> 00:21:06,950

 

673
00:21:06,950 --> 00:21:08,310

 

674
00:21:08,310 --> 00:21:08,320

 

675
00:21:08,320 --> 00:21:09,909


676
00:21:09,909 --> 00:21:12,390

 

677
00:21:12,390 --> 00:21:13,990

 

678
00:21:13,990 --> 00:21:16,070

 

679
00:21:16,070 --> 00:21:17,750

 

680
00:21:17,750 --> 00:21:19,830

 

681
00:21:19,830 --> 00:21:21,590

 

682
00:21:21,590 --> 00:21:23,669

 

683
00:21:23,669 --> 00:21:25,830

 

684
00:21:25,830 --> 00:21:27,270

 

685
00:21:27,270 --> 00:21:27,280

 

686
00:21:27,280 --> 00:21:28,830


687
00:21:28,830 --> 00:21:31,510

 

688
00:21:31,510 --> 00:21:34,230

 

689
00:21:34,230 --> 00:21:36,470

 

690
00:21:36,470 --> 00:21:38,870

 

691
00:21:38,870 --> 00:21:40,470

 

692
00:21:40,470 --> 00:21:42,549

 

693
00:21:42,549 --> 00:21:43,669

 

694
00:21:43,669 --> 00:21:45,750

 

695
00:21:45,750 --> 00:21:47,510

 

696
00:21:47,510 --> 00:21:48,950

 

697
00:21:48,950 --> 00:21:48,960

 

698
00:21:48,960 --> 00:21:51,750


699
00:21:51,750 --> 00:21:53,669

 

700
00:21:53,669 --> 00:21:57,990

 

701
00:21:57,990 --> 00:22:00,710

 

702
00:22:00,710 --> 00:22:04,630

 

703
00:22:04,630 --> 00:22:06,230

 

704
00:22:06,230 --> 00:22:08,070

 

705
00:22:08,070 --> 00:22:09,990

 

706
00:22:09,990 --> 00:22:11,590

 

707
00:22:11,590 --> 00:22:13,590

 

708
00:22:13,590 --> 00:22:15,430

 

709
00:22:15,430 --> 00:22:15,440

 

710
00:22:15,440 --> 00:22:17,750


711
00:22:17,750 --> 00:22:19,990

 

712
00:22:19,990 --> 00:22:22,630

 

713
00:22:22,630 --> 00:22:24,230

 

714
00:22:24,230 --> 00:22:26,390

 

715
00:22:26,390 --> 00:22:28,549

 

716
00:22:28,549 --> 00:22:29,590

 

717
00:22:29,590 --> 00:22:31,909

 

718
00:22:31,909 --> 00:22:34,390

 

719
00:22:34,390 --> 00:22:36,070

 

720
00:22:36,070 --> 00:22:37,830

 

721
00:22:37,830 --> 00:22:41,830

 

722
00:22:41,830 --> 00:22:44,070

 

723
00:22:44,070 --> 00:22:46,870

 

724
00:22:46,870 --> 00:22:49,350

 

725
00:22:49,350 --> 00:22:51,190

 

726
00:22:51,190 --> 00:22:51,200

 

727
00:22:51,200 --> 00:22:52,310


728
00:22:52,310 --> 00:22:52,320

 

729
00:22:52,320 --> 00:22:53,110


730
00:22:53,110 --> 00:22:56,310

 

731
00:22:56,310 --> 00:22:57,990

 

732
00:22:57,990 --> 00:23:00,470

 

733
00:23:00,470 --> 00:23:04,710

 

734
00:23:04,710 --> 00:23:07,029

 

735
00:23:07,029 --> 00:23:09,590

 

736
00:23:09,590 --> 00:23:12,710

 

737
00:23:12,710 --> 00:23:12,720

 

738
00:23:12,720 --> 00:23:14,710


739
00:23:14,710 --> 00:23:16,710

 

740
00:23:16,710 --> 00:23:18,230

 

741
00:23:18,230 --> 00:23:20,149

 

742
00:23:20,149 --> 00:23:22,390

 

743
00:23:22,390 --> 00:23:24,950

 

744
00:23:24,950 --> 00:23:26,870

 

745
00:23:26,870 --> 00:23:26,880

 

746
00:23:26,880 --> 00:23:27,990


747
00:23:27,990 --> 00:23:29,669

 

748
00:23:29,669 --> 00:23:31,750

 

749
00:23:31,750 --> 00:23:34,070

 

750
00:23:34,070 --> 00:23:36,470

 

751
00:23:36,470 --> 00:23:37,909

 

752
00:23:37,909 --> 00:23:39,190

 

753
00:23:39,190 --> 00:23:39,200

 

754
00:23:39,200 --> 00:23:40,870


755
00:23:40,870 --> 00:23:42,549

 

756
00:23:42,549 --> 00:23:45,909

 

757
00:23:45,909 --> 00:23:47,669

 

758
00:23:47,669 --> 00:23:50,230

 

759
00:23:50,230 --> 00:23:53,190

 

760
00:23:53,190 --> 00:23:55,110

 

761
00:23:55,110 --> 00:23:57,430

 

762
00:23:57,430 --> 00:23:58,789

 

763
00:23:58,789 --> 00:24:01,590

 

764
00:24:01,590 --> 00:24:03,990

 

765
00:24:03,990 --> 00:24:05,909

 

766
00:24:05,909 --> 00:24:08,310

 

767
00:24:08,310 --> 00:24:10,950

 

768
00:24:10,950 --> 00:24:13,590

 

769
00:24:13,590 --> 00:24:15,190

 

770
00:24:15,190 --> 00:24:16,870

 

771
00:24:16,870 --> 00:24:19,269

 

772
00:24:19,269 --> 00:24:21,990

 

773
00:24:21,990 --> 00:24:26,230

 

774
00:24:26,230 --> 00:24:27,750

 

775
00:24:27,750 --> 00:24:29,430

 

776
00:24:29,430 --> 00:24:31,990

 

777
00:24:31,990 --> 00:24:33,510

 

778
00:24:33,510 --> 00:24:35,510

 

779
00:24:35,510 --> 00:24:37,789

 

780
00:24:37,789 --> 00:24:40,549

 

781
00:24:40,549 --> 00:24:42,870

 

782
00:24:42,870 --> 00:24:45,269

 

783
00:24:45,269 --> 00:24:45,279

 

784
00:24:45,279 --> 00:24:46,789


785
00:24:46,789 --> 00:24:49,029

 

786
00:24:49,029 --> 00:24:50,870

 

787
00:24:50,870 --> 00:24:52,710

 

788
00:24:52,710 --> 00:24:54,470

 

789
00:24:54,470 --> 00:24:56,549

 

790
00:24:56,549 --> 00:24:59,510

 

791
00:24:59,510 --> 00:25:01,110

 

792
00:25:01,110 --> 00:25:03,750

 

793
00:25:03,750 --> 00:25:06,310

 

794
00:25:06,310 --> 00:25:08,950

 

795
00:25:08,950 --> 00:25:08,960

 

796
00:25:08,960 --> 00:25:10,310


797
00:25:10,310 --> 00:25:11,990

 

798
00:25:11,990 --> 00:25:14,470

 

799
00:25:14,470 --> 00:25:17,029

 

800
00:25:17,029 --> 00:25:17,039

 

801
00:25:17,039 --> 00:25:18,230


802
00:25:18,230 --> 00:25:20,950

 

803
00:25:20,950 --> 00:25:21,670

 

804
00:25:21,670 --> 00:25:21,680

 

805
00:25:21,680 --> 00:25:22,789


806
00:25:22,789 --> 00:25:24,870

 

807
00:25:24,870 --> 00:25:26,789

 

808
00:25:26,789 --> 00:25:28,390

 

809
00:25:28,390 --> 00:25:31,029

 

810
00:25:31,029 --> 00:25:32,950

 

811
00:25:32,950 --> 00:25:34,310

 

812
00:25:34,310 --> 00:25:36,390

 

813
00:25:36,390 --> 00:25:40,310

 

814
00:25:40,310 --> 00:25:42,789

 

815
00:25:42,789 --> 00:25:45,350

 

816
00:25:45,350 --> 00:25:47,350

 

817
00:25:47,350 --> 00:25:47,360

 

818
00:25:47,360 --> 00:25:48,630


819
00:25:48,630 --> 00:25:50,470

 

820
00:25:50,470 --> 00:25:52,870

 

821
00:25:52,870 --> 00:25:54,230

 

822
00:25:54,230 --> 00:25:56,070

 

823
00:25:56,070 --> 00:25:57,669

 

824
00:25:57,669 --> 00:26:00,070

 

825
00:26:00,070 --> 00:26:03,350

 

826
00:26:03,350 --> 00:26:05,350

 

827
00:26:05,350 --> 00:26:07,029

 

828
00:26:07,029 --> 00:26:09,190

 

829
00:26:09,190 --> 00:26:10,789

 

830
00:26:10,789 --> 00:26:12,630

 

831
00:26:12,630 --> 00:26:13,830

 

832
00:26:13,830 --> 00:26:15,190

 

833
00:26:15,190 --> 00:26:18,390

 

834
00:26:18,390 --> 00:26:20,630

 

835
00:26:20,630 --> 00:26:22,870

 

836
00:26:22,870 --> 00:26:27,909

 

837
00:26:27,909 --> 00:26:30,470

 

838
00:26:30,470 --> 00:26:33,029

 

839
00:26:33,029 --> 00:26:35,269

 

840
00:26:35,269 --> 00:26:37,590

 

841
00:26:37,590 --> 00:26:37,600

 

842
00:26:37,600 --> 00:26:38,390


843
00:26:38,390 --> 00:26:40,950

 

844
00:26:40,950 --> 00:26:43,350

 

845
00:26:43,350 --> 00:26:46,070

 

846
00:26:46,070 --> 00:26:47,750

 

847
00:26:47,750 --> 00:26:50,149

 

848
00:26:50,149 --> 00:26:53,830

 

849
00:26:53,830 --> 00:26:55,669

 

850
00:26:55,669 --> 00:26:57,110

 

851
00:26:57,110 --> 00:27:00,710

 

852
00:27:00,710 --> 00:27:02,630

 

853
00:27:02,630 --> 00:27:06,149

 

854
00:27:06,149 --> 00:27:08,870

 

855
00:27:08,870 --> 00:27:09,750

 

856
00:27:09,750 --> 00:27:09,760

 

857
00:27:09,760 --> 00:27:10,789


858
00:27:10,789 --> 00:27:12,789

 

859
00:27:12,789 --> 00:27:14,230

 

860
00:27:14,230 --> 00:27:16,710

 

861
00:27:16,710 --> 00:27:25,750

 

862
00:27:25,750 --> 00:27:29,430

 

863
00:27:29,430 --> 00:27:29,440

 

864
00:27:29,440 --> 00:27:31,750


865
00:27:31,750 --> 00:27:31,760

 

866
00:27:31,760 --> 00:27:33,350


867
00:27:33,350 --> 00:27:35,350

 

868
00:27:35,350 --> 00:27:37,350

 

869
00:27:37,350 --> 00:27:39,269

 

870
00:27:39,269 --> 00:27:41,190

 

871
00:27:41,190 --> 00:27:42,389

 

872
00:27:42,389 --> 00:27:43,750

 

873
00:27:43,750 --> 00:27:43,760

 

874
00:27:43,760 --> 00:27:45,750


875
00:27:45,750 --> 00:27:48,149

 

876
00:27:48,149 --> 00:27:50,710

 

877
00:27:50,710 --> 00:27:53,110

 

878
00:27:53,110 --> 00:27:55,830

 

879
00:27:55,830 --> 00:27:59,590

 

880
00:27:59,590 --> 00:28:03,269

 

881
00:28:03,269 --> 00:28:05,110

 

882
00:28:05,110 --> 00:28:06,630

 

883
00:28:06,630 --> 00:28:08,070

 

884
00:28:08,070 --> 00:28:12,310

 

885
00:28:12,310 --> 00:28:12,320

 

886
00:28:12,320 --> 00:28:13,190


887
00:28:13,190 --> 00:28:15,029

 

888
00:28:15,029 --> 00:28:16,470

 

889
00:28:16,470 --> 00:28:19,750

 

890
00:28:19,750 --> 00:28:21,990

 

891
00:28:21,990 --> 00:28:23,590

 

892
00:28:23,590 --> 00:28:23,600

 

893
00:28:23,600 --> 00:28:25,190


894
00:28:25,190 --> 00:28:25,200

 

895
00:28:25,200 --> 00:28:26,310


896
00:28:26,310 --> 00:28:28,549

 

897
00:28:28,549 --> 00:28:31,029

 

898
00:28:31,029 --> 00:28:33,029

 

899
00:28:33,029 --> 00:28:33,039

 

900
00:28:33,039 --> 00:28:34,230


901
00:28:34,230 --> 00:28:36,549

 

902
00:28:36,549 --> 00:28:39,430

 

903
00:28:39,430 --> 00:28:43,190

 

904
00:28:43,190 --> 00:28:45,430

 

905
00:28:45,430 --> 00:28:47,830

 

906
00:28:47,830 --> 00:28:50,710

 

907
00:28:50,710 --> 00:28:50,720

 

908
00:28:50,720 --> 00:28:51,510


909
00:28:51,510 --> 00:28:53,909

 

910
00:28:53,909 --> 00:28:55,830

 

911
00:28:55,830 --> 00:28:57,669

 

912
00:28:57,669 --> 00:28:59,430

 

913
00:28:59,430 --> 00:29:01,029

 

914
00:29:01,029 --> 00:29:03,110

 

915
00:29:03,110 --> 00:29:04,789

 

916
00:29:04,789 --> 00:29:05,909

 

917
00:29:05,909 --> 00:29:05,919

 

918
00:29:05,919 --> 00:29:07,110


919
00:29:07,110 --> 00:29:09,830

 

920
00:29:09,830 --> 00:29:12,789

 

921
00:29:12,789 --> 00:29:14,789

 

922
00:29:14,789 --> 00:29:17,350

 

923
00:29:17,350 --> 00:29:19,750

 

924
00:29:19,750 --> 00:29:21,669

 

925
00:29:21,669 --> 00:29:23,510

 

926
00:29:23,510 --> 00:29:25,590

 

927
00:29:25,590 --> 00:29:28,310

 

928
00:29:28,310 --> 00:29:30,310

 

929
00:29:30,310 --> 00:29:33,750

 

930
00:29:33,750 --> 00:29:33,760

 

931
00:29:33,760 --> 00:29:34,870


932
00:29:34,870 --> 00:29:35,830

 

933
00:29:35,830 --> 00:29:37,430

 

934
00:29:37,430 --> 00:29:39,430

 

935
00:29:39,430 --> 00:29:41,430

 

936
00:29:41,430 --> 00:29:43,430

 

937
00:29:43,430 --> 00:29:46,710

 

938
00:29:46,710 --> 00:29:49,990

 

939
00:29:49,990 --> 00:29:53,830

 

940
00:29:53,830 --> 00:29:57,909

 

941
00:29:57,909 --> 00:29:57,919

 

942
00:29:57,919 --> 00:29:58,789


943
00:29:58,789 --> 00:30:00,310

 

944
00:30:00,310 --> 00:30:02,950

 

945
00:30:02,950 --> 00:30:05,190

 

946
00:30:05,190 --> 00:30:07,110

 

947
00:30:07,110 --> 00:30:07,120

 

948
00:30:07,120 --> 00:30:09,750


949
00:30:09,750 --> 00:30:11,350

 

950
00:30:11,350 --> 00:30:12,950

 

951
00:30:12,950 --> 00:30:15,350

 

952
00:30:15,350 --> 00:30:18,310

 

953
00:30:18,310 --> 00:30:21,110

 

954
00:30:21,110 --> 00:30:22,710

 

955
00:30:22,710 --> 00:30:24,870

 

956
00:30:24,870 --> 00:30:25,909

 

957
00:30:25,909 --> 00:30:27,510

 

958
00:30:27,510 --> 00:30:29,669

 

959
00:30:29,669 --> 00:30:32,310

 

960
00:30:32,310 --> 00:30:34,870

 

961
00:30:34,870 --> 00:30:36,870

 

962
00:30:36,870 --> 00:30:39,110

 

963
00:30:39,110 --> 00:30:40,470

 

964
00:30:40,470 --> 00:30:43,990

 

965
00:30:43,990 --> 00:30:44,000

 

966
00:30:44,000 --> 00:30:45,590


967
00:30:45,590 --> 00:30:47,110

 

968
00:30:47,110 --> 00:30:51,029

 

969
00:30:51,029 --> 00:30:52,710

 

970
00:30:52,710 --> 00:30:55,430

 

971
00:30:55,430 --> 00:30:56,950

 

972
00:30:56,950 --> 00:30:59,669

 

973
00:30:59,669 --> 00:31:05,990

 

974
00:31:05,990 --> 00:31:08,310

 

975
00:31:08,310 --> 00:31:10,710

 

976
00:31:10,710 --> 00:31:13,269

 

977
00:31:13,269 --> 00:31:15,909

 

978
00:31:15,909 --> 00:31:18,310

 

979
00:31:18,310 --> 00:31:21,269

 

980
00:31:21,269 --> 00:31:23,430

 

981
00:31:23,430 --> 00:31:26,230

 

982
00:31:26,230 --> 00:31:29,509

 

983
00:31:29,509 --> 00:31:31,990

 

984
00:31:31,990 --> 00:31:33,990

 

985
00:31:33,990 --> 00:31:36,789

 

986
00:31:36,789 --> 00:31:40,630

 

987
00:31:40,630 --> 00:31:43,029

 

988
00:31:43,029 --> 00:31:46,789

 

989
00:31:46,789 --> 00:31:48,230

 

990
00:31:48,230 --> 00:31:48,240

 

991
00:31:48,240 --> 00:31:49,350


992
00:31:49,350 --> 00:31:51,269

 

993
00:31:51,269 --> 00:31:53,909

 

994
00:31:53,909 --> 00:31:58,710

 

995
00:31:58,710 --> 00:32:02,870

 

996
00:32:02,870 --> 00:32:04,630

 

997
00:32:04,630 --> 00:32:06,470

 

998
00:32:06,470 --> 00:32:09,269

 

999
00:32:09,269 --> 00:32:11,269

 

1000
00:32:11,269 --> 00:32:12,549

 

1001
00:32:12,549 --> 00:32:14,710

 

1002
00:32:14,710 --> 00:32:16,710

 

1003
00:32:16,710 --> 00:32:19,350

 

1004
00:32:19,350 --> 00:32:23,190

 

1005
00:32:23,190 --> 00:32:25,590

 

1006
00:32:25,590 --> 00:32:27,909

 

1007
00:32:27,909 --> 00:32:30,950

 

1008
00:32:30,950 --> 00:32:34,789

 

1009
00:32:34,789 --> 00:32:34,799

 

1010
00:32:34,799 --> 00:32:35,990


1011
00:32:35,990 --> 00:32:36,000

 

1012
00:32:36,000 --> 00:32:36,950


1013
00:32:36,950 --> 00:32:39,590

 

1014
00:32:39,590 --> 00:32:42,230

 

1015
00:32:42,230 --> 00:32:45,269

 

1016
00:32:45,269 --> 00:32:45,279

 

1017
00:32:45,279 --> 00:32:46,470


1018
00:32:46,470 --> 00:32:48,789

 

1019
00:32:48,789 --> 00:32:50,789

 

1020
00:32:50,789 --> 00:32:53,509

 

1021
00:32:53,509 --> 00:32:56,149

 

1022
00:32:56,149 --> 00:32:58,310

 

1023
00:32:58,310 --> 00:33:00,149

 

1024
00:33:00,149 --> 00:33:02,070

 

1025
00:33:02,070 --> 00:33:05,430

 

1026
00:33:05,430 --> 00:33:07,269

 

1027
00:33:07,269 --> 00:33:07,279

 

1028
00:33:07,279 --> 00:33:09,350


1029
00:33:09,350 --> 00:33:12,630

 

1030
00:33:12,630 --> 00:33:15,430

 

1031
00:33:15,430 --> 00:33:18,070

 

1032
00:33:18,070 --> 00:33:19,269

 

1033
00:33:19,269 --> 00:33:22,470

 

1034
00:33:22,470 --> 00:33:23,830

 

1035
00:33:23,830 --> 00:33:26,149

 

1036
00:33:26,149 --> 00:33:26,159

 

1037
00:33:26,159 --> 00:33:27,269


1038
00:33:27,269 --> 00:33:29,590

 

1039
00:33:29,590 --> 00:33:31,190

 

1040
00:33:31,190 --> 00:33:34,149

 

1041
00:33:34,149 --> 00:33:36,070

 

1042
00:33:36,070 --> 00:33:37,750

 

1043
00:33:37,750 --> 00:33:39,830

 

1044
00:33:39,830 --> 00:33:41,909

 

1045
00:33:41,909 --> 00:33:43,990

 

1046
00:33:43,990 --> 00:33:51,269

 

1047
00:33:51,269 --> 00:33:51,279

 

1048
00:33:51,279 --> 00:33:53,590


1049
00:33:53,590 --> 00:33:53,600

 

1050
00:33:53,600 --> 00:33:54,870


1051
00:33:54,870 --> 00:33:57,669

 

1052
00:33:57,669 --> 00:33:59,029

 

1053
00:33:59,029 --> 00:34:00,710

 

1054
00:34:00,710 --> 00:34:03,830

 

1055
00:34:03,830 --> 00:34:05,269

 

1056
00:34:05,269 --> 00:34:08,069

 

1057
00:34:08,069 --> 00:34:10,149

 

1058
00:34:10,149 --> 00:34:12,230

 

1059
00:34:12,230 --> 00:34:13,829

 

1060
00:34:13,829 --> 00:34:15,190

 

1061
00:34:15,190 --> 00:34:19,430

 

1062
00:34:19,430 --> 00:34:21,750

 

1063
00:34:21,750 --> 00:34:26,629

 

1064
00:34:26,629 --> 00:34:28,149

 

1065
00:34:28,149 --> 00:34:30,710

 

1066
00:34:30,710 --> 00:34:33,510

 

1067
00:34:33,510 --> 00:34:35,430

 

1068
00:34:35,430 --> 00:34:37,430

 

1069
00:34:37,430 --> 00:34:39,349

 

1070
00:34:39,349 --> 00:34:41,190

 

1071
00:34:41,190 --> 00:34:42,869

 

1072
00:34:42,869 --> 00:34:44,950

 

1073
00:34:44,950 --> 00:34:47,190

 

1074
00:34:47,190 --> 00:34:48,629

 

1075
00:34:48,629 --> 00:34:50,470

 

1076
00:34:50,470 --> 00:34:52,629

 

1077
00:34:52,629 --> 00:34:54,869

 

1078
00:34:54,869 --> 00:34:57,030

 

1079
00:34:57,030 --> 00:34:59,270

 

1080
00:34:59,270 --> 00:35:01,750

 

1081
00:35:01,750 --> 00:35:04,630

 

1082
00:35:04,630 --> 00:35:06,950

 

1083
00:35:06,950 --> 00:35:09,430

 

1084
00:35:09,430 --> 00:35:12,069

 

1085
00:35:12,069 --> 00:35:13,589

 

1086
00:35:13,589 --> 00:35:15,349

 

1087
00:35:15,349 --> 00:35:18,790

 

1088
00:35:18,790 --> 00:35:22,790

 

1089
00:35:22,790 --> 00:35:25,190

 

1090
00:35:25,190 --> 00:35:29,990

 

1091
00:35:29,990 --> 00:35:31,990

 

1092
00:35:31,990 --> 00:35:32,000

 

1093
00:35:32,000 --> 00:35:33,589


1094
00:35:33,589 --> 00:35:35,829

 

1095
00:35:35,829 --> 00:35:37,750

 

1096
00:35:37,750 --> 00:35:39,430

 

1097
00:35:39,430 --> 00:35:43,030

 

1098
00:35:43,030 --> 00:35:45,990

 

1099
00:35:45,990 --> 00:35:47,990

 

1100
00:35:47,990 --> 00:35:50,710

 

1101
00:35:50,710 --> 00:35:52,310

 

1102
00:35:52,310 --> 00:35:52,320

 

1103
00:35:52,320 --> 00:35:53,349


1104
00:35:53,349 --> 00:35:56,310

 

1105
00:35:56,310 --> 00:35:58,790

 

1106
00:35:58,790 --> 00:36:00,470

 

1107
00:36:00,470 --> 00:36:02,390

 

1108
00:36:02,390 --> 00:36:02,400

 

1109
00:36:02,400 --> 00:36:03,990


1110
00:36:03,990 --> 00:36:07,430

 

1111
00:36:07,430 --> 00:36:09,670

 

1112
00:36:09,670 --> 00:36:11,990

 

1113
00:36:11,990 --> 00:36:14,069

 

1114
00:36:14,069 --> 00:36:16,230

 

1115
00:36:16,230 --> 00:36:17,349

 

1116
00:36:17,349 --> 00:36:19,910

 

1117
00:36:19,910 --> 00:36:21,430

 

1118
00:36:21,430 --> 00:36:23,990

 

1119
00:36:23,990 --> 00:36:26,150

 

1120
00:36:26,150 --> 00:36:28,710

 

1121
00:36:28,710 --> 00:36:29,990

 

1122
00:36:29,990 --> 00:36:32,630

 

1123
00:36:32,630 --> 00:36:33,990

 

1124
00:36:33,990 --> 00:36:34,000

 

1125
00:36:34,000 --> 00:36:35,190


1126
00:36:35,190 --> 00:36:38,950

 

1127
00:36:38,950 --> 00:36:41,030

 

1128
00:36:41,030 --> 00:36:43,109

 

1129
00:36:43,109 --> 00:36:45,109

 

1130
00:36:45,109 --> 00:36:46,550

 

1131
00:36:46,550 --> 00:36:49,190

 

1132
00:36:49,190 --> 00:36:51,270

 

1133
00:36:51,270 --> 00:36:53,270

 

1134
00:36:53,270 --> 00:36:54,790

 

1135
00:36:54,790 --> 00:36:54,800

 

1136
00:36:54,800 --> 00:36:55,910


1137
00:36:55,910 --> 00:36:59,190

 

1138
00:36:59,190 --> 00:37:01,030

 

1139
00:37:01,030 --> 00:37:03,349

 

1140
00:37:03,349 --> 00:37:05,510

 

1141
00:37:05,510 --> 00:37:08,230

 

1142
00:37:08,230 --> 00:37:10,390

 

1143
00:37:10,390 --> 00:37:11,589

 

1144
00:37:11,589 --> 00:37:14,310

 

1145
00:37:14,310 --> 00:37:16,710

 

1146
00:37:16,710 --> 00:37:18,310

 

1147
00:37:18,310 --> 00:37:20,390

 

1148
00:37:20,390 --> 00:37:23,510

 

1149
00:37:23,510 --> 00:37:23,520

 

1150
00:37:23,520 --> 00:37:24,390


1151
00:37:24,390 --> 00:37:26,150

 

1152
00:37:26,150 --> 00:37:28,630

 

1153
00:37:28,630 --> 00:37:30,950

 

1154
00:37:30,950 --> 00:37:33,270

 

1155
00:37:33,270 --> 00:37:35,750

 

1156
00:37:35,750 --> 00:37:38,550

 

1157
00:37:38,550 --> 00:37:38,560

 

1158
00:37:38,560 --> 00:37:39,589


1159
00:37:39,589 --> 00:37:41,990

 

1160
00:37:41,990 --> 00:37:43,510

 

1161
00:37:43,510 --> 00:37:45,190

 

1162
00:37:45,190 --> 00:37:48,310

 

1163
00:37:48,310 --> 00:37:50,150

 

1164
00:37:50,150 --> 00:37:53,510

 

1165
00:37:53,510 --> 00:37:55,430

 

1166
00:37:55,430 --> 00:37:58,310

 

1167
00:37:58,310 --> 00:37:58,320

 

1168
00:37:58,320 --> 00:37:59,190


1169
00:37:59,190 --> 00:38:01,910

 

1170
00:38:01,910 --> 00:38:04,390

 

1171
00:38:04,390 --> 00:38:06,710

 

1172
00:38:06,710 --> 00:38:07,670

 

1173
00:38:07,670 --> 00:38:07,680

 

1174
00:38:07,680 --> 00:38:08,550


1175
00:38:08,550 --> 00:38:11,589

 

1176
00:38:11,589 --> 00:38:13,670

 

1177
00:38:13,670 --> 00:38:15,190

 

1178
00:38:15,190 --> 00:38:15,200

 

1179
00:38:15,200 --> 00:38:18,550


1180
00:38:18,550 --> 00:38:25,349

 

1181
00:38:25,349 --> 00:38:28,790

 

1182
00:38:28,790 --> 00:38:31,109

 

1183
00:38:31,109 --> 00:38:34,069

 

1184
00:38:34,069 --> 00:38:34,079

 

1185
00:38:34,079 --> 00:38:35,270


1186
00:38:35,270 --> 00:38:38,310

 

1187
00:38:38,310 --> 00:38:40,310

 

1188
00:38:40,310 --> 00:38:42,550

 

1189
00:38:42,550 --> 00:38:42,560

 

1190
00:38:42,560 --> 00:38:43,430


1191
00:38:43,430 --> 00:38:45,349

 

1192
00:38:45,349 --> 00:38:47,190

 

1193
00:38:47,190 --> 00:38:58,829

 

1194
00:38:58,829 --> 00:39:00,950

 

1195
00:39:00,950 --> 00:39:04,630

 

1196
00:39:04,630 --> 00:39:07,190

 

1197
00:39:07,190 --> 00:39:07,200

 

1198
00:39:07,200 --> 00:39:09,030


1199
00:39:09,030 --> 00:39:09,040

 

1200
00:39:09,040 --> 00:39:11,430


1201
00:39:11,430 --> 00:39:16,069

 

1202
00:39:16,069 --> 00:39:18,310

 

1203
00:39:18,310 --> 00:39:18,320

 

1204
00:39:18,320 --> 00:39:19,430


1205
00:39:19,430 --> 00:39:19,440

 

1206
00:39:19,440 --> 00:39:24,630


1207
00:39:24,630 --> 00:39:27,190

 

1208
00:39:27,190 --> 00:39:32,870

 

1209
00:39:32,870 --> 00:39:32,880

 

1210
00:39:32,880 --> 00:39:42,150


1211
00:39:42,150 --> 00:39:42,160

 

1212
00:39:42,160 --> 00:39:43,030


1213
00:39:43,030 --> 00:39:44,790

 

1214
00:39:44,790 --> 00:39:46,390

 

1215
00:39:46,390 --> 00:39:46,400

 

1216
00:39:46,400 --> 00:39:47,510


1217
00:39:47,510 --> 00:39:49,349

 

1218
00:39:49,349 --> 00:39:51,750

 

1219
00:39:51,750 --> 00:39:53,910

 

1220
00:39:53,910 --> 00:39:53,920

 

1221
00:39:53,920 --> 00:39:55,190


1222
00:39:55,190 --> 00:39:57,430

 

1223
00:39:57,430 --> 00:39:59,829

 

1224
00:39:59,829 --> 00:40:02,230

 

1225
00:40:02,230 --> 00:40:04,550

 

1226
00:40:04,550 --> 00:40:06,309

 

1227
00:40:06,309 --> 00:40:08,630

 

1228
00:40:08,630 --> 00:40:10,470

 

1229
00:40:10,470 --> 00:40:12,790

 

1230
00:40:12,790 --> 00:40:15,030

 

1231
00:40:15,030 --> 00:40:15,040

 

1232
00:40:15,040 --> 00:40:16,069


1233
00:40:16,069 --> 00:40:18,390

 

1234
00:40:18,390 --> 00:40:21,750

 

1235
00:40:21,750 --> 00:40:25,190

 

1236
00:40:25,190 --> 00:40:26,790

 

1237
00:40:26,790 --> 00:40:29,510

 

1238
00:40:29,510 --> 00:40:29,520

 

1239
00:40:29,520 --> 00:40:30,950


1240
00:40:30,950 --> 00:40:32,309

 

1241
00:40:32,309 --> 00:40:34,550

 

1242
00:40:34,550 --> 00:40:36,550

 

1243
00:40:36,550 --> 00:40:38,870

 

1244
00:40:38,870 --> 00:40:40,470

 

1245
00:40:40,470 --> 00:40:40,480

 

1246
00:40:40,480 --> 00:40:43,270


1247
00:40:43,270 --> 00:40:46,309

 

1248
00:40:46,309 --> 00:40:49,030

 

1249
00:40:49,030 --> 00:40:50,870

 

1250
00:40:50,870 --> 00:40:53,109

 

1251
00:40:53,109 --> 00:40:57,829

 

1252
00:40:57,829 --> 00:40:57,839

 

1253
00:40:57,839 --> 00:40:59,190


1254
00:40:59,190 --> 00:41:01,109

 

1255
00:41:01,109 --> 00:41:03,589

 

1256
00:41:03,589 --> 00:41:05,349

 

1257
00:41:05,349 --> 00:41:07,190

 

1258
00:41:07,190 --> 00:41:10,309

 

1259
00:41:10,309 --> 00:41:13,589

 

1260
00:41:13,589 --> 00:41:15,510

 

1261
00:41:15,510 --> 00:41:18,390

 

1262
00:41:18,390 --> 00:41:19,589

 

1263
00:41:19,589 --> 00:41:21,829

 

1264
00:41:21,829 --> 00:41:23,270

 

1265
00:41:23,270 --> 00:41:25,589

 

1266
00:41:25,589 --> 00:41:27,349

 

1267
00:41:27,349 --> 00:41:29,190

 

1268
00:41:29,190 --> 00:41:31,910

 

1269
00:41:31,910 --> 00:41:34,150

 

1270
00:41:34,150 --> 00:41:35,829

 

1271
00:41:35,829 --> 00:41:35,839

 

1272
00:41:35,839 --> 00:41:36,710


1273
00:41:36,710 --> 00:41:36,720

 

1274
00:41:36,720 --> 00:41:37,750


1275
00:41:37,750 --> 00:41:41,030

 

1276
00:41:41,030 --> 00:41:43,270

 

1277
00:41:43,270 --> 00:41:45,670

 

1278
00:41:45,670 --> 00:41:48,390

 

1279
00:41:48,390 --> 00:41:50,069

 

1280
00:41:50,069 --> 00:41:51,430

 

1281
00:41:51,430 --> 00:41:53,510

 

1282
00:41:53,510 --> 00:41:55,910

 

1283
00:41:55,910 --> 00:41:58,870

 

1284
00:41:58,870 --> 00:42:00,710

 

1285
00:42:00,710 --> 00:42:03,990

 

1286
00:42:03,990 --> 00:42:08,470

 

1287
00:42:08,470 --> 00:42:10,470

 

1288
00:42:10,470 --> 00:42:13,349

 

1289
00:42:13,349 --> 00:42:15,109

 

1290
00:42:15,109 --> 00:42:19,829

 

1291
00:42:19,829 --> 00:42:19,839

 

1292
00:42:19,839 --> 00:42:20,630


1293
00:42:20,630 --> 00:42:22,950

 

1294
00:42:22,950 --> 00:42:25,190

 

1295
00:42:25,190 --> 00:42:28,309

 

1296
00:42:28,309 --> 00:42:30,390

 

1297
00:42:30,390 --> 00:42:31,990

 

1298
00:42:31,990 --> 00:42:32,950

 

1299
00:42:32,950 --> 00:42:34,790

 

1300
00:42:34,790 --> 00:42:37,829

 

1301
00:42:37,829 --> 00:42:37,839

 

1302
00:42:37,839 --> 00:42:40,470


1303
00:42:40,470 --> 00:42:40,480

 

1304
00:42:40,480 --> 00:42:42,829


1305
00:42:42,829 --> 00:42:49,990

 

1306
00:42:49,990 --> 00:42:50,000

 

1307
00:42:50,000 --> 00:42:50,790


1308
00:42:50,790 --> 00:42:53,190

 

1309
00:42:53,190 --> 00:42:53,200

 

1310
00:42:53,200 --> 00:42:54,550


1311
00:42:54,550 --> 00:42:56,230

 

1312
00:42:56,230 --> 00:42:59,030

 

1313
00:42:59,030 --> 00:43:00,790

 

1314
00:43:00,790 --> 00:43:02,309

 

1315
00:43:02,309 --> 00:43:04,870

 

1316
00:43:04,870 --> 00:43:04,880

 

1317
00:43:04,880 --> 00:43:06,790


1318
00:43:06,790 --> 00:43:09,349

 

1319
00:43:09,349 --> 00:43:11,349

 

1320
00:43:11,349 --> 00:43:13,670

 

1321
00:43:13,670 --> 00:43:16,069

 

1322
00:43:16,069 --> 00:43:18,870

 

1323
00:43:18,870 --> 00:43:21,670

 

1324
00:43:21,670 --> 00:43:21,680

 

1325
00:43:21,680 --> 00:43:23,190


1326
00:43:23,190 --> 00:43:25,030

 

1327
00:43:25,030 --> 00:43:26,390

 

1328
00:43:26,390 --> 00:43:28,630

 

1329
00:43:28,630 --> 00:43:28,640

 

1330
00:43:28,640 --> 00:43:29,750


1331
00:43:29,750 --> 00:43:30,950

 

1332
00:43:30,950 --> 00:43:34,470

 

1333
00:43:34,470 --> 00:43:36,710

 

1334
00:43:36,710 --> 00:43:38,550

 

1335
00:43:38,550 --> 00:43:42,150

 

1336
00:43:42,150 --> 00:43:50,630

 

1337
00:43:50,630 --> 00:43:52,710

 

1338
00:43:52,710 --> 00:43:54,309

 

1339
00:43:54,309 --> 00:43:56,470

 

1340
00:43:56,470 --> 00:43:56,480

 

1341
00:43:56,480 --> 00:43:57,670


1342
00:43:57,670 --> 00:43:59,349

 

1343
00:43:59,349 --> 00:43:59,359

 

1344
00:43:59,359 --> 00:44:00,550


1345
00:44:00,550 --> 00:44:03,030

 

1346
00:44:03,030 --> 00:44:05,190

 

1347
00:44:05,190 --> 00:44:06,470

 

1348
00:44:06,470 --> 00:44:10,069

 

1349
00:44:10,069 --> 00:44:10,079

 

1350
00:44:10,079 --> 00:44:11,910


1351
00:44:11,910 --> 00:44:13,270

 

1352
00:44:13,270 --> 00:44:15,030

 

1353
00:44:15,030 --> 00:44:17,270

 

1354
00:44:17,270 --> 00:44:19,270

 

1355
00:44:19,270 --> 00:44:21,190

 

1356
00:44:21,190 --> 00:44:24,069

 

1357
00:44:24,069 --> 00:44:25,829

 

1358
00:44:25,829 --> 00:44:27,910

 

1359
00:44:27,910 --> 00:44:29,829

 

1360
00:44:29,829 --> 00:44:32,870

 

1361
00:44:32,870 --> 00:44:36,630

 

1362
00:44:36,630 --> 00:44:39,030

 

1363
00:44:39,030 --> 00:44:40,870

 

1364
00:44:40,870 --> 00:44:40,880

 

1365
00:44:40,880 --> 00:44:42,710


1366
00:44:42,710 --> 00:44:42,720

 

1367
00:44:42,720 --> 00:44:45,349


1368
00:44:45,349 --> 00:44:47,109

 

1369
00:44:47,109 --> 00:44:47,119

 

1370
00:44:47,119 --> 00:44:48,550


1371
00:44:48,550 --> 00:44:52,069

 

1372
00:44:52,069 --> 00:44:52,079

 

1373
00:44:52,079 --> 00:44:53,349


1374
00:44:53,349 --> 00:44:53,359

 

1375
00:44:53,359 --> 00:44:59,430


1376
00:44:59,430 --> 00:45:01,109

 

1377
00:45:01,109 --> 00:45:01,119

 

1378
00:45:01,119 --> 00:45:02,069


1379
00:45:02,069 --> 00:45:02,079

 

1380
00:45:02,079 --> 00:45:02,950


1381
00:45:02,950 --> 00:45:05,190

 

1382
00:45:05,190 --> 00:45:06,309

 

1383
00:45:06,309 --> 00:45:10,150

 

1384
00:45:10,150 --> 00:45:10,160

 

1385
00:45:10,160 --> 00:45:12,069


1386
00:45:12,069 --> 00:45:14,390

 

1387
00:45:14,390 --> 00:45:16,710

 

1388
00:45:16,710 --> 00:45:18,710

 

1389
00:45:18,710 --> 00:45:20,710

 

1390
00:45:20,710 --> 00:45:22,550

 

1391
00:45:22,550 --> 00:45:25,910

 

1392
00:45:25,910 --> 00:45:27,750

 

1393
00:45:27,750 --> 00:45:29,510

 

1394
00:45:29,510 --> 00:45:32,470

 

1395
00:45:32,470 --> 00:45:34,470

 

1396
00:45:34,470 --> 00:45:37,109

 

1397
00:45:37,109 --> 00:45:38,630

 

1398
00:45:38,630 --> 00:45:40,950

 

1399
00:45:40,950 --> 00:45:42,710

 

1400
00:45:42,710 --> 00:45:46,069

 

1401
00:45:46,069 --> 00:45:47,990

 

1402
00:45:47,990 --> 00:45:50,230

 

1403
00:45:50,230 --> 00:45:51,270

 

1404
00:45:51,270 --> 00:45:52,550

 

1405
00:45:52,550 --> 00:45:54,390

 

1406
00:45:54,390 --> 00:45:54,400

 

1407
00:45:54,400 --> 00:45:55,910


1408
00:45:55,910 --> 00:45:57,589

 

1409
00:45:57,589 --> 00:45:59,670

 

1410
00:45:59,670 --> 00:46:01,910

 

1411
00:46:01,910 --> 00:46:04,309

 

1412
00:46:04,309 --> 00:46:04,319

 

1413
00:46:04,319 --> 00:46:05,190


1414
00:46:05,190 --> 00:46:07,990

 

1415
00:46:07,990 --> 00:46:10,150

 

1416
00:46:10,150 --> 00:46:12,150

 

1417
00:46:12,150 --> 00:46:14,630

 

1418
00:46:14,630 --> 00:46:16,069

 

1419
00:46:16,069 --> 00:46:18,150

 

1420
00:46:18,150 --> 00:46:20,710

 

1421
00:46:20,710 --> 00:46:21,990

 

1422
00:46:21,990 --> 00:46:23,430

 

1423
00:46:23,430 --> 00:46:25,910

 

1424
00:46:25,910 --> 00:46:28,230

 

1425
00:46:28,230 --> 00:46:30,069

 

1426
00:46:30,069 --> 00:46:32,230

 

1427
00:46:32,230 --> 00:46:32,240

 

1428
00:46:32,240 --> 00:46:33,670


1429
00:46:33,670 --> 00:46:33,680

 

1430
00:46:33,680 --> 00:46:34,790


1431
00:46:34,790 --> 00:46:36,870

 

1432
00:46:36,870 --> 00:46:39,190

 

1433
00:46:39,190 --> 00:46:41,270

 

1434
00:46:41,270 --> 00:46:43,109

 

1435
00:46:43,109 --> 00:46:45,510

 

1436
00:46:45,510 --> 00:46:47,910

 

1437
00:46:47,910 --> 00:46:49,910

 

1438
00:46:49,910 --> 00:46:53,510

 

1439
00:46:53,510 --> 00:46:56,230

 

1440
00:46:56,230 --> 00:46:57,990

 

1441
00:46:57,990 --> 00:47:00,870

 

1442
00:47:00,870 --> 00:47:03,190

 

1443
00:47:03,190 --> 00:47:03,200

 

1444
00:47:03,200 --> 00:47:04,309


1445
00:47:04,309 --> 00:47:06,630

 

1446
00:47:06,630 --> 00:47:08,150

 

1447
00:47:08,150 --> 00:47:09,750

 

1448
00:47:09,750 --> 00:47:13,109

 

1449
00:47:13,109 --> 00:47:13,119

 

1450
00:47:13,119 --> 00:47:16,550


1451
00:47:16,550 --> 00:47:16,560

 

1452
00:47:16,560 --> 00:47:17,349


1453
00:47:17,349 --> 00:47:19,349

 

1454
00:47:19,349 --> 00:47:21,349

 

1455
00:47:21,349 --> 00:47:23,430

 

1456
00:47:23,430 --> 00:47:25,349

 

1457
00:47:25,349 --> 00:47:27,589

 

1458
00:47:27,589 --> 00:47:30,309

 

1459
00:47:30,309 --> 00:47:32,150

 

1460
00:47:32,150 --> 00:47:34,309

 

1461
00:47:34,309 --> 00:47:36,150

 

1462
00:47:36,150 --> 00:47:36,160

 

1463
00:47:36,160 --> 00:47:39,990


1464
00:47:39,990 --> 00:47:41,990

 

1465
00:47:41,990 --> 00:47:43,510

 

1466
00:47:43,510 --> 00:47:45,589

 

1467
00:47:45,589 --> 00:47:47,190

 

1468
00:47:47,190 --> 00:47:49,349

 

1469
00:47:49,349 --> 00:47:52,230

 

1470
00:47:52,230 --> 00:47:54,630

 

1471
00:47:54,630 --> 00:47:55,910

 

1472
00:47:55,910 --> 00:47:55,920

 

1473
00:47:55,920 --> 00:47:57,190


1474
00:47:57,190 --> 00:47:59,670

 

1475
00:47:59,670 --> 00:48:01,670

 

1476
00:48:01,670 --> 00:48:05,510

 

1477
00:48:05,510 --> 00:48:08,950

 

1478
00:48:08,950 --> 00:48:11,349

 

1479
00:48:11,349 --> 00:48:13,670

 

1480
00:48:13,670 --> 00:48:16,549

 

1481
00:48:16,549 --> 00:48:18,069

 

1482
00:48:18,069 --> 00:48:18,079

 

1483
00:48:18,079 --> 00:48:21,589


1484
00:48:21,589 --> 00:48:21,599

 

1485
00:48:21,599 --> 00:48:22,470


1486
00:48:22,470 --> 00:48:26,069

 

1487
00:48:26,069 --> 00:48:27,670

 

1488
00:48:27,670 --> 00:48:29,510

 

1489
00:48:29,510 --> 00:48:33,670

 

1490
00:48:33,670 --> 00:48:35,829

 

1491
00:48:35,829 --> 00:48:37,829

 

1492
00:48:37,829 --> 00:48:39,430

 

1493
00:48:39,430 --> 00:48:41,589

 

1494
00:48:41,589 --> 00:48:43,910

 

1495
00:48:43,910 --> 00:48:46,309

 

1496
00:48:46,309 --> 00:48:48,230

 

1497
00:48:48,230 --> 00:48:50,390

 

1498
00:48:50,390 --> 00:48:52,549

 

1499
00:48:52,549 --> 00:48:54,790

 

1500
00:48:54,790 --> 00:48:55,829

 

1501
00:48:55,829 --> 00:48:57,829

 

1502
00:48:57,829 --> 00:48:57,839

 

1503
00:48:57,839 --> 00:48:58,950


1504
00:48:58,950 --> 00:49:00,710

 

1505
00:49:00,710 --> 00:49:03,270

 

1506
00:49:03,270 --> 00:49:05,349

 

1507
00:49:05,349 --> 00:49:07,349

 

1508
00:49:07,349 --> 00:49:09,430

 

1509
00:49:09,430 --> 00:49:11,990

 

1510
00:49:11,990 --> 00:49:13,910

 

1511
00:49:13,910 --> 00:49:15,109

 

1512
00:49:15,109 --> 00:49:19,109

 

1513
00:49:19,109 --> 00:49:20,950

 

1514
00:49:20,950 --> 00:49:22,549

 

1515
00:49:22,549 --> 00:49:24,390

 

1516
00:49:24,390 --> 00:49:26,710

 

1517
00:49:26,710 --> 00:49:28,870

 

1518
00:49:28,870 --> 00:49:29,829

 

1519
00:49:29,829 --> 00:49:29,839

 

1520
00:49:29,839 --> 00:49:31,910


1521
00:49:31,910 --> 00:49:33,510

 

1522
00:49:33,510 --> 00:49:34,870

 

1523
00:49:34,870 --> 00:49:34,880

 

1524
00:49:34,880 --> 00:49:35,990


1525
00:49:35,990 --> 00:49:36,000

 

1526
00:49:36,000 --> 00:49:37,109


1527
00:49:37,109 --> 00:49:37,119

 

1528
00:49:37,119 --> 00:49:40,790


1529
00:49:40,790 --> 00:49:42,829

 

1530
00:49:42,829 --> 00:49:46,870

 

1531
00:49:46,870 --> 00:49:49,349

 

1532
00:49:49,349 --> 00:49:51,510

 

1533
00:49:51,510 --> 00:49:51,520

 

1534
00:49:51,520 --> 00:49:53,109


1535
00:49:53,109 --> 00:49:54,790

 

1536
00:49:54,790 --> 00:49:57,270

 

1537
00:49:57,270 --> 00:49:59,750

 

1538
00:49:59,750 --> 00:50:01,430

 

1539
00:50:01,430 --> 00:50:03,430

 

1540
00:50:03,430 --> 00:50:03,440

 

1541
00:50:03,440 --> 00:50:05,349


1542
00:50:05,349 --> 00:50:07,349

 

1543
00:50:07,349 --> 00:50:09,030

 

1544
00:50:09,030 --> 00:50:10,710

 

1545
00:50:10,710 --> 00:50:13,670

 

1546
00:50:13,670 --> 00:50:13,680

 

1547
00:50:13,680 --> 00:50:14,870


1548
00:50:14,870 --> 00:50:16,630

 

1549
00:50:16,630 --> 00:50:18,470

 

1550
00:50:18,470 --> 00:50:21,109

 

1551
00:50:21,109 --> 00:50:22,630

 

1552
00:50:22,630 --> 00:50:22,640

 

1553
00:50:22,640 --> 00:50:23,910


1554
00:50:23,910 --> 00:50:24,950

 

1555
00:50:24,950 --> 00:50:27,190

 

1556
00:50:27,190 --> 00:50:28,549

 

1557
00:50:28,549 --> 00:50:30,710

 

1558
00:50:30,710 --> 00:50:32,790

 

1559
00:50:32,790 --> 00:50:35,190

 

1560
00:50:35,190 --> 00:50:35,200

 

1561
00:50:35,200 --> 00:50:38,829


1562
00:50:38,829 --> 00:50:41,670

 

1563
00:50:41,670 --> 00:50:43,750

 

1564
00:50:43,750 --> 00:50:47,510

 

1565
00:50:47,510 --> 00:50:49,670

 

1566
00:50:49,670 --> 00:50:51,910

 

1567
00:50:51,910 --> 00:50:51,920

 

1568
00:50:51,920 --> 00:50:53,190


1569
00:50:53,190 --> 00:50:55,030

 

1570
00:50:55,030 --> 00:50:57,430

 

1571
00:50:57,430 --> 00:50:59,109

 

1572
00:50:59,109 --> 00:51:00,870

 

1573
00:51:00,870 --> 00:51:00,880

 

1574
00:51:00,880 --> 00:51:02,150


1575
00:51:02,150 --> 00:51:04,069

 

1576
00:51:04,069 --> 00:51:06,230

 

1577
00:51:06,230 --> 00:51:08,710

 

1578
00:51:08,710 --> 00:51:08,720

 

1579
00:51:08,720 --> 00:51:10,309


1580
00:51:10,309 --> 00:51:11,589

 

1581
00:51:11,589 --> 00:51:13,990

 

1582
00:51:13,990 --> 00:51:15,829

 

1583
00:51:15,829 --> 00:51:17,990

 

1584
00:51:17,990 --> 00:51:20,150

 

1585
00:51:20,150 --> 00:51:22,230

 

1586
00:51:22,230 --> 00:51:23,990

 

1587
00:51:23,990 --> 00:51:27,750

 

1588
00:51:27,750 --> 00:51:30,950

 

1589
00:51:30,950 --> 00:51:30,960

 

1590
00:51:30,960 --> 00:51:31,910


1591
00:51:31,910 --> 00:51:33,829

 

1592
00:51:33,829 --> 00:51:35,109

 

1593
00:51:35,109 --> 00:51:37,670

 

1594
00:51:37,670 --> 00:51:39,270

 

1595
00:51:39,270 --> 00:51:40,950

 

1596
00:51:40,950 --> 00:51:44,470

 

1597
00:51:44,470 --> 00:51:46,470

 

1598
00:51:46,470 --> 00:51:48,309

 

1599
00:51:48,309 --> 00:51:48,319

 

1600
00:51:48,319 --> 00:51:49,109


1601
00:51:49,109 --> 00:51:51,589

 

1602
00:51:51,589 --> 00:51:53,589

 

1603
00:51:53,589 --> 00:51:54,710

 

1604
00:51:54,710 --> 00:51:57,750

 

1605
00:51:57,750 --> 00:51:59,990

 

1606
00:51:59,990 --> 00:52:01,750

 

1607
00:52:01,750 --> 00:52:01,760

 

1608
00:52:01,760 --> 00:52:03,109


1609
00:52:03,109 --> 00:52:04,710

 

1610
00:52:04,710 --> 00:52:07,510

 

1611
00:52:07,510 --> 00:52:08,950

 

1612
00:52:08,950 --> 00:52:10,230

 

1613
00:52:10,230 --> 00:52:11,510

 

1614
00:52:11,510 --> 00:52:11,520

 

1615
00:52:11,520 --> 00:52:12,950


1616
00:52:12,950 --> 00:52:15,750

 

1617
00:52:15,750 --> 00:52:19,030

 

1618
00:52:19,030 --> 00:52:21,109

 

1619
00:52:21,109 --> 00:52:23,190

 

1620
00:52:23,190 --> 00:52:23,200

 

1621
00:52:23,200 --> 00:52:23,910


1622
00:52:23,910 --> 00:52:25,829

 

1623
00:52:25,829 --> 00:52:25,839

 

1624
00:52:25,839 --> 00:52:26,870


1625
00:52:26,870 --> 00:52:28,069

 

1626
00:52:28,069 --> 00:52:30,950

 

1627
00:52:30,950 --> 00:52:34,069

 

1628
00:52:34,069 --> 00:52:36,790

 

1629
00:52:36,790 --> 00:52:39,190

 

1630
00:52:39,190 --> 00:52:39,200

 

1631
00:52:39,200 --> 00:52:40,390


1632
00:52:40,390 --> 00:52:42,230

 

1633
00:52:42,230 --> 00:52:44,630

 

1634
00:52:44,630 --> 00:52:46,549

 

1635
00:52:46,549 --> 00:52:48,710

 

1636
00:52:48,710 --> 00:52:52,230

 

1637
00:52:52,230 --> 00:52:52,240

 

1638
00:52:52,240 --> 00:52:55,589


1639
00:52:55,589 --> 00:52:55,599

 

1640
00:52:55,599 --> 00:52:56,790


1641
00:52:56,790 --> 00:52:59,270

 

1642
00:52:59,270 --> 00:53:00,870

 

1643
00:53:00,870 --> 00:53:03,030

 

1644
00:53:03,030 --> 00:53:04,950

 

1645
00:53:04,950 --> 00:53:07,349

 

1646
00:53:07,349 --> 00:53:09,510

 

1647
00:53:09,510 --> 00:53:13,109

 

1648
00:53:13,109 --> 00:53:16,069

 

1649
00:53:16,069 --> 00:53:17,750

 

1650
00:53:17,750 --> 00:53:19,829

 

1651
00:53:19,829 --> 00:53:21,750

 

1652
00:53:21,750 --> 00:53:24,390

 

1653
00:53:24,390 --> 00:53:26,790

 

1654
00:53:26,790 --> 00:53:28,870

 

1655
00:53:28,870 --> 00:53:31,030

 

1656
00:53:31,030 --> 00:53:33,190

 

1657
00:53:33,190 --> 00:53:34,470

 

1658
00:53:34,470 --> 00:53:36,230

 

1659
00:53:36,230 --> 00:53:37,990

 

1660
00:53:37,990 --> 00:53:40,230

 

1661
00:53:40,230 --> 00:53:40,240

 

1662
00:53:40,240 --> 00:53:41,750


1663
00:53:41,750 --> 00:53:41,760

 

1664
00:53:41,760 --> 00:53:43,030


1665
00:53:43,030 --> 00:53:46,390

 

1666
00:53:46,390 --> 00:53:48,470

 

1667
00:53:48,470 --> 00:53:50,710

 

1668
00:53:50,710 --> 00:53:52,870

 

1669
00:53:52,870 --> 00:53:56,630

 

1670
00:53:56,630 --> 00:53:56,640

 

1671
00:53:56,640 --> 00:53:59,030


1672
00:53:59,030 --> 00:53:59,040

 

1673
00:53:59,040 --> 00:54:02,069


1674
00:54:02,069 --> 00:54:02,079

 

1675
00:54:02,079 --> 00:54:03,510


1676
00:54:03,510 --> 00:54:04,950

 

1677
00:54:04,950 --> 00:54:07,190

 

1678
00:54:07,190 --> 00:54:07,200

 

1679
00:54:07,200 --> 00:54:08,309


1680
00:54:08,309 --> 00:54:10,790

 

1681
00:54:10,790 --> 00:54:12,390

 

1682
00:54:12,390 --> 00:54:13,750

 

1683
00:54:13,750 --> 00:54:15,589

 

1684
00:54:15,589 --> 00:54:17,190

 

1685
00:54:17,190 --> 00:54:19,430

 

1686
00:54:19,430 --> 00:54:21,270

 

1687
00:54:21,270 --> 00:54:23,670

 

1688
00:54:23,670 --> 00:54:23,680

 

1689
00:54:23,680 --> 00:54:24,549


1690
00:54:24,549 --> 00:54:26,069

 

1691
00:54:26,069 --> 00:54:26,079

 

1692
00:54:26,079 --> 00:54:28,150


1693
00:54:28,150 --> 00:54:30,390

 

1694
00:54:30,390 --> 00:54:30,400

 

1695
00:54:30,400 --> 00:54:32,710


1696
00:54:32,710 --> 00:54:35,030

 

1697
00:54:35,030 --> 00:54:36,549

 

1698
00:54:36,549 --> 00:54:40,230

 

1699
00:54:40,230 --> 00:54:41,670

 

1700
00:54:41,670 --> 00:54:44,390

 

1701
00:54:44,390 --> 00:54:46,230

 

1702
00:54:46,230 --> 00:54:47,829

 

1703
00:54:47,829 --> 00:54:51,510

 

1704
00:54:51,510 --> 00:54:53,349

 

1705
00:54:53,349 --> 00:54:55,190

 

1706
00:54:55,190 --> 00:54:57,829

 

1707
00:54:57,829 --> 00:54:59,829

 

1708
00:54:59,829 --> 00:55:01,270

 

1709
00:55:01,270 --> 00:55:03,270

 

1710
00:55:03,270 --> 00:55:05,109

 

1711
00:55:05,109 --> 00:55:06,710

 

1712
00:55:06,710 --> 00:55:08,069

 

1713
00:55:08,069 --> 00:55:09,670

 

1714
00:55:09,670 --> 00:55:11,589

 

1715
00:55:11,589 --> 00:55:13,589

 

1716
00:55:13,589 --> 00:55:15,750

 

1717
00:55:15,750 --> 00:55:17,510

 

1718
00:55:17,510 --> 00:55:20,789

 

1719
00:55:20,789 --> 00:55:20,799

 

1720
00:55:20,799 --> 00:55:23,109


1721
00:55:23,109 --> 00:55:23,119

 

1722
00:55:23,119 --> 00:55:23,910


1723
00:55:23,910 --> 00:55:23,920

 

1724
00:55:23,920 --> 00:55:25,349


1725
00:55:25,349 --> 00:55:25,359

 

1726
00:55:25,359 --> 00:55:26,390


1727
00:55:26,390 --> 00:55:29,030

 

1728
00:55:29,030 --> 00:55:31,990

 

1729
00:55:31,990 --> 00:55:34,710

 

1730
00:55:34,710 --> 00:55:37,270

 

1731
00:55:37,270 --> 00:55:39,750

 

1732
00:55:39,750 --> 00:55:41,430

 

1733
00:55:41,430 --> 00:55:43,349

 

1734
00:55:43,349 --> 00:55:45,910

 

1735
00:55:45,910 --> 00:55:47,349

 

1736
00:55:47,349 --> 00:55:47,359

 

1737
00:55:47,359 --> 00:55:48,950


1738
00:55:48,950 --> 00:55:48,960

 

1739
00:55:48,960 --> 00:55:49,910


1740
00:55:49,910 --> 00:55:51,990

 

1741
00:55:51,990 --> 00:55:55,190

 

1742
00:55:55,190 --> 00:55:57,990

 

1743
00:55:57,990 --> 00:56:00,549

 

1744
00:56:00,549 --> 00:56:02,069

 

1745
00:56:02,069 --> 00:56:02,079

 

1746
00:56:02,079 --> 00:56:03,190


1747
00:56:03,190 --> 00:56:04,710

 

1748
00:56:04,710 --> 00:56:06,630

 

1749
00:56:06,630 --> 00:56:08,309

 

1750
00:56:08,309 --> 00:56:11,270

 

1751
00:56:11,270 --> 00:56:12,390

 

1752
00:56:12,390 --> 00:56:13,750

 

1753
00:56:13,750 --> 00:56:15,670

 

1754
00:56:15,670 --> 00:56:18,470

 

1755
00:56:18,470 --> 00:56:20,390

 

1756
00:56:20,390 --> 00:56:22,470

 

1757
00:56:22,470 --> 00:56:26,309

 

1758
00:56:26,309 --> 00:56:27,829

 

1759
00:56:27,829 --> 00:56:30,390

 

1760
00:56:30,390 --> 00:56:34,309

 

1761
00:56:34,309 --> 00:56:35,990

 

1762
00:56:35,990 --> 00:56:38,150

 

1763
00:56:38,150 --> 00:56:40,789

 

1764
00:56:40,789 --> 00:56:43,510

 

1765
00:56:43,510 --> 00:56:46,630

 

1766
00:56:46,630 --> 00:56:48,789

 

1767
00:56:48,789 --> 00:56:51,829

 

1768
00:56:51,829 --> 00:56:54,789

 

1769
00:56:54,789 --> 00:56:56,150

 

1770
00:56:56,150 --> 00:56:57,589

 

1771
00:56:57,589 --> 00:56:59,750

 

1772
00:56:59,750 --> 00:57:03,109

 

1773
00:57:03,109 --> 00:57:03,119

 

1774
00:57:03,119 --> 00:57:03,990


1775
00:57:03,990 --> 00:57:04,000

 

1776
00:57:04,000 --> 00:57:05,910


1777
00:57:05,910 --> 00:57:08,069

 

1778
00:57:08,069 --> 00:57:09,990

 

1779
00:57:09,990 --> 00:57:13,030

 

1780
00:57:13,030 --> 00:57:15,910

 

1781
00:57:15,910 --> 00:57:17,270

 

1782
00:57:17,270 --> 00:57:18,549

 

1783
00:57:18,549 --> 00:57:20,230

 

1784
00:57:20,230 --> 00:57:22,710

 

1785
00:57:22,710 --> 00:57:26,150

 

1786
00:57:26,150 --> 00:57:27,349

 

1787
00:57:27,349 --> 00:57:28,870

 

1788
00:57:28,870 --> 00:57:31,349

 

1789
00:57:31,349 --> 00:57:34,309

 

1790
00:57:34,309 --> 00:57:36,390

 

1791
00:57:36,390 --> 00:57:37,750

 

1792
00:57:37,750 --> 00:57:39,829

 

1793
00:57:39,829 --> 00:57:42,230

 

1794
00:57:42,230 --> 00:57:43,990

 

1795
00:57:43,990 --> 00:57:45,430

 

1796
00:57:45,430 --> 00:57:47,829

 

1797
00:57:47,829 --> 00:57:50,230

 

1798
00:57:50,230 --> 00:57:51,670

 

1799
00:57:51,670 --> 00:57:53,670

 

1800
00:57:53,670 --> 00:57:56,710

 

1801
00:57:56,710 --> 00:57:59,430

 

1802
00:57:59,430 --> 00:58:00,789

 

1803
00:58:00,789 --> 00:58:00,799

 

1804
00:58:00,799 --> 00:58:02,630


1805
00:58:02,630 --> 00:58:04,630

 

1806
00:58:04,630 --> 00:58:07,190

 

1807
00:58:07,190 --> 00:58:09,270

 

1808
00:58:09,270 --> 00:58:11,109

 

1809
00:58:11,109 --> 00:58:13,430

 

1810
00:58:13,430 --> 00:58:14,870

 

1811
00:58:14,870 --> 00:58:16,789

 

1812
00:58:16,789 --> 00:58:19,030

 

1813
00:58:19,030 --> 00:58:22,710

 

1814
00:58:22,710 --> 00:58:22,720

 

1815
00:58:22,720 --> 00:58:24,870


1816
00:58:24,870 --> 00:58:26,230

 

1817
00:58:26,230 --> 00:58:27,750

 

1818
00:58:27,750 --> 00:58:28,710

 

1819
00:58:28,710 --> 00:58:30,069

 

1820
00:58:30,069 --> 00:58:32,069

 

1821
00:58:32,069 --> 00:58:33,829

 

1822
00:58:33,829 --> 00:58:33,839

 

1823
00:58:33,839 --> 00:58:34,710


1824
00:58:34,710 --> 00:58:36,309

 

1825
00:58:36,309 --> 00:58:39,670

 

1826
00:58:39,670 --> 00:58:41,990

 

1827
00:58:41,990 --> 00:58:45,030

 

1828
00:58:45,030 --> 00:58:45,040

 

1829
00:58:45,040 --> 00:58:46,549


1830
00:58:46,549 --> 00:58:48,470

 

1831
00:58:48,470 --> 00:58:49,910

 

1832
00:58:49,910 --> 00:58:51,270

 

1833
00:58:51,270 --> 00:58:54,150

 

1834
00:58:54,150 --> 00:58:58,549

 

1835
00:58:58,549 --> 00:58:58,559

 

1836
00:58:58,559 --> 00:59:00,230


1837
00:59:00,230 --> 00:59:04,549

 

1838
00:59:04,549 --> 00:59:06,150

 

1839
00:59:06,150 --> 00:59:07,510

 

1840
00:59:07,510 --> 00:59:09,910

 

1841
00:59:09,910 --> 00:59:11,750

 

1842
00:59:11,750 --> 00:59:13,510

 

1843
00:59:13,510 --> 00:59:13,520

 

1844
00:59:13,520 --> 00:59:14,470


1845
00:59:14,470 --> 00:59:16,630

 

1846
00:59:16,630 --> 00:59:18,470

 

1847
00:59:18,470 --> 00:59:20,309

 

1848
00:59:20,309 --> 00:59:21,750

 

1849
00:59:21,750 --> 00:59:23,190

 

1850
00:59:23,190 --> 00:59:25,670

 

1851
00:59:25,670 --> 00:59:27,990

 

1852
00:59:27,990 --> 00:59:29,589

 

1853
00:59:29,589 --> 00:59:31,589

 

1854
00:59:31,589 --> 00:59:34,470

 

1855
00:59:34,470 --> 00:59:37,349

 

1856
00:59:37,349 --> 00:59:40,549

 

1857
00:59:40,549 --> 00:59:40,559

 

1858
00:59:40,559 --> 00:59:41,589


1859
00:59:41,589 --> 00:59:43,670

 

1860
00:59:43,670 --> 00:59:44,950

 

1861
00:59:44,950 --> 00:59:46,870

 

1862
00:59:46,870 --> 00:59:49,430

 

1863
00:59:49,430 --> 00:59:51,510

 

1864
00:59:51,510 --> 00:59:53,109

 

1865
00:59:53,109 --> 00:59:53,119

 

1866
00:59:53,119 --> 00:59:55,030


1867
00:59:55,030 --> 00:59:57,510

 

1868
00:59:57,510 --> 00:59:59,109

 

1869
00:59:59,109 --> 01:00:01,510

 

1870
01:00:01,510 --> 01:00:03,910

 

1871
01:00:03,910 --> 01:00:07,270

 

1872
01:00:07,270 --> 01:00:10,069

 

1873
01:00:10,069 --> 01:00:12,950

 

1874
01:00:12,950 --> 01:00:17,190

 

1875
01:00:17,190 --> 01:00:18,470

 

1876
01:00:18,470 --> 01:00:20,789

 

1877
01:00:20,789 --> 01:00:23,030

 

1878
01:00:23,030 --> 01:00:25,430

 

1879
01:00:25,430 --> 01:00:27,030

 

1880
01:00:27,030 --> 01:00:27,040

 

1881
01:00:27,040 --> 01:00:28,789


1882
01:00:28,789 --> 01:00:28,799

 

1883
01:00:28,799 --> 01:00:30,390


1884
01:00:30,390 --> 01:00:30,400

 

1885
01:00:30,400 --> 01:00:31,430


1886
01:00:31,430 --> 01:00:33,670

 

1887
01:00:33,670 --> 01:00:36,150

 

1888
01:00:36,150 --> 01:00:37,910

 

1889
01:00:37,910 --> 01:00:41,190

 

1890
01:00:41,190 --> 01:00:42,710

 

1891
01:00:42,710 --> 01:00:45,030

 

1892
01:00:45,030 --> 01:00:47,349

 

1893
01:00:47,349 --> 01:00:48,710

 

1894
01:00:48,710 --> 01:00:48,720

 

1895
01:00:48,720 --> 01:00:50,230


1896
01:00:50,230 --> 01:00:51,589

 

1897
01:00:51,589 --> 01:00:53,190

 

1898
01:00:53,190 --> 01:00:53,200

 

1899
01:00:53,200 --> 01:00:54,390


1900
01:00:54,390 --> 01:00:54,400

 

1901
01:00:54,400 --> 01:00:55,349


1902
01:00:55,349 --> 01:00:57,910

 

1903
01:00:57,910 --> 01:01:00,470

 

1904
01:01:00,470 --> 01:01:03,270

 

1905
01:01:03,270 --> 01:01:03,280

 

1906
01:01:03,280 --> 01:01:04,230


1907
01:01:04,230 --> 01:01:06,150

 

1908
01:01:06,150 --> 01:01:08,549

 

1909
01:01:08,549 --> 01:01:09,910

 

1910
01:01:09,910 --> 01:01:11,829

 

1911
01:01:11,829 --> 01:01:14,069

 

1912
01:01:14,069 --> 01:01:14,079

 

1913
01:01:14,079 --> 01:01:15,510


1914
01:01:15,510 --> 01:01:18,549

 

1915
01:01:18,549 --> 01:01:18,559

 

1916
01:01:18,559 --> 01:01:20,230


1917
01:01:20,230 --> 01:01:22,950

 

1918
01:01:22,950 --> 01:01:24,390

 

1919
01:01:24,390 --> 01:01:26,710

 

1920
01:01:26,710 --> 01:01:28,069

 

1921
01:01:28,069 --> 01:01:30,870

 

1922
01:01:30,870 --> 01:01:32,630

 

1923
01:01:32,630 --> 01:01:34,150

 

1924
01:01:34,150 --> 01:01:37,990

 

1925
01:01:37,990 --> 01:01:41,109

 

1926
01:01:41,109 --> 01:01:43,589

 

1927
01:01:43,589 --> 01:01:43,599

 

1928
01:01:43,599 --> 01:01:45,109


1929
01:01:45,109 --> 01:01:48,150

 

1930
01:01:48,150 --> 01:01:49,990

 

1931
01:01:49,990 --> 01:01:51,750

 

1932
01:01:51,750 --> 01:01:51,760

 

1933
01:01:51,760 --> 01:01:53,270


1934
01:01:53,270 --> 01:01:54,870

 

1935
01:01:54,870 --> 01:01:56,789

 

1936
01:01:56,789 --> 01:01:59,190

 

1937
01:01:59,190 --> 01:02:00,870

 

1938
01:02:00,870 --> 01:02:02,789

 

1939
01:02:02,789 --> 01:02:05,109

 

1940
01:02:05,109 --> 01:02:07,190

 

1941
01:02:07,190 --> 01:02:09,510

 

1942
01:02:09,510 --> 01:02:11,270

 

1943
01:02:11,270 --> 01:02:13,910

 

1944
01:02:13,910 --> 01:02:17,670

 

1945
01:02:17,670 --> 01:02:19,750

 

1946
01:02:19,750 --> 01:02:19,760

 

1947
01:02:19,760 --> 01:02:20,549


1948
01:02:20,549 --> 01:02:22,069

 

1949
01:02:22,069 --> 01:02:23,829

 

1950
01:02:23,829 --> 01:02:25,990

 

1951
01:02:25,990 --> 01:02:27,829

 

1952
01:02:27,829 --> 01:02:29,750

 

1953
01:02:29,750 --> 01:02:29,760

 

1954
01:02:29,760 --> 01:02:30,950


1955
01:02:30,950 --> 01:02:33,349

 

1956
01:02:33,349 --> 01:02:34,870

 

1957
01:02:34,870 --> 01:02:36,630

 

1958
01:02:36,630 --> 01:02:38,470

 

1959
01:02:38,470 --> 01:02:41,109

 

1960
01:02:41,109 --> 01:02:43,510

 

1961
01:02:43,510 --> 01:02:45,430

 

1962
01:02:45,430 --> 01:02:47,670

 

1963
01:02:47,670 --> 01:02:49,829

 

1964
01:02:49,829 --> 01:02:52,069

 

1965
01:02:52,069 --> 01:02:53,910

 

1966
01:02:53,910 --> 01:02:55,190

 

1967
01:02:55,190 --> 01:02:57,190

 

1968
01:02:57,190 --> 01:02:58,390

 

1969
01:02:58,390 --> 01:03:00,069

 

1970
01:03:00,069 --> 01:03:00,079

 

1971
01:03:00,079 --> 01:03:00,950


1972
01:03:00,950 --> 01:03:04,950

 

1973
01:03:04,950 --> 01:03:06,789

 

1974
01:03:06,789 --> 01:03:09,589

 

1975
01:03:09,589 --> 01:03:11,829

 

1976
01:03:11,829 --> 01:03:11,839

 

1977
01:03:11,839 --> 01:03:14,470


1978
01:03:14,470 --> 01:03:16,870

 

1979
01:03:16,870 --> 01:03:18,470

 

1980
01:03:18,470 --> 01:03:20,470

 

1981
01:03:20,470 --> 01:03:21,829

 

1982
01:03:21,829 --> 01:03:22,950

 

1983
01:03:22,950 --> 01:03:24,870

 

1984
01:03:24,870 --> 01:03:28,549

 

1985
01:03:28,549 --> 01:03:30,390

 

1986
01:03:30,390 --> 01:03:32,230

 

1987
01:03:32,230 --> 01:03:33,510

 

1988
01:03:33,510 --> 01:03:33,520

 

1989
01:03:33,520 --> 01:03:34,789


1990
01:03:34,789 --> 01:03:36,150

 

1991
01:03:36,150 --> 01:03:38,549

 

1992
01:03:38,549 --> 01:03:39,670

 

1993
01:03:39,670 --> 01:03:39,680

 

1994
01:03:39,680 --> 01:03:41,349


1995
01:03:41,349 --> 01:03:43,510

 

1996
01:03:43,510 --> 01:03:43,520

 

1997
01:03:43,520 --> 01:03:44,710


1998
01:03:44,710 --> 01:03:44,720

 

1999
01:03:44,720 --> 01:03:46,309


2000
01:03:46,309 --> 01:03:48,150

 

2001
01:03:48,150 --> 01:03:50,470

 

2002
01:03:50,470 --> 01:03:52,710

 

2003
01:03:52,710 --> 01:03:54,789

 

2004
01:03:54,789 --> 01:03:56,710

 

2005
01:03:56,710 --> 01:03:59,270

 

2006
01:03:59,270 --> 01:04:01,190

 

2007
01:04:01,190 --> 01:04:03,029

 

2008
01:04:03,029 --> 01:04:04,549

 

2009
01:04:04,549 --> 01:04:06,870

 

2010
01:04:06,870 --> 01:04:06,880

 

2011
01:04:06,880 --> 01:04:10,870


2012
01:04:10,870 --> 01:04:12,870

 

2013
01:04:12,870 --> 01:04:16,230

 

2014
01:04:16,230 --> 01:04:18,710

 

2015
01:04:18,710 --> 01:04:20,630

 

2016
01:04:20,630 --> 01:04:23,190

 

2017
01:04:23,190 --> 01:04:23,200

 

2018
01:04:23,200 --> 01:04:24,069


2019
01:04:24,069 --> 01:04:24,079

 

2020
01:04:24,079 --> 01:04:25,029


2021
01:04:25,029 --> 01:04:27,750

 

2022
01:04:27,750 --> 01:04:27,760

 

2023
01:04:27,760 --> 01:04:29,029


2024
01:04:29,029 --> 01:04:32,470

 

2025
01:04:32,470 --> 01:04:34,870

 

2026
01:04:34,870 --> 01:04:37,670

 

2027
01:04:37,670 --> 01:04:42,789

 

2028
01:04:42,789 --> 01:04:44,390

 

2029
01:04:44,390 --> 01:04:47,190

 

2030
01:04:47,190 --> 01:04:49,430

 

2031
01:04:49,430 --> 01:04:51,109

 

2032
01:04:51,109 --> 01:04:53,029

 

2033
01:04:53,029 --> 01:04:55,829

 

2034
01:04:55,829 --> 01:04:57,750

 

2035
01:04:57,750 --> 01:05:00,309

 

2036
01:05:00,309 --> 01:05:01,670

 

2037
01:05:01,670 --> 01:05:02,870

 

2038
01:05:02,870 --> 01:05:04,870

 

2039
01:05:04,870 --> 01:05:06,470

 

2040
01:05:06,470 --> 01:05:10,309

 

2041
01:05:10,309 --> 01:05:13,109

 

2042
01:05:13,109 --> 01:05:16,710

 

2043
01:05:16,710 --> 01:05:19,349

 

2044
01:05:19,349 --> 01:05:20,829

 

2045
01:05:20,829 --> 01:05:23,990

 

2046
01:05:23,990 --> 01:05:26,630

 

2047
01:05:26,630 --> 01:05:28,150

 

2048
01:05:28,150 --> 01:05:31,829

 

2049
01:05:31,829 --> 01:05:31,839

 

2050
01:05:31,839 --> 01:05:35,510


2051
01:05:35,510 --> 01:05:37,589

 

2052
01:05:37,589 --> 01:05:39,190

 

2053
01:05:39,190 --> 01:05:39,200

 

2054
01:05:39,200 --> 01:05:40,309


2055
01:05:40,309 --> 01:05:42,309

 

2056
01:05:42,309 --> 01:05:44,789

 

2057
01:05:44,789 --> 01:05:47,270

 

2058
01:05:47,270 --> 01:05:49,109

 

2059
01:05:49,109 --> 01:05:52,309

 

2060
01:05:52,309 --> 01:05:53,990

 

2061
01:05:53,990 --> 01:05:56,390

 

2062
01:05:56,390 --> 01:05:57,829

 

2063
01:05:57,829 --> 01:05:59,349

 

2064
01:05:59,349 --> 01:06:01,750

 

2065
01:06:01,750 --> 01:06:04,390

 

2066
01:06:04,390 --> 01:06:06,150

 

2067
01:06:06,150 --> 01:06:06,160

 

2068
01:06:06,160 --> 01:06:07,270


2069
01:06:07,270 --> 01:06:10,710

 

2070
01:06:10,710 --> 01:06:15,109

 

2071
01:06:15,109 --> 01:06:16,710

 

2072
01:06:16,710 --> 01:06:18,549

 

2073
01:06:18,549 --> 01:06:19,910

 

2074
01:06:19,910 --> 01:06:22,710

 

2075
01:06:22,710 --> 01:06:22,720

 

2076
01:06:22,720 --> 01:06:23,589


2077
01:06:23,589 --> 01:06:25,829

 

2078
01:06:25,829 --> 01:06:28,549

 

2079
01:06:28,549 --> 01:06:31,270

 

2080
01:06:31,270 --> 01:06:33,510

 

2081
01:06:33,510 --> 01:06:35,430

 

2082
01:06:35,430 --> 01:06:38,150

 

2083
01:06:38,150 --> 01:06:40,309

 

2084
01:06:40,309 --> 01:06:43,270

 

2085
01:06:43,270 --> 01:06:47,349

 

2086
01:06:47,349 --> 01:06:49,829

 

2087
01:06:49,829 --> 01:06:52,309

 

2088
01:06:52,309 --> 01:06:52,319

 

2089
01:06:52,319 --> 01:06:53,270


2090
01:06:53,270 --> 01:06:55,670

 

2091
01:06:55,670 --> 01:06:57,750

 

2092
01:06:57,750 --> 01:06:57,760

 

2093
01:06:57,760 --> 01:06:58,870


2094
01:06:58,870 --> 01:07:00,230

 

2095
01:07:00,230 --> 01:07:00,240

 

2096
01:07:00,240 --> 01:07:01,270


2097
01:07:01,270 --> 01:07:02,870

 

2098
01:07:02,870 --> 01:07:04,390

 

2099
01:07:04,390 --> 01:07:06,390

 

2100
01:07:06,390 --> 01:07:07,910

 

2101
01:07:07,910 --> 01:07:10,470

 

2102
01:07:10,470 --> 01:07:14,390

 

2103
01:07:14,390 --> 01:07:17,109

 

2104
01:07:17,109 --> 01:07:19,910

 

2105
01:07:19,910 --> 01:07:22,069

 

2106
01:07:22,069 --> 01:07:24,950

 

2107
01:07:24,950 --> 01:07:26,150

 

2108
01:07:26,150 --> 01:07:30,630

 

2109
01:07:30,630 --> 01:07:32,710

 

2110
01:07:32,710 --> 01:07:35,670

 

2111
01:07:35,670 --> 01:07:37,349

 

2112
01:07:37,349 --> 01:07:40,309

 

2113
01:07:40,309 --> 01:07:42,470

 

2114
01:07:42,470 --> 01:07:44,150

 

2115
01:07:44,150 --> 01:07:46,069

 

2116
01:07:46,069 --> 01:07:47,670

 

2117
01:07:47,670 --> 01:07:49,270

 

2118
01:07:49,270 --> 01:07:52,230

 

2119
01:07:52,230 --> 01:07:54,950

 

2120
01:07:54,950 --> 01:07:57,430

 

2121
01:07:57,430 --> 01:07:57,440

 

2122
01:07:57,440 --> 01:08:00,150


2123
01:08:00,150 --> 01:08:02,950

 

2124
01:08:02,950 --> 01:08:04,870

 

2125
01:08:04,870 --> 01:08:07,430

 

2126
01:08:07,430 --> 01:08:10,309

 

2127
01:08:10,309 --> 01:08:12,950

 

2128
01:08:12,950 --> 01:08:14,470

 

2129
01:08:14,470 --> 01:08:16,309

 

2130
01:08:16,309 --> 01:08:19,430

 

2131
01:08:19,430 --> 01:08:21,829

 

2132
01:08:21,829 --> 01:08:24,390

 

2133
01:08:24,390 --> 01:08:26,709

 

2134
01:08:26,709 --> 01:08:30,470

 

2135
01:08:30,470 --> 01:08:33,430

 

2136
01:08:33,430 --> 01:08:35,749

 

2137
01:08:35,749 --> 01:08:37,749

 

2138
01:08:37,749 --> 01:08:39,510

 

2139
01:08:39,510 --> 01:08:41,110

 

2140
01:08:41,110 --> 01:08:43,430

 

2141
01:08:43,430 --> 01:08:45,669

 

2142
01:08:45,669 --> 01:08:47,510

 

2143
01:08:47,510 --> 01:08:49,510

 

2144
01:08:49,510 --> 01:08:52,709

 

2145
01:08:52,709 --> 01:08:55,030

 

2146
01:08:55,030 --> 01:08:56,950

 

2147
01:08:56,950 --> 01:08:58,470

 

2148
01:08:58,470 --> 01:09:01,189

 

2149
01:09:01,189 --> 01:09:02,630

 

2150
01:09:02,630 --> 01:09:04,950

 

2151
01:09:04,950 --> 01:09:06,309

 

2152
01:09:06,309 --> 01:09:08,390

 

2153
01:09:08,390 --> 01:09:10,309

 

2154
01:09:10,309 --> 01:09:11,990

 

2155
01:09:11,990 --> 01:09:12,000

 

2156
01:09:12,000 --> 01:09:12,950


2157
01:09:12,950 --> 01:09:12,960

 

2158
01:09:12,960 --> 01:09:13,669


2159
01:09:13,669 --> 01:09:16,950

 

2160
01:09:16,950 --> 01:09:18,309

 

2161
01:09:18,309 --> 01:09:20,870

 

2162
01:09:20,870 --> 01:09:23,349

 

2163
01:09:23,349 --> 01:09:28,630

 

2164
01:09:28,630 --> 01:09:30,070

 

2165
01:09:30,070 --> 01:09:32,149

 

2166
01:09:32,149 --> 01:09:34,309

 

2167
01:09:34,309 --> 01:09:37,269

 

2168
01:09:37,269 --> 01:09:39,110

 

2169
01:09:39,110 --> 01:09:39,120

 

2170
01:09:39,120 --> 01:09:40,550


2171
01:09:40,550 --> 01:09:40,560

 

2172
01:09:40,560 --> 01:09:41,829


2173
01:09:41,829 --> 01:09:43,269

 

2174
01:09:43,269 --> 01:09:43,279

 

2175
01:09:43,279 --> 01:09:46,829


2176
01:09:46,829 --> 01:09:50,229

 

2177
01:09:50,229 --> 01:09:52,950

 

2178
01:09:52,950 --> 01:09:52,960

 

2179
01:09:52,960 --> 01:09:54,709


2180
01:09:54,709 --> 01:09:56,709

 

2181
01:09:56,709 --> 01:09:56,719

 

2182
01:09:56,719 --> 01:09:58,709


2183
01:09:58,709 --> 01:10:01,350

 

2184
01:10:01,350 --> 01:10:05,030

 

2185
01:10:05,030 --> 01:10:05,040

 

2186
01:10:05,040 --> 01:10:06,229


2187
01:10:06,229 --> 01:10:07,750

 

2188
01:10:07,750 --> 01:10:11,030

 

2189
01:10:11,030 --> 01:10:13,270

 

2190
01:10:13,270 --> 01:10:15,030

 

2191
01:10:15,030 --> 01:10:16,390

 

2192
01:10:16,390 --> 01:10:19,510

 

2193
01:10:19,510 --> 01:10:22,550

 

2194
01:10:22,550 --> 01:10:23,669

 

2195
01:10:23,669 --> 01:10:23,679

 

2196
01:10:23,679 --> 01:10:24,870


2197
01:10:24,870 --> 01:10:24,880

 

2198
01:10:24,880 --> 01:10:26,310


2199
01:10:26,310 --> 01:10:26,320

 

2200
01:10:26,320 --> 01:10:27,270


2201
01:10:27,270 --> 01:10:29,270

 

2202
01:10:29,270 --> 01:10:30,950

 

2203
01:10:30,950 --> 01:10:30,960

 

2204
01:10:30,960 --> 01:10:33,430


2205
01:10:33,430 --> 01:10:37,350

 

2206
01:10:37,350 --> 01:10:40,070

 

2207
01:10:40,070 --> 01:10:42,229

 

2208
01:10:42,229 --> 01:10:44,950

 

2209
01:10:44,950 --> 01:10:47,830

 

2210
01:10:47,830 --> 01:10:51,910

 

2211
01:10:51,910 --> 01:10:53,750

 

2212
01:10:53,750 --> 01:10:55,669

 

2213
01:10:55,669 --> 01:10:59,590

 

2214
01:10:59,590 --> 01:11:01,189

 

2215
01:11:01,189 --> 01:11:03,430

 

2216
01:11:03,430 --> 01:11:05,990

 

2217
01:11:05,990 --> 01:11:08,149

 

2218
01:11:08,149 --> 01:11:10,709

 

2219
01:11:10,709 --> 01:11:13,350

 

2220
01:11:13,350 --> 01:11:15,189

 

2221
01:11:15,189 --> 01:11:17,830

 

2222
01:11:17,830 --> 01:11:17,840

 

2223
01:11:17,840 --> 01:11:19,110


2224
01:11:19,110 --> 01:11:21,110

 

2225
01:11:21,110 --> 01:11:21,120

 

2226
01:11:21,120 --> 01:11:21,430


2227
01:11:21,430 --> 01:11:21,440

 

2228
01:11:21,440 --> 01:11:23,990


2229
01:11:23,990 --> 01:11:25,830

 

2230
01:11:25,830 --> 01:11:28,149

 

2231
01:11:28,149 --> 01:11:29,590

 

2232
01:11:29,590 --> 01:11:31,189

 

2233
01:11:31,189 --> 01:11:33,430

 

2234
01:11:33,430 --> 01:11:35,910

 

2235
01:11:35,910 --> 01:11:38,830

 

2236
01:11:38,830 --> 01:11:43,910

 

2237
01:11:43,910 --> 01:11:48,229

 

2238
01:11:48,229 --> 01:11:51,430

 

2239
01:11:51,430 --> 01:11:54,550

 

2240
01:11:54,550 --> 01:12:00,310

 

2241
01:12:00,310 --> 01:12:00,320

 

2242
01:12:00,320 --> 01:12:01,430


2243
01:12:01,430 --> 01:12:05,990

 

2244
01:12:05,990 --> 01:12:06,000

 

2245
01:12:06,000 --> 01:12:07,990


2246
01:12:07,990 --> 01:12:08,000

 

2247
01:12:08,000 --> 01:12:10,950


2248
01:12:10,950 --> 01:12:12,950

 

2249
01:12:12,950 --> 01:12:12,960

 

2250
01:12:12,960 --> 01:12:13,830


2251
01:12:13,830 --> 01:12:13,840

 

2252
01:12:13,840 --> 01:12:16,390


2253
01:12:16,390 --> 01:12:18,390

 

2254
01:12:18,390 --> 01:12:18,400

 

2255
01:12:18,400 --> 01:12:19,510


2256
01:12:19,510 --> 01:12:19,520

 

2257
01:12:19,520 --> 01:12:23,350


2258
01:12:23,350 --> 01:12:23,360

 

2259
01:12:23,360 --> 01:12:24,550


2260
01:12:24,550 --> 01:12:27,270

 

2261
01:12:27,270 --> 01:12:29,590

 

2262
01:12:29,590 --> 01:12:32,550

 

2263
01:12:32,550 --> 01:12:32,560

 

2264
01:12:32,560 --> 01:12:34,830


2265
01:12:34,830 --> 01:12:40,630

 

2266
01:12:40,630 --> 01:12:43,830

 

2267
01:12:43,830 --> 01:12:46,470

 

2268
01:12:46,470 --> 01:12:49,750

 

2269
01:12:49,750 --> 01:12:49,760

 

2270
01:12:49,760 --> 01:12:51,189


2271
01:12:51,189 --> 01:12:51,199

 

2272
01:12:51,199 --> 01:12:52,229


2273
01:12:52,229 --> 01:12:52,239

 

2274
01:12:52,239 --> 01:12:53,030


2275
01:12:53,030 --> 01:12:54,630

 

2276
01:12:54,630 --> 01:12:57,270

 

2277
01:12:57,270 --> 01:13:00,470

 

2278
01:13:00,470 --> 01:13:03,830

 

2279
01:13:03,830 --> 01:13:06,070

 

2280
01:13:06,070 --> 01:13:07,910

 

2281
01:13:07,910 --> 01:13:09,830

 

2282
01:13:09,830 --> 01:13:12,070

 

2283
01:13:12,070 --> 01:13:12,080

 

2284
01:13:12,080 --> 01:13:13,430


2285
01:13:13,430 --> 01:13:14,950

 

2286
01:13:14,950 --> 01:13:16,790

 

2287
01:13:16,790 --> 01:13:18,550

 

2288
01:13:18,550 --> 01:13:20,630

 

2289
01:13:20,630 --> 01:13:24,310

 

2290
01:13:24,310 --> 01:13:26,149

 

2291
01:13:26,149 --> 01:13:29,830

 

2292
01:13:29,830 --> 01:13:32,470

 

2293
01:13:32,470 --> 01:13:35,270

 

2294
01:13:35,270 --> 01:13:35,280

 

2295
01:13:35,280 --> 01:13:37,830


2296
01:13:37,830 --> 01:13:37,840

 

2297
01:13:37,840 --> 01:13:38,790


2298
01:13:38,790 --> 01:13:41,990

 

2299
01:13:41,990 --> 01:13:44,070

 

2300
01:13:44,070 --> 01:13:47,030

 

2301
01:13:47,030 --> 01:13:49,030

 

2302
01:13:49,030 --> 01:13:50,950

 

2303
01:13:50,950 --> 01:13:52,709

 

2304
01:13:52,709 --> 01:13:54,229

 

2305
01:13:54,229 --> 01:13:56,390

 

2306
01:13:56,390 --> 01:13:56,400

 

2307
01:13:56,400 --> 01:13:59,110


2308
01:13:59,110 --> 01:14:00,870

 

2309
01:14:00,870 --> 01:14:04,790

 

2310
01:14:04,790 --> 01:14:06,709

 

2311
01:14:06,709 --> 01:14:08,390

 

2312
01:14:08,390 --> 01:14:10,630

 

2313
01:14:10,630 --> 01:14:12,470

 

2314
01:14:12,470 --> 01:14:14,229

 

2315
01:14:14,229 --> 01:14:16,790

 

2316
01:14:16,790 --> 01:14:18,870

 

2317
01:14:18,870 --> 01:14:20,870

 

2318
01:14:20,870 --> 01:14:23,750

 

2319
01:14:23,750 --> 01:14:23,760

 

2320
01:14:23,760 --> 01:14:25,110


2321
01:14:25,110 --> 01:14:27,350

 

2322
01:14:27,350 --> 01:14:27,360

 

2323
01:14:27,360 --> 01:14:28,790


2324
01:14:28,790 --> 01:14:31,669

 

2325
01:14:31,669 --> 01:14:31,679

 

2326
01:14:31,679 --> 01:14:32,950


2327
01:14:32,950 --> 01:14:34,470

 

2328
01:14:34,470 --> 01:14:37,430

 

2329
01:14:37,430 --> 01:14:39,350

 

2330
01:14:39,350 --> 01:14:41,189

 

2331
01:14:41,189 --> 01:14:44,390

 

2332
01:14:44,390 --> 01:14:48,630

 

2333
01:14:48,630 --> 01:14:48,640

 

2334
01:14:48,640 --> 01:14:50,709


2335
01:14:50,709 --> 01:14:52,550

 

2336
01:14:52,550 --> 01:14:54,630

 

2337
01:14:54,630 --> 01:14:56,790

 

2338
01:14:56,790 --> 01:14:56,800

 

2339
01:14:56,800 --> 01:14:57,669


2340
01:14:57,669 --> 01:14:59,189

 

2341
01:14:59,189 --> 01:15:01,830

 

2342
01:15:01,830 --> 01:15:04,390

 

2343
01:15:04,390 --> 01:15:06,229

 

2344
01:15:06,229 --> 01:15:06,239

 

2345
01:15:06,239 --> 01:15:07,110


2346
01:15:07,110 --> 01:15:08,870

 

2347
01:15:08,870 --> 01:15:10,709

 

2348
01:15:10,709 --> 01:15:10,719

 

2349
01:15:10,719 --> 01:15:12,310


2350
01:15:12,310 --> 01:15:14,790

 

2351
01:15:14,790 --> 01:15:16,950

 

2352
01:15:16,950 --> 01:15:16,960

 

2353
01:15:16,960 --> 01:15:17,990


2354
01:15:17,990 --> 01:15:20,070

 

2355
01:15:20,070 --> 01:15:21,669

 

2356
01:15:21,669 --> 01:15:23,350

 

2357
01:15:23,350 --> 01:15:25,590

 

2358
01:15:25,590 --> 01:15:29,270

 

2359
01:15:29,270 --> 01:15:31,350

 

2360
01:15:31,350 --> 01:15:32,870

 

2361
01:15:32,870 --> 01:15:35,189

 

2362
01:15:35,189 --> 01:15:36,870

 

2363
01:15:36,870 --> 01:15:39,110

 

2364
01:15:39,110 --> 01:15:41,669

 

2365
01:15:41,669 --> 01:15:43,510

 

2366
01:15:43,510 --> 01:15:46,149

 

2367
01:15:46,149 --> 01:15:47,990

 

2368
01:15:47,990 --> 01:15:50,229

 

2369
01:15:50,229 --> 01:15:51,990

 

2370
01:15:51,990 --> 01:15:52,000

 

2371
01:15:52,000 --> 01:15:52,950


2372
01:15:52,950 --> 01:15:55,110

 

2373
01:15:55,110 --> 01:15:55,120

 

2374
01:15:55,120 --> 01:15:56,149


2375
01:15:56,149 --> 01:15:58,149

 

2376
01:15:58,149 --> 01:16:01,110

 

2377
01:16:01,110 --> 01:16:02,470

 

2378
01:16:02,470 --> 01:16:05,110

 

2379
01:16:05,110 --> 01:16:06,790

 

2380
01:16:06,790 --> 01:16:08,310

 

2381
01:16:08,310 --> 01:16:09,750

 

2382
01:16:09,750 --> 01:16:11,750

 

2383
01:16:11,750 --> 01:16:11,760

 

2384
01:16:11,760 --> 01:16:12,470


2385
01:16:12,470 --> 01:16:13,990

 

2386
01:16:13,990 --> 01:16:15,750

 

2387
01:16:15,750 --> 01:16:18,390

 

2388
01:16:18,390 --> 01:16:19,910

 

2389
01:16:19,910 --> 01:16:22,149

 

2390
01:16:22,149 --> 01:16:23,990

 

2391
01:16:23,990 --> 01:16:24,000

 

2392
01:16:24,000 --> 01:16:25,590


2393
01:16:25,590 --> 01:16:27,590

 

2394
01:16:27,590 --> 01:16:29,189

 

2395
01:16:29,189 --> 01:16:30,470

 

2396
01:16:30,470 --> 01:16:32,830

 

2397
01:16:32,830 --> 01:16:36,470

 

2398
01:16:36,470 --> 01:16:38,149

 

2399
01:16:38,149 --> 01:16:39,830

 

2400
01:16:39,830 --> 01:16:39,840

 

2401
01:16:39,840 --> 01:16:40,950


2402
01:16:40,950 --> 01:16:43,189

 

2403
01:16:43,189 --> 01:16:44,870

 

2404
01:16:44,870 --> 01:16:47,430

 

2405
01:16:47,430 --> 01:16:50,070

 

2406
01:16:50,070 --> 01:16:54,550

 

2407
01:16:54,550 --> 01:16:55,990

 

2408
01:16:55,990 --> 01:16:57,430

 

2409
01:16:57,430 --> 01:16:59,669

 

2410
01:16:59,669 --> 01:17:04,550

 

2411
01:17:04,550 --> 01:17:06,950

 

2412
01:17:06,950 --> 01:17:09,350

 

2413
01:17:09,350 --> 01:17:11,030

 

2414
01:17:11,030 --> 01:17:13,830

 

2415
01:17:13,830 --> 01:17:13,840

 

2416
01:17:13,840 --> 01:17:15,350


2417
01:17:15,350 --> 01:17:17,830

 

2418
01:17:17,830 --> 01:17:21,430

 

2419
01:17:21,430 --> 01:17:24,070

 

2420
01:17:24,070 --> 01:17:26,149

 

2421
01:17:26,149 --> 01:17:29,189

 

2422
01:17:29,189 --> 01:17:32,070

 

2423
01:17:32,070 --> 01:17:32,080

 

2424
01:17:32,080 --> 01:17:33,750


2425
01:17:33,750 --> 01:17:35,270

 

2426
01:17:35,270 --> 01:17:36,310

 

2427
01:17:36,310 --> 01:17:38,709

 

2428
01:17:38,709 --> 01:17:40,149

 

2429
01:17:40,149 --> 01:17:44,630

 

2430
01:17:44,630 --> 01:17:47,030

 

2431
01:17:47,030 --> 01:17:48,790

 

2432
01:17:48,790 --> 01:17:51,510

 

2433
01:17:51,510 --> 01:17:53,270

 

2434
01:17:53,270 --> 01:17:54,950

 

2435
01:17:54,950 --> 01:17:57,830

 

2436
01:17:57,830 --> 01:18:00,470

 

2437
01:18:00,470 --> 01:18:00,480

 

2438
01:18:00,480 --> 01:18:01,270


2439
01:18:01,270 --> 01:18:03,350

 

2440
01:18:03,350 --> 01:18:05,910

 

2441
01:18:05,910 --> 01:18:08,310

 

2442
01:18:08,310 --> 01:18:08,320

 

2443
01:18:08,320 --> 01:18:10,149


2444
01:18:10,149 --> 01:18:10,159

 

2445
01:18:10,159 --> 01:18:11,669


2446
01:18:11,669 --> 01:18:11,679

 

2447
01:18:11,679 --> 01:18:13,750


2448
01:18:13,750 --> 01:18:16,070

 

2449
01:18:16,070 --> 01:18:17,990

 

2450
01:18:17,990 --> 01:18:19,990

 

2451
01:18:19,990 --> 01:18:22,310

 

2452
01:18:22,310 --> 01:18:22,320

 

2453
01:18:22,320 --> 01:18:23,510


2454
01:18:23,510 --> 01:18:23,520

 

2455
01:18:23,520 --> 01:18:24,470


2456
01:18:24,470 --> 01:18:27,669

 

2457
01:18:27,669 --> 01:18:29,750

 

2458
01:18:29,750 --> 01:18:32,229

 

2459
01:18:32,229 --> 01:18:32,239

 

2460
01:18:32,239 --> 01:18:35,030


2461
01:18:35,030 --> 01:18:37,350

 

2462
01:18:37,350 --> 01:18:37,360

 

2463
01:18:37,360 --> 01:18:38,390


2464
01:18:38,390 --> 01:18:40,470

 

2465
01:18:40,470 --> 01:18:43,350

 

2466
01:18:43,350 --> 01:18:45,030

 

2467
01:18:45,030 --> 01:18:47,990

 

2468
01:18:47,990 --> 01:18:49,990

 

2469
01:18:49,990 --> 01:18:51,910

 

2470
01:18:51,910 --> 01:18:54,790

 

2471
01:18:54,790 --> 01:18:57,189

 

2472
01:18:57,189 --> 01:19:01,270

 

2473
01:19:01,270 --> 01:19:03,910

 

2474
01:19:03,910 --> 01:19:03,920

 

2475
01:19:03,920 --> 01:19:05,990


2476
01:19:05,990 --> 01:19:07,830

 

2477
01:19:07,830 --> 01:19:11,510

 

2478
01:19:11,510 --> 01:19:14,790

 

2479
01:19:14,790 --> 01:19:14,800

 

2480
01:19:14,800 --> 01:19:15,990


2481
01:19:15,990 --> 01:19:19,669

 

2482
01:19:19,669 --> 01:19:19,679

 

2483
01:19:19,679 --> 01:19:20,950


2484
01:19:20,950 --> 01:19:22,470

 

2485
01:19:22,470 --> 01:19:25,030

 

2486
01:19:25,030 --> 01:19:26,550

 

2487
01:19:26,550 --> 01:19:29,350

 

2488
01:19:29,350 --> 01:19:29,360

 

2489
01:19:29,360 --> 01:19:31,189


2490
01:19:31,189 --> 01:19:33,510

 

2491
01:19:33,510 --> 01:19:35,110

 

2492
01:19:35,110 --> 01:19:37,030

 

2493
01:19:37,030 --> 01:19:39,030

 

2494
01:19:39,030 --> 01:19:39,040

 

2495
01:19:39,040 --> 01:19:40,149


2496
01:19:40,149 --> 01:19:42,550

 

2497
01:19:42,550 --> 01:19:45,910

 

2498
01:19:45,910 --> 01:19:48,229

 

2499
01:19:48,229 --> 01:19:49,430

 

2500
01:19:49,430 --> 01:19:51,750

 

2501
01:19:51,750 --> 01:19:53,270

 

2502
01:19:53,270 --> 01:19:55,350

 

2503
01:19:55,350 --> 01:19:57,030

 

2504
01:19:57,030 --> 01:19:59,350

 

2505
01:19:59,350 --> 01:20:00,790

 

2506
01:20:00,790 --> 01:20:02,950

 

2507
01:20:02,950 --> 01:20:07,430

 

2508
01:20:07,430 --> 01:20:07,440

 

2509
01:20:07,440 --> 01:20:08,550


2510
01:20:08,550 --> 01:20:11,270

 

2511
01:20:11,270 --> 01:20:13,110

 

2512
01:20:13,110 --> 01:20:14,870

 

2513
01:20:14,870 --> 01:20:14,880

 

2514
01:20:14,880 --> 01:20:16,229


2515
01:20:16,229 --> 01:20:17,830

 

2516
01:20:17,830 --> 01:20:17,840

 

2517
01:20:17,840 --> 01:20:18,790


2518
01:20:18,790 --> 01:20:20,229

 

2519
01:20:20,229 --> 01:20:21,910

 

2520
01:20:21,910 --> 01:20:23,110

 

2521
01:20:23,110 --> 01:20:24,390

 

2522
01:20:24,390 --> 01:20:24,400

 

2523
01:20:24,400 --> 01:20:25,270


2524
01:20:25,270 --> 01:20:26,709

 

2525
01:20:26,709 --> 01:20:29,590

 

2526
01:20:29,590 --> 01:20:29,600

 

2527
01:20:29,600 --> 01:20:30,470


2528
01:20:30,470 --> 01:20:33,990

 

2529
01:20:33,990 --> 01:20:35,669

 

2530
01:20:35,669 --> 01:20:37,830

 

2531
01:20:37,830 --> 01:20:40,629

 

2532
01:20:40,629 --> 01:20:41,990

 

2533
01:20:41,990 --> 01:20:42,000

 

2534
01:20:42,000 --> 01:20:43,030


2535
01:20:43,030 --> 01:20:44,149

 

2536
01:20:44,149 --> 01:20:47,110

 

2537
01:20:47,110 --> 01:20:48,629

 

2538
01:20:48,629 --> 01:20:50,390

 

2539
01:20:50,390 --> 01:20:52,709

 

2540
01:20:52,709 --> 01:20:54,870

 

2541
01:20:54,870 --> 01:20:57,669

 

2542
01:20:57,669 --> 01:20:57,679

 

2543
01:20:57,679 --> 01:20:58,790


2544
01:20:58,790 --> 01:21:00,390

 

2545
01:21:00,390 --> 01:21:01,830

 

2546
01:21:01,830 --> 01:21:01,840

 

2547
01:21:01,840 --> 01:21:03,030


2548
01:21:03,030 --> 01:21:05,189

 

2549
01:21:05,189 --> 01:21:07,110

 

2550
01:21:07,110 --> 01:21:09,510

 

2551
01:21:09,510 --> 01:21:11,669

 

2552
01:21:11,669 --> 01:21:15,110

 

2553
01:21:15,110 --> 01:21:15,990

 

2554
01:21:15,990 --> 01:21:17,990

 

2555
01:21:17,990 --> 01:21:21,030

 

2556
01:21:21,030 --> 01:21:23,350

 

2557
01:21:23,350 --> 01:21:25,430

 

2558
01:21:25,430 --> 01:21:27,189

 

2559
01:21:27,189 --> 01:21:29,910

 

2560
01:21:29,910 --> 01:21:33,669

 

2561
01:21:33,669 --> 01:21:36,149

 

2562
01:21:36,149 --> 01:21:38,229

 

2563
01:21:38,229 --> 01:21:38,239

 

2564
01:21:38,239 --> 01:21:39,590


2565
01:21:39,590 --> 01:21:39,600

 

2566
01:21:39,600 --> 01:21:41,110


2567
01:21:41,110 --> 01:21:44,310

 

2568
01:21:44,310 --> 01:21:46,149

 

2569
01:21:46,149 --> 01:21:48,629

 

2570
01:21:48,629 --> 01:21:50,709

 

2571
01:21:50,709 --> 01:21:53,110

 

2572
01:21:53,110 --> 01:21:55,669

 

2573
01:21:55,669 --> 01:21:57,590

 

2574
01:21:57,590 --> 01:21:59,750

 

2575
01:21:59,750 --> 01:22:01,990

 

2576
01:22:01,990 --> 01:22:03,750

 

2577
01:22:03,750 --> 01:22:05,270

 

2578
01:22:05,270 --> 01:22:07,590

 

2579
01:22:07,590 --> 01:22:09,910

 

2580
01:22:09,910 --> 01:22:11,510

 

2581
01:22:11,510 --> 01:22:13,990

 

2582
01:22:13,990 --> 01:22:17,110

 

2583
01:22:17,110 --> 01:22:19,350

 

2584
01:22:19,350 --> 01:22:21,750

 

2585
01:22:21,750 --> 01:22:24,149

 

2586
01:22:24,149 --> 01:22:26,310

 

2587
01:22:26,310 --> 01:22:28,310

 

2588
01:22:28,310 --> 01:22:30,790

 

2589
01:22:30,790 --> 01:22:30,800

 

2590
01:22:30,800 --> 01:22:33,590


2591
01:22:33,590 --> 01:22:35,830

 

2592
01:22:35,830 --> 01:22:39,350

 

2593
01:22:39,350 --> 01:22:41,350

 

2594
01:22:41,350 --> 01:22:42,870

 

2595
01:22:42,870 --> 01:22:45,830

 

2596
01:22:45,830 --> 01:22:45,840

 

2597
01:22:45,840 --> 01:22:46,950


2598
01:22:46,950 --> 01:22:46,960

 

2599
01:22:46,960 --> 01:22:49,110


2600
01:22:49,110 --> 01:22:49,120

 

2601
01:22:49,120 --> 01:22:50,709


2602
01:22:50,709 --> 01:22:53,510

 

2603
01:22:53,510 --> 01:22:53,520

 

2604
01:22:53,520 --> 01:22:54,470


2605
01:22:54,470 --> 01:22:55,830

 

2606
01:22:55,830 --> 01:22:57,270

 

2607
01:22:57,270 --> 01:22:59,189

 

2608
01:22:59,189 --> 01:23:01,030

 

2609
01:23:01,030 --> 01:23:03,030

 

2610
01:23:03,030 --> 01:23:05,189

 

2611
01:23:05,189 --> 01:23:08,390

 

2612
01:23:08,390 --> 01:23:08,400

 

2613
01:23:08,400 --> 01:23:09,590


2614
01:23:09,590 --> 01:23:11,030

 

2615
01:23:11,030 --> 01:23:13,270

 

2616
01:23:13,270 --> 01:23:16,070

 

2617
01:23:16,070 --> 01:23:18,790

 

2618
01:23:18,790 --> 01:23:22,229

 

2619
01:23:22,229 --> 01:23:24,149

 

2620
01:23:24,149 --> 01:23:26,390

 

2621
01:23:26,390 --> 01:23:28,950

 

2622
01:23:28,950 --> 01:23:28,960

 

2623
01:23:28,960 --> 01:23:31,189


2624
01:23:31,189 --> 01:23:31,199

 

2625
01:23:31,199 --> 01:23:33,030


2626
01:23:33,030 --> 01:23:34,550

 

2627
01:23:34,550 --> 01:23:36,709

 

2628
01:23:36,709 --> 01:23:38,629

 

2629
01:23:38,629 --> 01:23:38,639

 

2630
01:23:38,639 --> 01:23:39,750


2631
01:23:39,750 --> 01:23:41,990

 

2632
01:23:41,990 --> 01:23:44,709

 

2633
01:23:44,709 --> 01:23:44,719

 

2634
01:23:44,719 --> 01:23:46,470


2635
01:23:46,470 --> 01:23:49,110

 

2636
01:23:49,110 --> 01:23:51,430

 

2637
01:23:51,430 --> 01:23:52,870

 

2638
01:23:52,870 --> 01:23:55,430

 

2639
01:23:55,430 --> 01:23:59,110

 

2640
01:23:59,110 --> 01:24:01,110

 

2641
01:24:01,110 --> 01:24:02,870

 

2642
01:24:02,870 --> 01:24:04,470

 

2643
01:24:04,470 --> 01:24:06,470

 

2644
01:24:06,470 --> 01:24:09,189

 

2645
01:24:09,189 --> 01:24:11,510

 

2646
01:24:11,510 --> 01:24:11,520

 

2647
01:24:11,520 --> 01:24:13,350


2648
01:24:13,350 --> 01:24:17,750

 

2649
01:24:17,750 --> 01:24:17,760

 

2650
01:24:17,760 --> 01:24:18,790


2651
01:24:18,790 --> 01:24:18,800

 

2652
01:24:18,800 --> 01:24:19,510


2653
01:24:19,510 --> 01:24:21,270

 

2654
01:24:21,270 --> 01:24:23,669

 

2655
01:24:23,669 --> 01:24:26,629

 

2656
01:24:26,629 --> 01:24:28,070

 

2657
01:24:28,070 --> 01:24:31,110

 

2658
01:24:31,110 --> 01:24:33,350

 

2659
01:24:33,350 --> 01:24:35,510

 

2660
01:24:35,510 --> 01:24:35,520

 

2661
01:24:35,520 --> 01:24:36,950


2662
01:24:36,950 --> 01:24:39,189

 

2663
01:24:39,189 --> 01:24:41,910

 

2664
01:24:41,910 --> 01:24:44,310

 

2665
01:24:44,310 --> 01:24:45,750

 

2666
01:24:45,750 --> 01:24:49,270

 

2667
01:24:49,270 --> 01:24:51,110

 

2668
01:24:51,110 --> 01:24:53,510

 

2669
01:24:53,510 --> 01:24:53,520

 

2670
01:24:53,520 --> 01:24:55,110


2671
01:24:55,110 --> 01:24:55,120

 

2672
01:24:55,120 --> 01:24:57,110


2673
01:24:57,110 --> 01:24:59,270

 

2674
01:24:59,270 --> 01:25:00,950

 

2675
01:25:00,950 --> 01:25:02,870

 

2676
01:25:02,870 --> 01:25:05,510

 

2677
01:25:05,510 --> 01:25:08,390

 

2678
01:25:08,390 --> 01:25:10,790

 

2679
01:25:10,790 --> 01:25:13,990

 

2680
01:25:13,990 --> 01:25:16,390

 

2681
01:25:16,390 --> 01:25:17,750

 

2682
01:25:17,750 --> 01:25:17,760

 

2683
01:25:17,760 --> 01:25:19,590


2684
01:25:19,590 --> 01:25:21,270

 

2685
01:25:21,270 --> 01:25:21,280

 

2686
01:25:21,280 --> 01:25:22,070


2687
01:25:22,070 --> 01:25:24,310

 

2688
01:25:24,310 --> 01:25:27,189

 

2689
01:25:27,189 --> 01:25:30,709

 

2690
01:25:30,709 --> 01:25:30,719

 

2691
01:25:30,719 --> 01:25:31,590


2692
01:25:31,590 --> 01:25:33,590

 

2693
01:25:33,590 --> 01:25:36,229

 

2694
01:25:36,229 --> 01:25:41,480