15 points Problem 3

Consider the 2-server problem in the plane R2. Show that

the following locally greedy on-line algorithm has no

bounded competitive ratio:

Greedy algorithm for 2-server -- to respond to a request that is

not covered by a server, move the server that is nearest to the

request.

Consider 3 points p1, p2, p3 where the initial configuration

has servers on p1, p2. Suppose d(p1, p2) = 10, d(p1, p3) = 10,

and d(p2, p3) = 1.

Let the request sequence R be (p2, p3, p2, p3 …) N times

Opt(R) = 10

Greedy(R) = N

As N goes to infinity, the cost of Greedy over the cost of Opt is

unbounded. Therefore Greedy has no bounded competetive

ratio

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