Randomized Competitive Analysis for Two Server Problems Wolfgang Bein Kazuo Iwama Jun Kawahara

k-server problem Goal: Minimize the total distance

k-server problem

……… Greedy does not work,

2-server 3-point problem

a b c Adversary (always malicious): cababacb…… Opt: cababacb…… one move per two inputs Algorithm exists whose CR = 2.0

k-server: Known Facts Introduction of the problem [Manasse, McGeoch, Sleator 90] Lower bound: k [MMS90] General upper bound: 2k-1 [Koutsoupias, Papadimitriou 94] k-server conjecture –true for 2-servers, line, trees, fixed k+1 or K+2 points, …… –still open for 3 server 7 points

Randomized k-server Very little is known for general cases Even for 2-servers (CR=2 for det. case): –On the line [Bartal, Chrobak, Lamore 98] –Cross polytope space [Bein et al. 08] –Specific 3 points: Can use LP to derive an optimal algorithm (but nothing was given about the CR) [Lund, Reingold 94] –Almost nothing is known about its CR for a general metric space

Randomized 2-server 3-point a b c Adversary is malicious: c…… Select a server (a or b) at random Adversary’s attempt fails with prob. 0.5

Our Contribution For (general) 2-server 3-point problem, we prove that CR < Well below 2.0 (=the lower bound for the deterministic case): Superiority of randomization for the server problem Our approach is very brute force

The Idea We can assume a triangle in the plain wolg. For a specific triangle, its algorithm can be given as a (finite) state diagram, which can be derived by LP [LR94] Calculation of its CR is not hard. Just try many (different shaped) triangles, then…..

C L C LR 1 1 RL R R 12

The Idea We can assume a triangle in the plain wolg. For a specific triangle, its algorithm can be given as a (finite) state diagram, which can be derived by LP [LR94] Calculation of its CR is not hard. Just try many (different shaped) triangles, then…..

Testing Many Triangles …… CR= almost the same but CR= Approximation Lemma Line Lemma

Approximation Lemma 1.0 Proof

Line Lemma

Using Approximation Lemma a 1 b 2

Using Line Lemma a 1 b 2 decreasing 2

a 1 b 2 finite set of squares (triangles) Our algorithm = algs for squares + alg for the line

a 1 b 2

Computer Program a 1 b 2

a 1 b 2

a 1 b 2 + some heuristics

Some Data Conjecture: Current bound: –13,285 squares, d=1/256~1 Small area [5/4, 7/4, 1/16]: –69 squares, d=1/64~1/128 Small area [7/4, 9/4, 1/4]: –555 squares, d=1/2048~1/64 (5/4, 7/4) 1/16

Proof for Line Lemma

Final Remarks Strong conjecture that the real CR is e/(e-1). Analytical proof? Extension of our approach to, say, the 4- point case. Many very interesting open problems in the online server problem.

Thanks!