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

2. k-server problem Goal: Minimize the total distance

3. k-server problem

9. ……… Greedy does not work,

10. 2-server 3-point problem

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

15. 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

16. 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

17. 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

18. Our Contribution For (general) 2-server 3-point problem, we prove that CR < 1.5897. Well below 2.0 (=the lower bound for the deterministic case): Superiority of randomization for the server problem Our approach is very brute force

19. 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…..

20. 11 15 4 11 15 4 1 3 2 3 1 C L C LR 1 1 RL R R 12

21. 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…..

22. Testing Many Triangles …… CR=1.5 1.531.4891.533 1.536 almost the same but CR=1.89 1.537 1.0 Approximation Lemma Line Lemma

23. Approximation Lemma 1.0 Proof

24. Line Lemma

25. Using Approximation Lemma a 1 b 2

26. Using Line Lemma a 1 b 2 decreasing 2

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

28. a 1 b 2

29. Computer Program a 1 b 2

30. a 1 b 2

31. a 1 b 2 + some heuristics

32. Some Data Conjecture: 1.5819 Current bound: 1.5897 –13,285 squares, d=1/256~1 Small area [5/4, 7/4, 1/16]: 1.5784 –69 squares, d=1/64~1/128 Small area [7/4, 9/4, 1/4]: 1.5825 –555 squares, d=1/2048~1/64 (5/4, 7/4) 1/16

33. Proof for Line Lemma

35. 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.

36. Thanks!