Randomized Competitive Analysis for Two Server Problems

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

4. k-server problem

5. k-server problem

6. k-server problem

7. k-server problem

8. k-server problem

9. k-server problem ……… Greedy does not work,

10. 2-server 3-point problem

11. 2-server 3-point problem

12. 2-server 3-point problem

13. 2-server 3-point problem

14. 2-server 3-point problem a b c Algorithm exists whose CR = 2.0 Adversary (always malicious): cababacb…… Opt: cababacb…… one move per two inputs

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. R L 1 1 1 C 11 R 15 4 R 15 L 11 15 L R 4 15 C 1 3 1 2 2 3

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 1.53 1.489 1.533 CR=1.5 1.0 1.536 …… almost the same but CR=1.89 Approximation Lemma 1.537 Line Lemma

23. Approximation Lemma 1.0 1.0 Proof

24. Line Lemma

25. Using Approximation Lemma b 2 a 1

26. Using Line Lemma b 2 2 decreasing a 1

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

28. b 2 a 1

29. Computer Program b 2 a 1

30. Computer Program b 2 a 1

31. Computer Program b 2 + some heuristics a 1

32. (5/4, 7/4) 1/16 SomeData • 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

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!