1. The k-server Problem Study Group: Randomized Algorithm Presented by Ray Lam August 16, 2003

2. 2 Outline 1. Background and problem definition 2. The Harmonic k-server Algorithm 3. Proving the claimed performance of the algorithm

3. Background And Problem Definition

4. 4 The Metric Space Definition: A metric space M = (V, d) consists of a set of points V with a distance function d:V R satisfying the following properties: d(u,v) 0 for all u, v V. d(u,v) = 0 iff u = v. d(u,v) = d(v,u) for all u, v V. d(u,v) + d(v,w) d(u,w) for all u, v, w V.

5. 5 The Metric Space Think of it as a complete weighted graph Weight corresponds to distance between points 1 3 4 2 2 2 3 2 1 1

6. 6 The k-server Problem k servers in the metric space Located at particular points Request of service Happens at the points To serve the request: move a server to the point of request A request sequence, where is a point in M, is a finite sequence of requests

7. 7 The k-server Problem Two competing algorithms An adversary offline algorithm An online algorithm to be designed The adversary algorithm Knows all of right from the beginning and serves them optimally with his own k servers Thus it is offline

8. 8 The k-server Problem Algorithm to be designed Online Only knows the next request and has to serve it immediately Cost measure Total distance moved by all the servers to serve : total cost incurred by the optimal offline algorithm

9. 9 The k-server Problem Let denote the cost of algorithm A on request sequence. Definition: A randomized algorithm A is c- competitive (compared to the optimal offline algorithm), if for all starting configurations there is a real a, independent of, such that

10. 10 Lower Bound of Performance Theorem: For any metric space, the competitive ratio of the k-server problem is at least k (i.e. k- competitive). Note: This lower bound holds for any randomized algorithm against an optimal online adversary The proof is skipped

11. The Harmonic k-server Algorithm

12. 12 The Harmonic Algorithm Suppose node r makes a request The algorithm works as follows: Let d i be the distance from server i to the request node r If any d i = 0, do nothing (server i will serve the request; no server moves) Else, use server i with probability inversely proportional to d i......

13. 13 The Harmonic Algorithm i.e. let and choose server i with probability. We denote the Harmonic k-server algorithm by Harmonic or H in the following slides Eddie Grove proved that H is -competitive for all.

14. Eddie Grove’s Proof Showing H is -competitive

15. 15 Process of Serving Requests Let be a request sequence of length m Let be the i th request Think of the process of serving requests as follows: For each request, first the adversary moves a server, if necessary, to serve the request Then H “flips a coin” (takes a decision at random according to the pdf mentioned) to choose a server to serve

16. 16 Process of Serving Requests In this way, we have 2m phases Odd phase (phase ): adversary serves Even phase (phase 2i ): H serves Let Dj be the distance moved by the server during phase j Odd j : Distance moved by adversary’s server Even j : Distance moved by H ’s server

17. 17 Introducing the Potential Function To analyze, a function is used Define to be the value of at the end of phase t. is chosen in such a fashion that the following three conditions hold: 1. 2., where c k is the constant to be determined later 3. Referred as Condition (1), (2) and (3) in the following slides

18. 18 Introducing the Potential Function What means? From Vijay Gupta’s lecture: represents the amount of work that H can be forced to do if the offline servers do not move My intuition: “Potential energy”, reserved by adversary moves, consumed by H ’s moves Why introduce ? Lemma: If Condition (1), (2) and (3) hold, then H is c k -competitive.

19. 19 Lemma from 3 Conditions Proof:

20. 20 Lemma from 3 Conditions Now, (1) (2)

21. 21 Lemma from 3 Conditions Using Equation (1) and (2), we have Put Also, by the linearity of expectation, we have But, from Condition (1), Hence,

22. 22 More Notations k offline and k online servers Lower-case letter: online server Capital letter: offline server Perfect matchings M between online and offline servers Denote by M(x) the mate of x Initial condition: every online server coincides with one offline server i.e. In the 0 th phase, d(x, M(x)) = 0 for each online server x

23. 23 Matching M Each time an online server moves, update matching M Example Request placed at offline server A with M(a) = A Online server b, with M(b) = B, moves to the request at A Change matching to: M(b) = A, M(a) = B Matching unchanged for all other servers

24. 24 Active Set Idea of active set is central to the proof Call OFF the set of all k offline servers For and any online server x, the radius of about x is AS(x), the active set of x, is the with largest minimizing

25. 25 Active Set Example k = 4 All offline servers shown; only online server a shown M(a) = A Let Two possible minimizing AS(x) = {A,B,D} 1 1 2 5 a A B C D

26. 26 Active Set Any minimizing set must contain all offline servers within distance of x Intuitively, the active set includes offline servers close to x in comparison to d(x,M(x)) For convenience: Definition:

27. The Potential Function All the 3 conditions satisfied?

28. 28 The Potential Function Definition: The potential function is computed as: Condition (1) is satisfied:, hence, is always non-negative At t=0, every online server and its matched offline server at identical point,

29. 29 Notes before Analysis Condition (2) corresponds to an adversary move Condition (3) corresponds to a Harmonic move Analyzing an (generic) adversary move and a (generic) Harmonic move completes the proof

30. 30 Notes before Analysis In the following analysis, a request is placed at some point Let A be the offline server moved in response to the request, with M(a)=A Let b be the online server moved in response to the request, with M(b)=B Unless otherwise specified, all expressions describe configuration BEFORE the movement Abuse notation: same variable for a server and the point it occupies

31. 31 Analysis of Adversary Moves Let Z be the place of request A moves a distance D 2i+1 to Z in phase 2i+1 Consider the set of servers, Physical meaning: online server with A inside its active set, and now A moves out of its active set boundary For won’t increase

32. 32 Analysis of Adversary Moves Indexing all y h as follows: If a in, y 0 =a ; else no y 0 For h>0, index y h such that When an offline server moves a distance D 2i+1 increases by at most for all Other terms do not increase

33. 33 Analysis of Adversary Moves To estimate the increase in potential, we need to estimate S(y h ) Let Y h be the offline server matched to y h Lemma: For h>1,

34. 34 Analysis of Adversary Moves Proof: Let. Hence Distance from y h to any Y j in T h is bounded by Hence,

35. 35 Analysis of Adversary Moves By the minimality in the definition of, we have Hence

36. 36 Analysis of Adversary Moves The increase in potential due to a move by an offline server of distance D 2i+1 is at most Condition (2) is satisfied with competitive ratio

37. 37 Analysis of Harmonic Moves Three cases Case 1: a serves the request at A (i.e. b is identical to a ) Case 2: B is close to a, Case 3: B is at distance greater than R(a) from a, We will describe sets NS(x) for which AFTER update matching M

38. 38 Harmonic Moves: Case 1 Case 1: a serves the request at A AFTER the move, goes to zero Nothing else is changed Chance is Expected change in potential

39. 39 Harmonic Moves: Case 2 Case 2: B is close to a, For, let NS(x)=AS(x). NS(b)={A} Terms for unaffected Potential decreases by at least This term is dropped in an inequality in later proof

40. 40 Harmonic Moves: Case 3 Case 3: B is at distance greater than R(a) from a, Call B i the offline server that is i th closest to a among offline servers at a distance more than R(a) from a Break any ties arbitrarily Let B l = B Call b i the online server matched to B i b l = b Let d l =d(A,b l )

41. 41 Harmonic Moves: Case 3 For R(a,NS(a)) will be at most Now Since, we have

42. 42 Harmonic Moves: Case 3 Only and changes Expected increase in potential at most The increase happens for each l between 1 and k-S(a)

43. 43 Analysis of Harmonic Moves It remains to show that satisfies Condition (3) From previous results, we see that

44. 44 Analysis of Harmonic Moves The identity, proves that This completes the proof that the Harmonic algorithm is -competitive for all

45. 45 Reference V. Gupta, “CS497 SHT Spring 1999 Prof. Shang-Hua Teng Lecture 12: 2nd March, 1999,” Mar. 1999 E.F. Grove, “The Harmonic online k-server algorithm is competitive,” Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, 1991