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The k-server Problem

The k-server Problem. Study Group: Randomized Algorithm Presented by Ray Lam August 16, 2003. Outline. Background and problem definition The Harmonic k-server Algorithm Proving the claimed performance of the algorithm. Background. And Problem Definition. The Metric Space.

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The k-server Problem

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

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

  3. Background And Problem Definition

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

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

  13. The Harmonic Algorithm • i.e. letand 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. Process of Serving Requests • Let be a request sequence of length m • Let be the ith 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. 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. 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: • , where ck is the constant to be determined later • Referred as Condition(1), (2) and (3) in the following slides

  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 ck-competitive.

  19. Lemma from 3 Conditions • Proof:

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

  21. Lemma from 3 Conditions • Using Equation(1) and (2), we havePutAlso, by the linearity of expectation, we haveBut, from Condition (1),Hence,

  22. More Notations • k offline and k online servers • Lower-case letter: online serverCapital 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 0th phase, d(x, M(x)) = 0 for each online server x

  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. 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. 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} B C 5 1 A 1 a 2 D

  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: • Definition:

  27. The Potential Function All the 3 conditions satisfied?

  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. 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. 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. Analysis of Adversary Moves • Let Z be the place of request • A moves a distance D2i+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. Analysis of Adversary Moves • Indexing all yh as follows: • If a in , y0=a; else no y0 • For h>0, index yh such that • When an offline server moves a distance D2i+1 • increases by at most for all • Other terms do not increase

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

  34. Analysis of Adversary Moves • Proof:Let . HenceDistance from yh to any Yj in Th is bounded byHence,

  35. Analysis of Adversary Moves • By the minimality in the definition of , we haveHence

  36. Analysis of Adversary Moves • The increase in potential due to a move by an offline server of distance D2i+1 is at most • Condition (2) is satisfied with competitive ratio

  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. 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. 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. Harmonic Moves: Case 3 • Case 3: B is at distance greater than R(a) from a, • Call Bi the offline server that is ith closest to a among offline servers at a distance more than R(a) from a • Break any ties arbitrarily • Let Bl = B • Call bi the online server matched to Bi • bl = b • Let dl=d(A,bl)

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

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

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

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

  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

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