1 / 45

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.

isaiah
Download Presentation

The k-server Problem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related