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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

The k-server Problem

Study Group: Randomized Algorithm

Presented by Ray Lam

August 16, 2003

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


And Problem Definition

the metric space
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.
the metric space1
The Metric Space
  • Think of it as a complete weighted graph
  • Weight corresponds to distance between points











the k server problem1
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
the k server problem2
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
the k server problem3
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
the k server problem4
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
lower bound of performance
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
the harmonic algorithm
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......
the harmonic algorithm1
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 .
eddie grove s proof

Eddie Grove’s Proof

Showing H is -competitive

process of serving requests
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
process of serving requests1
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
introducing the potential function
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
introducing the potential function1
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.
lemma from 3 conditions2
Lemma from 3 Conditions
  • Using Equation(1) and (2), we havePutAlso, by the linearity of expectation, we haveBut, from Condition (1),Hence,
more notations
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
matching m
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
active set
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
active set1
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}










active set2
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:
the potential function

The Potential Function

All the 3 conditions satisfied?

the potential function1
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,
notes before analysis
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
notes before analysis1
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
analysis of adversary moves
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
analysis of adversary moves1
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
analysis of adversary moves2
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,
analysis of adversary moves3
Analysis of Adversary Moves
  • Proof:Let . HenceDistance from yh to any Yj in Th is bounded byHence,
analysis of adversary moves4
Analysis of Adversary Moves
  • By the minimality in the definition of , we haveHence
analysis of adversary moves5
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
analysis of harmonic moves
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
harmonic moves case 1
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
harmonic moves case 2
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
harmonic moves case 3
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)
harmonic moves case 31
Harmonic Moves: Case 3
  • For
  • R(a,NS(a)) will be at most
  • Now
  • Since , we have
harmonic moves case 32
Harmonic Moves: Case 3
  • Only and changes
  • Expected increase in potential at most
  • The increase happens for each l between 1 andk-S(a)
analysis of harmonic moves1
Analysis of Harmonic Moves
  • It remains to show that satisfies Condition(3)
  • From previous results, we see that
analysis of harmonic moves2
Analysis of Harmonic Moves
  • The identity,proves that
  • This completes the proof that the Harmonic algorithm is -competitive for all
  • 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