The k-server Problem

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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

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
• 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 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

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

Showing H is -competitive

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

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

All the 3 conditions satisfied?

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
• 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 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
• 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
• 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
• 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,
• Proof:Let . HenceDistance from yh to any Yj in Th is bounded byHence,
• By the minimality in the definition of , we haveHence
• 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
• 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
• 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
• 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
• 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 3
• For
• R(a,NS(a)) will be at most
• Now
• Since , we have
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 Moves
• It remains to show that satisfies Condition(3)
• From previous results, we see that
Analysis of Harmonic Moves
• The identity,proves that
• This completes the proof that the Harmonic algorithm is -competitive for all
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