The k-server Problem

Study Group: Randomized Algorithm

Presented by Ray LamAugust 16, 2003

2

Outline

1. Background and problem definition

2. The Harmonic k-server Algorithm

3. Proving the claimed performance of the algorithm

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

1 3

4

2 22

3

2

1

1

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...21 i

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

)(OPT

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

)(A

.)()(E a, for allOPTcσA

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

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. 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 .

kdddN

111

21

N

di/1

2k kk k 224/5

Eddie Grove’s Proof

Showing H is -competitive kk k 224/5

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 fo

llows: For each request , first the adversary moves a s

erver, 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

i

i

i

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

12 i ii

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 ck is the constant to be determined later

3. Referred as Condition (1), (2) and (3) in the

following slides

)(t

0(0) 0)( and for all tt12)22()12( ik Dcii

0122E 2 iiD i

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:

(3))2( 0

)22()12(E)12()2(EE

)22()12()12()2(

)22()2(

0

122

122

122

0

and tion from Condi

DciiiiDX

DciiiiD

DciiDX

, andLet X

ikii

iki

ikii

20

Lemma from 3 Conditions

Now,

)0()2() '() '(

)0()2(

)22()2(

1

1 1122

1122

1

mcostsadversaryccostsHX

mDcD

DciiDX

k

m

ii

m

i

m

iiki

m

iiki

m

ii

0EE ,0E 11

m

ii

m

iii XXXSince

(1)

(2)

21

Lemma from 3 Conditions

Using Equation (1) and (2), we have

PutAlso, by the linearity of expectation, we have

But, from Condition (1),

Hence,

)1( 0)0( Conditionfrom

0)0()2() () '(E mcostsadversary'ccostsH k

)2(E)( 'E mOPTcon σcostsH k

0)2(E0)2( mm

)( 'E OPTconcostsH k

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 serv

ers 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 reque

st 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 radi

us of about x is

AS(x), the active set of x, is the with largest minimizing

OFF YYxdxR ),(max),(

)(x

OFFxM

xRkxLet k )(

2

),(min2)(

25

Active Set

Example k = 4 All offline servers shown; only on

line server a shown M(a) = A Let Two possible minimizing

AS(x) = {A,B,D}

5),( ,,, aRCBA

11

2

5

a

A

B C

D

)(a

162

224)(:,,

162

124)(:,

34

24

aDBA

aBA

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:

),( xR

)()( xASxS ))(,()( xASxRxR

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,

xserversonline

x

)(

0)0(

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 reques

t, with M(a)=A Let b be the online server moved in response to the reques

t, 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

)(),( )( hhhh yRZydandyASAy

)( , xx

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

),()(),()( 11 AydyRAydyR hhhh

)( hy)(

12

22

hySik D

k hy

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,

)( 1)( 00 yS A is in AsinceyS

)( 2)( 111 y are in AS A and YsinceyS

hyS h )(

34

Analysis of Adversary Moves

Proof:Let . Hence

Distance from yh to any Yj in Th is bounded by

Hence,

hjYAT jh 1 1hTh

),(),(),( jjjh YydyAdAyd

)(3

)(),(),(

)(),(max),(

),(),(),(max),(

1

1

h

hhh

jjhjh

jjjhhjhh

yR

yRAydAyd

yRAydAyd

YydyAdAydTyR

35

Analysis of Adversary Moves

By the minimality in the definition of , we have

Hence

)( hy

1)( 2

)(3

2

),(

2

)(

hh

Thh

ySh yRTyRyR

hh

hyS h )(

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

kk

ikh

hki kDkD 1

1212

12 24

522

4

1

2

12

2 ,224

5 kforkkc k

k

37

Analysis of Harmonic Moves

Three cases Case 1: a serves the request at A (i.e. b is identica

l 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

)(aASB

)(aASB

)()( xNSxM

)(2

))(,(2)(

xNS

k xNSxRkx

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

)(a

),(

1

AadN

)()( 2

2

2

)(2

),(

1aS

k

aS

k

N

kaRk

AadN

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

bx )(aASB

bx )(b

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 amo

ng 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)

)(aASB

41

Harmonic Moves: Case 3

For R(a,NS(a)) will be at most

Now Since , we have

)()( ,, xASxNSbax l AbNS l )(

)(1)()( li bASliBaASaNS

)()()(),(),( llll bRdaRbRbAdAad

laSaNS )()(

)()( aNSbAS l )(,)(max)( bSlaSaNS

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)

)(a )( lb

laS

k

laSl

bSl

bSl

aSlaSl

k

bSl

aSbSlaSll

l

k

N

k

dbRbRaRaR

dN

k

bRaRbRdaR

dN

k

ll

ll

)(

)()()()()(

)()()(,)(max

2

2

22

)(

2

)(

2

)(

2

)(2

2

)(

2

)(

2

)()(2

43

Analysis of Harmonic Moves

It remains to show that satisfies Condition (3)

From previous results, we see that

0122E 2 iiD i

)(1)()(

)(1)()(

2

2

2

2

2

2

2

2

2

2

),(

),(

122

122E

aSkllaS

k

aS

k

aSkllaS

k

aS

k

xserversonline

i

i

N

k

N

k

N

k

N

k

N

k

xAdN

xAd

iiEDE

iiD

44

Analysis of Harmonic Moves

The identity,

proves that This completes the proof that the Harmonic

algorithm is -competitive for all

)(

)()(

)(1)(

2

2

212

22

2

2

aS

k

kaSaS

k

aSkl

laS

k

k

kk

kk

0122E 2 iiD i

2k kk k 224/5

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