An optimal on-line algorithm for metrical task system - CS - Huji

An Optimal

On-Line

Algorithm

for Metrical

Task System

ALLAN

BORODIN

Universip

of Toronto,

NATHAN Hebrew

Toronto,

Ontano,

Canada

LINIAL University,

Jerasalem,

Israel,

and IBM

Almaden

Research

Center,

San Jose, California

AND MICHAEL Universip

E.

SAKS

of Callfomia

at San Diego,

San Diego,

California

In practice, almost all dynamic systems require decisions to be made on-line, without full knowledge of their future impact on the system. A general model for the processing of sequences of tasks is introduced, and a general on-line decnion algorithm is developed. It is shown that, for an important class of special cases, this algorithm is optimal among all on-line algorithms. Specifically, a task system (S. d) for processing sequences of tasks consists of a set S of states and a cost matrix d where d(i, j) is the cost of changing from state i to state j (we assume that d satisfies the triangle inequality and all diagonal entries are f)). The cost of processing a given task Tk of tasks is a depends on the state of the system. A schedule for a sequence T1, T2,..., ‘equence sl,s~, ..., Sk of states where s~ is the state in which T’ is processed; the cost of a schedule is the sum of all task processing costs and state transition costs incurred. An on-line scheduling algorithm is one that chooses s, only knowing T1 Tz ~.. T’. Such an algorithm is w-competitive if, on any input task sequence, its cost is within an additive constant of w times the optimal offline schedule cost. The competitive ratio w(S, d) is the infimum w for which there is a w-competitive on-line scheduling algorithm for (S, d). It is shown that w(S, d) = 2 ISI – 1 for eoery task system in which d is symmetric, and w(S, d) = 0(1 S]2 ) for every task system. Finally, randomized on-line scheduling algorithms are introduced. It is shown that for the uniform task system (in which d(i, j) = 1 for all i, j), the expected competitive ratio w(S, d) = O(log 1s1).

Abstract.

A. Borodin was supported in part while he was a Lady Davis visiting professor at the Hebrew University in Jerusalem. M. E. Sakswas supported in part by National Science Foundation grants DMS 87-03541, CCR 89-11388 and Air Force Office of Scientific Research grant AFOSR-0271. Authors’ addresses: A. Borodin, Department of Computer Science. University of Toronto. Toronto, Ontario, Canada M5S 1A4: N. Linial, Institute of Computer Science and Mathematics, Hebrew University, Jerusalem, Israel: M. E. Saks, Department of Computer Science, University of California at San Diego, San Diego, CA. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. G 1992 ACM 0004-5411/92/1000-0745 $01.50 Journalof theAssoc]at]on for Computmg Machinery, Vol 39,No 4,October1992,pp 745-763

A. BORODIN ET AL.

746 Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms Nonnumerical Algorithms and Problems—sequencutg and scheduling: tics]: probabdcstzc algont}vns (mcludmg Monte Carlo )

and Problem Complexity]: [Probability and Statis-

G.3

General Terms: Algorithms, Performance, Theory Additional Key Words and Phrases: Competitive analysis, on-line algorithms

1. Introduction Computers It is usually example,

are typically faced with streams of tasks that they have to perform. the case that a task can be performed in more than one way. For

the system

same piece

of data

The decision system

may have several may reside

as to how to perform

in two

ways:

(1) the

files

a particular

cost

performed and (2) the state affect the cost of subsequent

possible

in several of the

of the tasks.

paging

task affects

present

system

schemes

available

each of which task

upon

or the

can be accessed.

the efficiency

depends

completion

of the

on how

of the

it is

task

may

There are many other situations in which present actions influence our well being in the future. In economics, one often encounters situations in which the choice between present alternatives influences the future costs/benefits. This is, for

example,

selecting

the

among

The difficulty

case in choosing

investment is, of course,

the past and the current measure taken

the

plans

quality

a proposed economists

approach

we take

here,

decision is to devise

was first

optimal

“clairvoyant” no

strategy

probabilistic

that

about

more

papem

approach

studied

in

spans. only on how to usually

model

of the theory

of the

of Markov

in the seminal

the performance of a strategy with the performance of the knowledge the

future

“worst-case” measure of quality. A numb?r of recent this paper analyze various computational problems from 15, 19, 22], and many

based

even clear

The

explicitly

has complete

assumptions

our decision

point

example,

and time

some probabilistic

paper of Sleator and Tarjan [21], is to compare that operates with no knowledge of the future requires

for

rates

It is not

strategy.

is the starting

which

strategy,

interest

that we have to make

future and act on this basis. This decision processes [13]. The

different

task we have to perform.

of

by mathematical

an investment

with

have appeared

of the future. and

This

is therefore

a

articles that preceded this point of view [2, 7,

subsequently.

More precisely, consider a system for processing tasks that can be configured into any of a possible set S of states (we take S to be the set {1, 2, . . . . n}). For any two states i # j, there is a positive transition cost d(i, j) associated with changing from i to j. We always assume that the distance matrix d satisfies the entries are triangle inequality, d(i, j) + d(j. k) > d(i, k) and that the diagonal O. The pair (S, d) is called a task system. For any task T, the cost of processing T is a function of the state of the system so we can view T as a vector T = (T(l),

T(2),...,

ing the task while ofapair{TITz a specified T = TITZ

T(n))

where

in state j. An

T(j)

is the (possibly

instance

infinite)

T of the scheduling

cost of processproblem

consists

.“” T“’, so} where TIT1 .”, Tm is a sequence of tasks and SO is initial state. T is called a task sequence and usually we write simply . . . Tm, where it is understood that there is a specified initial state

SO. (For most of what we do, the particular value of SO is unimportant.) A sclzedule for processing a sequence T = T 1T z . . . T’n of tasks is a function o: {O,. . . . m} + S where u(i) is the state in which T’ is processed (and u(0) = s(,).

Optimal

On-Line

Algorithm

The cost of schedule processing

It will

747

u on T is the sum of all state transition

u)

=

~d(a(i ,=1

be convenient

processed

during

transition

u

times.

a state

to view

the time

schedule

which

Task Systems

costs and the task

costs: c(T;

1.) A

for Metrical

ith

task

1
w~(A).

Let ?-m

c~(T1T2

=

““. T~)

and gm = CO(T1T2 It

is enough

then

there

to show

that

is a constant

for

all

. . . T~).

w G W’,

w > w T(A ). So suppose w G WA; r~ – wg,,l s KW for all m, which implies:

KW such that

r~ — 2n – 1, as desired.

w(A)

> 2n

Remark. Intuitively, the reason that the lower bound on the competitive ratio improves as e decreases is that smaller tasks reduce the amount of information available to the on-line algorithm at the times it must make decisions.

Compare

a sequence

(i.e., the task whose The

cost for the off-line

The

on-line

repeated

of k repeats

costs are obtained

algorithm

algorithm fares better

k times

except

tasks are all equal

to T.

that

from

is easily with

kT:

it is “given

of the task T with those

a single

of T on multiplying

seen to be the same in both Its situation a guarantee”

task

cases.

is the same as with that

the

kT

by k).

following

T k

sequence of tasks PROOF OF THEOREM 2.2. Let T’T2 . . . be the (infinite) produced by M(E) and let o be the schedule produced by A. Each e-elementary task T’ has nonzero cost only in state m(i – 1). Let SI, s?, 1$3,. . . and t1 2~f - 1 is incurred by B

of A(T),

A( ~ *) is in u.

lS1I > L M

2MI be the maximum

modified weight in MSTI. Assume first at u or at a vertex in Sz. Then the portion of the cycle spent in SI consists of 2 ~–}’1 cycles of ~l. By the induction hypothesis, B incurs a cost of 2~’ -1 in each cycle, for a total cost at least 2~- 1. If instead ~* begins at a vertex s + u in SI then the argument is that

the traversal

almost a visit

crosses

that

during

inequality, incurs

and

6. An Algorithm

on-line matching on-line

time since

algorithm upper algorithm

its traversals

moves

our

s‘ until

this exactly with

2, we ratio

B

lower

from bound

A( ~ *) is in Sz, the relevant

in state

we can treat

Section

(u, u ), completes this

while

B waits

competitive

~” begins

the same except that each traversal of ~ ~ by A(~) is interrupted by to Sz, and then resumed later. Consider the moves of B while

xl(~’)

In

and MSTI,

lS1I = 1, so S1 = {u}. Then, definition

(ii)

a cost d(u, u)

of r*.

state in S2 (or vice versa). Then, since MST transition costs at least d(.u, z]) > 2~- 1. Case 2.

weight

incurs

A(7 *) returns

saw

that

Ratio

(2n

the

cruel task

that

anticipates

the cruel

bound

against

guarantees

s“.

B disregards

(~, u). Say

By

the

the

task

triangle cost,

cost to B is at least as much moves

it

as if

to s“. Thus,

– l)x(d)

taskmaster system.

In

strategy this

taskmaster

adversary.

a 21Z – 1 ratio

the case of asymmetric task systems, To motivate the on-line algorithm,

to

to u, and then

in any metrical

that

for

s‘

❑

as before.

Competitive

of ~z and crosses

state

We

forces

a 2n – 1

section,

we present

strategy

and achieves

then

for any input

show

that

sequence

the

an a

same

of tasks. In

a factor of +(d) creeps into the analysis. assume for now that the adversary follows

the cruel taskmaster strategy (for some arbitrarily small e). In this case, each successive task has cost E in the state currently occupied by the on-line algorithm and cost O in every other state. Thus, the entire history of the system (inchzcling the tasks) depends only on the sequence SO,s,, S2,... of states occupied by the on-line algorithm and the transition times tl, tz, . . . Where tl is convenient to the time s, is entered. Instead of using the t, ‘s, it is more parametrize the system history by CO,c1, Cz, . . . where c~ is the cost incurred by the on-line algorithm while in state Sk; here CL = (tk+~ – tkk. The choice of {Sk} and {c~} constitutes a nearly oblivious on-line algorithm. How do we choose {Sk} and {c~}? The key is to understand how the off-line cost changes during some time interval (;, i + At). For a given task sequence T and an arbitrary time

t, let

@,: S -

R be the

off-line cost function; that is, @t(s) is the minimum cost of processing T up to time t, given that the system is in state s at time t. In the present case, where the adversary follows the cruel taskmaster strategy, consider a time ; at which the on-line algorithm is in state ;. Then, for

Optimal

On-Line

sufficiently can

small

stay

off-line

Algorithm At,

in state

@f+Js)

s during

algorithm

tional

for Metrical

can

either

for

interval

occupy

cost of ~At, or occupy

f at time

s +$

(;,; ~ during

some other

since

+ At).

state

the

For

(;,;

off-line

+ At),

s during

algorithm

s = f, note incurring

(;, i + At)

that

the

an addi-

and move

to

~ + At. Thus,

rein{+;(f)+ dt,:n+~(s)

O;+.,(f) = This

= 0;(s) the

755

Task Systems

expression

indicates

that

+,(f)

increases

;)).

+ d(s,

with

t Up until

the time

? such

that there is some state s for which ~;(s) < +;(;) – d(s, i). From that time o% the off-line cost remains constant until the next transition by the on-line algorithm. This suggests the following strategy by the on-line algorithm: stay in S until the time i such that there is some state s such that +;(s) < @f($) – d(s,;)

and then

adversa~ nearly izing

move

is a cruel

oblivious it so that

We define define reader

to state

s. As noted

taskmaster,

this strategy

one. We now describe

this on-line

it applies

the sequences

so

defined

=

defined

At the same time,

the functions

oblivious

we also

is given, for all

s.

f~_l(s)

+ d(s,

fk-l(s)

c~_l

The nearly

general-

Sk. ~ and fk _ ~:

{ fk-,(sk) Having

explicitly,

the as a

off-line cost up to the time the on-line to the off-line algorithm being in state s

Sk = state s minimizing fk(s)

that

from S to the reals. For intuition, the in the case that the tasks are given by the

= o

f,(s) having

algorithm

{Sk} and {c~} inductively.

a sequence { f~} of functions may find it useful to note that

k >1,

if we assume

can be “recomputed”

to any task sequence.

cruel taskmaster, f~(s) is the optimal algorithm moves into state sk, subject at that time. We have:

For

previously,

then

+ ~(s~>sk-1) { fk}

and states

=fk(sk_l)

algorithm

if

s+

if

s ‘s&l

{sk}, we define

–fk_l(s&

defined

s~_l) s&~ } for

“

k z 1,

l).

by {sk} and {c~} above

is denoted

Aj.

We prove: THEOREM PROOF.

6.1.

A:

has competitive

Let T = TITQ

< m be the (continuous)

ratio

at most

(2n

– l)o(d).

““o Tm be any task sequence. Let tl< t2 c times at which A: prescribes state transitions,

“””c t, that

is,

the on-line algorithm enters sk at time tk. Let hk(s) be the cost incurred by an optimal off-line algorithm up to time tk,subject to its being in state s at time tk. (In terms of the function @ defined before, hk(s) = @t$.s).) Let }ZL = be the optimal off-line cost up to time tk. We want to compare rein, ~s hk(S) to prove hk to a~, the cost of the on-line algorithm up to time tk.Specifically, Theorem 6.1 it suffices to prove that for some constant L independent of k,

‘k2 (2n

(6.1) -a~)v(d)

- “

A. BORODIN ET AL.

756 Note

first

that

ak = D~ + CL_,

where

and k

CL =

We need

three

lemmas:

LEMMA

6.2.

hk(s)

> ~~(s) for alls

LEMMA

6.3.

~k(s)

–~~(s’)

LEMMA

6.4.

Let

To deduce

EC,. 1=[)

(6.1) from

h~>

= S.

s d(s’,

s) @

the lemmas,

minfA(s) SES

s’ = S.

we have from

~,z~

>

alls,

~[FL

-

Lemma

2n. maxci(i,

6.2 and 6.3: J)],

1 Fk – B, 22?1–1 where

B is a constant

independent

of k. Using

Lemma

6.4 to substitute

for FL

gives (6.2)

Now

the distance

around

the cycle

so, SI, ...,

sk, SO k

&7ys1,s1 _,)

+d(so,

s,)

,=~ and by the definition

of q(d), k z ,=,

Using

this in (6.2), we get

this is at least

d(sL_J,.s,) +(d)

=

‘h +(d)

“

Optimal

On-Line

Since

~(d)

Algorithm

>1,

for Metrical

we have

C~_ ~ + D/*(d)

‘k> where

L is independent

(2n

757

Task Systems > a/4(d).

-a~)v(d)

of k, as required

Thus,

- “ to complete

the proof

of Theorem

6.1. It remains

to prove

PROOF OF LEMMA k >1,

the lemmas. 6.2.

By induction

on k; for k = O the result

is trivial.

For

we have h~(s)

if s # sk_l.

If s = sk_l,

hk(s~_l)

since either

> min

interval

(in

=f~(s)

>fk_l(S)

we have

hk_l(Sk_l)

the off-line

[t’_~,t’) or makes hk_ l(S)).

{

> h~_l(s)

+

algorithm

min (hk_l(s) s#s~_,

C’_~,

is in state

+ d(s,

sk _ ~ throughout

s’_~)))

the time

a transition from some state s to s~_ ~ during case the cost incurred prior to that transition

which

,

interval

that time is at least

NOW h~_l(s~_l)

by the induction

+ c~_~ >f’_,(s~_~)

hypothesis

+ c’_~

and the definition

‘f~-l(s~)

of c’-

‘f’(s~-~), ~, while

+ d(s’>s~-,)

‘fk(sk-1) by the induction

hypothesis

PROOF OF LEMMA

6.3.

and the definition By induction

s’)

s’-,)

‘f’(s’)

+ d(s’,

‘f’(s’)

s~-1)

s~-l)

(by definition

of s’)

(by definition

Of f’(s’)),

s # s~_l, ‘f’-l(s)
s~-1) s’-l)

(by the induction

hypothesis) ❑

A. BORODIN ET AL.

758 PROOF OF LEMMA 2d(s1,

so). For

(x 2

6.4.

k >1,

H

fk(~) +fk(~k) – 2

5# s,’

Now,

By

induction

it is enough

z

zfk(~k-1) equals

For

On-Line

of Section

k = 1, both

.fk-1(~) +“fk-l(~k-1) so the left-hand –fk-l(sk)

sides

equal

that

)

‘CL-1

side is equal

+~(~~l~k-1).

to

–fk-l(sk-1)> ❑

c~_ ~ + d(,s~, s~_ ~), by definition.

7. Probabilistic The result

k.

s#sk_,

~~(s) = f~ _ ~(s) if s # s~_,,

which

on

to show, by induction,

Algorithms

2 shows that,

in a general

task system, we cannot

hope for

a competitive ratio better than 2n – 1. The ability of the adversaty to force this ratio is highly dependent on its knowledge of the current state of the algorithm. This suggests that the performance can be improved by randomizing some of the decisions made by the on-line algorithm. A randomized on-line scheduling algorithm can be defined simply as a probability distribution on the set of all deterministic algorithms. The adversary, knowing this distribution, selects

the

sequence

deterministic

then

each successive the

notation

2A(T)

=

of tasks the

adversary

can

task.) The resulting of

Section

1) we

~ UC(T; U) . pr(~

probability

to be processed.2

that

A follows

predict

schedule are

(Note with

u on input

if the

algorithm

is

the

response

to

certainty

u is a random

interested

IT) as it relates

that

in

the

variable

relationship

to CO(T), where

T. The expected

and (using between

pr(cr IT) denotes

competitive

ratio

the of A,

E(A), is the infimum over all w such that F~(T) – WC()(T) is bounded above for any finite task sequence, and the randomized competitive ratio, W(s, d), of the task system rithms A. The

above

(S, d) is the definition

game described T’,..., T1, the For

a given

random

infimum

of

iZ(A ) over

can be described

in Section 2. The ith state first i states u(0), u(l),...,

sequence

variable

of

and

the

i tasks,

the

adversary

all randomized

in terms

on-line

algo-

of the scheduler-taskmaster

u(i) is a function of the first i tasks ~(i – 1), and some random bits.

schedule selects

followed the

up to that

(i + l)st

task

time

knowing

is a the

distribution of this random variable, but not the actual schedule that was followed. It is this uncertainty that gives randomized algorithms a potential advantage over purely deterministic ones. Thus. far we have been unable to analyze the expected competitive ratio for arbitrary task systems. However, for the uniform task system, the system where all state transitions have identical determine the expected competitive randomization provides a substantial that

the competitive

ratio

H(n)= 1 + 1/2+ 1/3+ in n and 1 + in ~z.

(say unit) cost, we have been able to ratio to within a factor of 2. In this case, reduction in the competitive ratio. (Recall

for a deterministic algorithm It is well known ““. + l/n.

2 Such an adversary is now called an “oblivious in Raghavan and Snir [18] and further studied

is at least 2 n – 1.) Let that H(n) is between

adversary”. Adaptive adversaries were introduced in Ben-David et al. [1],

Optimal

On-Line

THEOREM

Algorithm

7.1.

for Metrical

If (S, d) is the uniform H(n)

PROOF

OF THE

randomized algorithm proceeds

Task Systems

on-line

UPPER

s iZ(S,

BOUND.

algorithms

that achieves in a sequence

759

task system on n states, then d)

< 2H(Jz).

Note

first

that

and thus it suffices

the desired expected of phases. The time

Lemma

3.1

to describe

extends

to

a continuous-time

competitive ratio. The algorithm at which the ith phase begins is

denoted t,_ ~. For a state s, let A, be the algorithm that stays in state s for the entire algorithm. A state s is said to be saturated for phase i at time t > t, – 1, if by As during the time interval [t, _ ~, t] is at least 1. The ith the cost incurrecl

t,when

phase ends at the instant a phase until

the

that

on-line

state. Note probability

proceeds

saturated

as follows:

and then

jump

for that phase.

remain

in the

randomly

During

same

state

to an unsaturated

that at the end of a phase, the on-line algorithm moves with to some state, with all states again becoming unsaturated.

To analyze phase,

algorithm

state becomes

all states are saturated

the expected

any off-line

competitive

algorithm

incurs

ratio

of this

algorithm

a cost of at least

1, either

equal

during

a single

because

it makes

a state transition or because it remains in some state during the entire phase and that state becomes saturated. Hence, it suffices to show that the expected cost

of the

number

on-line

of state

algorithm

transitions

Let

f(k)

be the

the end of the phase

given

that

is at most until

2H( n).

expected

there

are k

is unsaturated states remaining. Note f(1) = 1. For k > 1, since the algorithm in each unsaturated state with equal probability, the probability that the next state to become

saturated

+ l/k, so that f(k) ing costs + transition

results

PROOF OF THE LOWER Neumann

minimax

performance be obtained the

task

sequence) on

BOUND.

theorem

for

and

algorithm the

expected

proving on this

LEMMA

Yao

7.2.

[23]

competitive

Thus,

has

noted

games,

that

a lower

bound

on the

We use this

Let

= f(k

(task

– 1)

process-

using

the

bound

ratio

of

expected

principle

on

a randomized

let TJ denote

the first

E denote

expectation

cost

to derive

von the

of any a lower

algorithm

on

a

j tasks.

Let (S, d) be a task system and D be a probabili~ task sequences.

f(k)

in these (and most other) models can distribution on the input (in this case,

a lower input.

uniform task system. If T is an infinite task sequence,

set of infinite

is l\k.

cost of the phase ❑

two-person

of randomized algorithms by choosing any probability

deterministic bound

in a transition

= H(k) and the expected costs) is at most 2H(n).

with

measure respect

suppose that E(cO(TJ )) tends to infinity with j. Let mJ denote the minimum deterministic on-line algorithms A of E(cti(T])). Then

on the

to D and overall

mJ ii(S,

d)

> limsup ~+~

E(cO(T’))

“

PROOF. By the definition of i7(S, d), there exists a randomized algorithm R and a constant c such that for all infinite task sequences T and integers j, CR(TJ ) – W(S, d)cO(TJ ) < c. Averaging over the distribution D, we obtain that for each j, E(cR(T])) – iiXS, d) E(cO(TJ)) < c or iiXS, d) > E(c~(TJ))\ E(CO(TJ)) – C/E(CO(TI)). Since E(c~(TJ)) > ml and E(cO(TJ)) tends to infinity we obtain lim sup m,\E(cO(T1)) < EJ(S, d). ❑

A. BORODIN ET AL.

760 Now

for

s c S, define

cost of 1 in state sequences uniformly

U, to be the unit

s and O in any other

obtained by generating from the unit elementary

and E(cO(T’))

< j/(nH(n))

elementary

state.

Let

task with

a processing

D be the distribution

on task

each successive task independently and tasks. We show that for each j, m, > j/n 7.2, we obtain ‘ Applying Lemma

+ 0(1).

j/n i7(S,

d)

= H(n),

> lim sup ,4= J/’(~~(~l)

+ 0(1))

as required. Ide~tify a sequence of unit elementa~ that define each task. Note that sl, s~, ..., deterministic

on-line

algorithm

A, then,

tasks with the sequence of states if u is the schedule produced by a

for

each time

step j, A incurs

a cost

of 1 at time step j if SJ = cr(j – 1). This happens with probability l/n, and hence the expected cost of processing each task is at least l/n and rnj > j\n. Now

for each time

sequence off-line unless

i, let

a(i) be the least index greater than i such that the contains all states of S. Consider the following

s~+l,st+z,...,s~(i~

algorithm: letting U,, denote the ith m(i – 1) =s,, in which case let u(i)

off-line strategy fact.) We claim this strategy algorithm and incurs

after

the first

incurs

j tasks, as a function

a cost of one exactly

zero cost at other

times.

the total

Then,

X~ s j}

and the

c1 = max{k:

distributed

{c,: j > 1} is a renewal Theorem

3.8]),

the

limit

compute

Exp( Y~ ), note

random

process. of the that

at those

Let

at which

identically

i do not change state is in fact the optimal

for each such task sequence, although we do not need that that the expected cost (with respect to the distribution D) of

let X~ be the time dent

task, at time = s.(,). (This

cost incurred quantities:

time

– 1) = S,,

up to time

j and

reaches

k.

Y~ = X~ – X~ _ ~ are indepenwhich

Exp(cJ )/j

Y~ is the

u(i

by the algorithm

By the elementary ratio

i such that

C-Jbe the cost incurred

variables,

+ 0(1).The

of j, is j/(nH(rz)) times

means renewal

exists until

and

that

the sequence

theorem

(see [20,

is l/Exp(Y~).

a sequence

of

To

randomly

generated states contains all of the states. When i of the states have appeared in the sequence the probability that the next state is a new state is (n – i)/n and hence the expected number of states between the ith and (i + l)st has appeared is n/(rz – i). Summing on i yields Exp(Y~) = nH(n). ❑ Upper and lower task systems would is for the system

state

bounds on the expected competitive ratio for other specific be of interest. In particular, we would like to know what it in which d is the graph distance on a cycle; that is,

d(s,, s]) = min(lj – il, n – lj – ii). We suspect that there is an O(log n) upper bound for any task system on n states, but the analysis of iZ(,S. d) for arbitrary task systems seems to be a challenging problem.

8. Conclusion We have introduced an abstract framework for studying the efficiency of on-line algorithms relative to optimal off-line algorithms. With regard to deterministic algorithms, we have established an optimal 2n – 1 bound on the competitive ratio for any n state metrical task system. For randomized algorithms, we have only succeeded in analyzing the expected competitive ratio for the uniform task system.

Optimal

On-Line

Algorithm

for Metrical

Task Systems

761

One must ask about the “utility” of such an abstract model. (For the utility of an even more abstract model, we refer the reader to Ben-David et al. [1]). As stated

in the

unduly

pessimistic

utility

of a general

application model,

introduction,

techniques

One

such

optimal

our model

model

of results,

Manasse

our

in that

should

but

one

not

suggest

appealing

setting

very

for

consider

is

the

concrete

k-server satisfy

extent

of

the

direct

to which

the

settings.

model

requests

is

that

in terms

the

in more

servers

algorithms

tasks. We believe

be measured

also results

k mobile

deterministic

arbitraxy

only

should

and results

et al. [16]. Here,

bound

allows

introduced that

come

by in the

form of a command edges have lengths

to place a server on one of n nodes in a graph G whose that satisfy the triangle inequality. (In their definition, the

edge

not

lengths

need

be symmetric

but

since

all the

results

in [16]

assume

symmetry, we make this assumption to simplify the discussion.) In fact, the model permits n to be infinite as, for example, when one takes G to be all of Rd. As observed

in [16], there

k server

problem

problem

is equivalent

tasks

(i.e.,

and ~ in that

relation

task system

those

that

remaining

are defined

by a given

k server

tasks

are

studies

with

restricted

the

matrix) to

problem

arbitrary

competitil’e

ratio.

to task systems

that

bound

the proof

by using

n – 1 is the

of Theorem when

applying

2.2 could

tasks

k server

the same adversary

2.2 and then

bound

show

of a n = k + 1 node,

is derived

our Theorem lower

et al. [16]

in

problem

where

the

elementary tasks. For both problems, one Our 2n – 1 upper bound then immediately

Indeed

ratio

is zero

and in this case the n = k + 1

is equivalent

(with or without to the k server

Manasse

that

of the k server

(where servers can either move to where the costs of such excursions

applies in the n = k + 1, k server lower bound only applies directly competitive

k server

the tasks are restricted

a cost vector

A variant

is also introduced

excursion

the n = k + 1 node,

an n = k + 1 node, where

have

state).

called the k-server with excursions problem service a request or else make an excursion node,

between

To be precise,

to an n state

to be ~-elementary n – 1 states

is a strong

and task systems.

a nice

optimal

problem.

bound

Their

(i.e., the cruel

averaging

be modified

are restricted

excursions) case but the with excursion problem.

taskmaster)

argument.

so as to yield

to ~-elementary

for

as in

(Alternatively,

the optimal

tasks.)

the

n – 1 lower

The

n – 1

k = n – 1

upper bound requires a new and intuitively appealing algorithm. By a quite different algorithm, Manasse et al. [16] also prove that for all k = 2 server problems, the competitive ratio is 2 and they make the following intriguing conjecture:

the

competitive

ratio

for

every

k-server

problem

(without

excur-

from the previous discussion since sions) is k. A lower bound of k is immediate such a bound can be derived when one restricts the adversary to n = k + 1 nodes. Our task system framework provides a little positive reinforcement for the conjecture the

competitive

which possible induced

(at

least)

in that ratio

our deterministic

upper

for

k-server

any

is independent

positions of the state transition

n-node, of the

edge

bound

can be used to show that

problem lengths.

is at most Here

servers becomes a state of the task costs inherit the triangle inequality

each

2(Y) of

the

– 1 (i!)

system and the and symmetry

properties. A node request becomes a task whose components are either zero (if the state contains the requested node) or infinite (if the state does not contain the requested node). Although a state transition may represent moving more than one server, as noted in [16], we need only move the server that covers the request and treat the state as representing the virtual position of

A. BORODIN ET AL.

762 servers

while

simulation,

remembering

our Theorem

the

physical

position

6.1 also provides

of all

servers.

some information

By the

same

for nonsymmetric

k-server problems. We note that there is now an impressive literature concerning the k-server conjecture, including optimal results on specific server systems (e.g., [5]. [61), as well as results yielding upper arbitrary k-server systems (see [10] and [12]). We note however general, be formulated the position but

(personal

of the servers, communication)

with

the

edge

(exponential

in k) for

that the k-server with excursion problem cannot, in as a task system problem. For now, if a state represents the cost of a task is not just

also of how we reached

grow

bounds

this state. has shown

lengths.

Indeed, that

for

a function

the competitive

(However,

we

of the state

n > k + 2, Shai Ben-David

should

ratio note

can be made

that

example uses excursion costs which do not satisfy the triangle inequality. suggestion is to redefine the problem so that servers can “move and rather than “move or serve”. This one server by Chung et al. [7].

point

of view

Our results on the expected competitive the unit-cost task system has a very nice problem that is better known as the paging problem,

Fiat

competitive ized

et al. [9] have established

ratio

algorithm.

case when

using

n = k + 1, their

7.1. A direct

modification

algorithm of our

establish an H(k) lower bound ized on-line paging algorithms. constructed number of

to note

upper

(and

possibly

that

in the

becomes

lower

) One serve”

in the case of

ratio of randomized algorithms for analogue in the unit-cost k-server or caching problem. For the paging

a 2H(k)

a very appealing

It is interesting

is investigated

to

Ben-David’s

bound

bound quite

on the expected

practical)

(admittedly

the algorithm in Theorem

random-

nonpractical) of our Theorem

7.1 can be used to

for the expected competitive ratio of randomSubsequently, McGeoch and Sleator [17] have

a randomized algorithm with a matching upper other important results on paging problems,

between randomization and memory, ized algorithms in the context of

bound of H(k). A on the trade-off

and on a more general study competitive ratios for on-line

of randomalgorithms

(against various types of adversaries) can be found in Raghavan and Snir [18], Coppersmith et al. [8], and Ben-David et al. [1]. The papers of Manasse et al. [19], Fiat et al. [9], and Raghavan and Snir [18] and the original papers by Sleator and Tarjan [21] and Karlin et al. [15] argue for the development of a theory (or rather theories) for competitive analysis. Such a study

seems fundamental

to many

addition to computer science and, alternative models and performance

other

disciplines

as such, we measures.

expect

to

(e.g., economics)

in

see a number

of

The idea of developing a general model for on-line task ACKNOWLEDGMENTS. systems was introduced to us in various conversations with Fan Chung, Lam-y Rudolph, Danny Sleator, and Marc Snir. Don Coppersmith suggested that randomization may be useful in this context. We gratefully acknowledge their contributions to this paper. Finally, we thank the referees and Gordon Turpin for their careful reading and suggestions concerning the submitted manuscript.

REFERENCES 1. BEN-DAYIIZ S., BORODIN,A., KARP, R.. TARDOS,G., AND WIGDERSON,A.

On the power of on in online algorithms. In F’roceedufgs of the 22nd Annual A CM Syrnposmm of Computing (Baltimore, Md., May 12-14). ACM, New York, 1990, pp. 379-386.

randomization Theov

Optimal

On-Line

Algorithm

for Metrical

Task Systems

763

2. BENTLEY, J. L., AND MCGEOCH, C. C. Amortized analyses of self-organizing sequential ACM 28, 4 (Apr. 1985), 404–411. search heuristics. Comrnun. 3. BITNER, J. R. Heuristics that dynamically alter data structures to reduce their access time. Ph.D. dissertation. Univ. Illinois, 1976. that dynamically organize data structures. ,SL4M J. Cornput. 8 4. BITNER, J. R. Heuristics (1979), 82-110. S. New results on sezver 5. CHROBAK.M., KARLOFF,H. J., PAYNE, T., AND VISHWANA’ZTZAN, problems. NAM J. Disc. Math. 4, 2 (1991), 172-181. 6. CHROBAK,M., AND LARMORE,L. An optimal on-line algorithm for the server problem on trees. SIAM J. CornpLtt. 20 (1991), 144-148. A dynamic location problem for graphs. 7. CHUNG,F. R. K., GRAHAM, R. L., AND SAKS, M. Combinzzforica 9 (1989), 111-131. D., DOYLE, P., RAGHAVAN,P., AND SNIR, M. Random walks on weighted 8. COPPERSMITH, of the 22nd Arznual ACM graphs and applications to on-line algorithms. In Proceedings Symposium on Theory of Cornputmg (Baltimore, Md., May 12-14). ACM, New York, 1990, pp. 369-378. 9. FIAT, A.. KARP, R., LUBY, M., MCGEOCH, L., SLEATOR,D., AND YOUNG, N. Competitive paging algorithms. J. Algotithnzs 12 (1991), 685-699. k-Server Algorithms. In Proceedings of the 10. FIAT, A., RABANI,Y., AND RAVID, Y. Competitive 31st Annual Symposium on Foundations of Computer Science. IEEE, New York, 1990, pp. 454-463. Exegesis of self-organizing linear search. 11. GONNET, G., MUNRO, J. I., AND SUWANDA,H. SIAM J. Conzput. 10, 3 (1981), 613-637. 12. GROVE,E. The harmonic online k-server algorithm is competitive. In Proceedings of the 23rd Azzrzual ACM Symposium on Theo~ of Computing (New Orleans, La., May 6–8). ACM, New York, 1991, pp. 260-266. 13. HEYMAN,D., AND SOBEL,M. Stochastic Models in Operations Research. vol. II. McGraw-Hill, New York, 1984. Optimal list order under partial memory constraints. J. Appl. 14. KAN, Y. C., AND Ross, S. M. Prob. 15.

17 (1980),

caching. 16.

Algotithmica

3, 1 (1988),

11 (1990),

snoopy

Competitive

algorithms

for server problems.

208-230.

MCGEOCH,L. A., ANDSLEATOR,D. D. Algon”thmica,

Competitive

79–119.

MANASSE,M., MCGEOCH,L., AND SLEATOR,D. J. Algorithms

17.

1004-1015.

KARLIN, A. R., MANASSE,M. S., RUDOLPH,L., AND SLEATOR,D. D.

A strongly

competitive

randomized

paging algorithm.

to appear.

in on-line algorithms”, Au18. RAGHAVAN,P., AND SNIR, M. “Memozy versus randomization tomata, Languages and Programming, Lecture Notes in Computer Science v. 372, SpringerVerlag, New York, 1989, pp. 687-703. 19. RIVEST, R. On self-organizing sequential search heuristics. Commun. ACM 19, 2 (Feb. 1976), 63-67. Models with Optimization Applications. Holden-Day, Calif. 1970. 20. ROSS,S. Applied Probabili~ efficiency of list Update and Paging rules. 21. SLEATOR,D. D., AND TARJAN, R. E. Amortized Commun. 22.

ACM

28 (1985),

TARJAN, R. E.

Amortized

202-208.

computational

complexity.

SZAM

J. Alg.

Disc.

Math.

6 (1985),

306-318. 23.

YAO, A.

Probabilistic

ings of the 18th Annual

computations: Symposwm

Towards

a unified

on Foundations

measure

of Computer

of complexity. Science.

In Proceed-

IEEE, New York

1977, PP. 222-227. RECEIVEDJULY 1988; REVISEDAPRIL 1991; ACCEPTEDMAY 1991

Journalof theAssoclatlon for Computmg Machinery, Vol.39,No.4,October1992