Lower

Bounds

Competitive

Algorithms

for

Online

Scheduling

of

Unit-Size

Tasks

to

Related

Machines

In

this

paper

we

study

the

problem

of

assigning

unit-size

tasks

to

related

machines

when

only

limited

online

information

is

provided

to

each

task

This

is

a

general

framework

whose

special

cases

are

the

classical

multiple-choice

games

for

the

assignment

of

unit-size

tasks

to

identical

machines

The

latter

case

was

the

subject

of

intensive

research

for

the

last

decade

The

problem

is

intriguing

in

the

sense

that

the

natural

extensions

of

the

greedy

oblivious

schedulers

which

are

known

to

achieve

near-optimal

performance

in

the

case

of

identical

machines

are

proved

to

perform

quite

poorly

in

the

case

of

the

related

machines

In

this

work

we

present

a

rather

surprising

lower

bound

stating

that

any

oblivious

scheduler

that

assigns

an

arbitrary

number

of

tasks

to

n

related

machines

would

need

log

n

loglog

n

polls

of

machine

loads

per

task

in

order

to

achieve

a

constant

competitive

ratio

versus

the

optimum

offline

assignment

of

the

same

input

sequence

to

these

machines

On

the

other

hand

we

prove

that

the

missing

information

for

an

oblivious

scheduler

to

perform

almost

optimally

is

the

amount

of

tasks

to

be

inserted

into

the

system

In

particular

we

provide

an

oblivious

scheduler

that

only

uses

O

-LRB-

loglog

n

-RRB-

polls

along

with

the

additional

information

of

the

size

of

the

input

sequence

in

order

to

achieve

a

constant

competitive

ratio

vs.

the

optimum

offline

assignment

The

philosophy

of

this

scheduler

is

based

on

an

interesting

exploitation

of

the

slowfit

concept

-LRB-

-LSB-

-RSB-

for

a

survey

see

-LSB-

-RSB-

-RRB-

for

the

assignment

of

the

tasks

to

the

related

machines

despite

the

restrictions

on

the

provided

online

information

in

combination

with

a

layered

in-This

work

has

been

partially

supported

by

the

IST

Program

of

the

European

Union

under

contract

number

-LRB-

ALCOM-FT

-RRB-

A

full

version

of

this

paper

is

available

at

spyros

Permission

to

make

digital

or

hard

copies

of

all

or

part

of

this

work

for

personal

or

classroom

use

is

granted

without

fee

provided

that

copies

are

not

made

or

distributed

for

profit

or

commercial

advantage

and

that

copies

bear

this

notice

and

the

full

citation

on

the

first

page

To

copy

otherwise

to

republish

to

post

on

servers

or

to

redistribute

to

lists

requires

prior

specific

permission

and/or

a

fee

STOC

May

Montreal

Quebec

Canada

Copyright

ACM

duction

argument

for

bounding

the

tails

of

the

number

of

tasks

passing

from

slower

to

faster

machines

We

finally

use

this

oblivious

scheduler

as

the

core

of

an

adaptive

scheduler

that

does

not

demand

the

knowledge

of

the

input

sequence

and

yet

achieves

almost

the

same

performance

The

problem

of

the

Online

Assignment

of

Tasks

to

Related

Machines

is

defined

as

follows

there

are

n

machines

possibly

of

different

speeds

that

are

determined

by

a

speed

vector

c

-LRB-

C

C

n

-RRB-

IR

n

and

an

input

sequence

of

m

independent

tasks

to

be

assigned

to

these

machines

The

tasks

arrive

sequentially

along

with

their

associated

weights

-LRB-

positive

reals

-RRB-

and

have

to

be

assigned

immediately

and

uniquely

to

the

machines

of

the

system

The

size

of

the

input

sequence

as

well

as

the

weights

of

the

tasks

are

determined

by

an

oblivious

adversary

-LRB-

denoted

here

by

ADV

-RRB-

Each

task

has

to

be

assigned

upon

its

arrival

to

one

of

the

machines

using

the

following

information

-LRB-

possibly

a

portion

of

-RRB-

the

online

information

of

current

status

of

the

system

the

offline

information

of

the

machine

speeds

and

its

own

weight

The

tasks

are

considered

to

be

of

infinite

duration

-LRB-

permanent

tasks

-RRB-

and

no

preemption

is

allowed

The

cost

of

an

online

scheduler

ALG

for

the

assignment

of

an

input

sequence

of

tasks

-LRB-

denoted

by

ALG

-LRB-

-RRB-

-RRB-

is

the

maximum

load

eventually

appearing

in

the

system

In

case

that

a

random-ized

scheduler

is

taken

into

account

then

the

cost

of

the

scheduler

is

the

expectation

of

the

corresponding

random

variable

The

quality

of

an

online

scheduler

is

compared

vs.

the

optimum

offline

assignment

of

the

same

input

sequence

to

the

n

machines

We

denote

the

optimum

offline

cost

for

by

ADV

-LRB-

-RRB-

That

is

we

consider

the

competitive

ratio

-LRB-

or

performance

guarantee

-RRB-

to

be

the

quality

measure

-LRB-

eg

see

-LSB-

-RSB-

-RRB-

Definition

An

online

scheduler

ALG

is

said

to

achieve

a

competitive

ratio

of

parameters

-LRB-

a

-RRB-

if

for

any

input

sequence

the

relation

connecting

its

own

cost

ALG

-LRB-

-RRB-

and

the

optimum

offline

cost

of

ADV

are

related

by

ALG

-LRB-

-RRB-

a

ADV

-LRB-

-RRB-

ALG

is

strictly

a-competitive

if

ALG

-LRB-

-RRB-

a

ADV

-LRB-

-RRB-

In

this

work

we

study

the

consequences

of

providing

only

some

portion

of

the

online

information

to

a

scheduler

That

is

we

focus

our

interest

on

the

case

where

each

task

is

capable

of

checking

the

online

status

only

by

a

-LRB-

small

wrt

n

-RRB-

number

d

of

polls

from

the

n

machines

In

this

case

the

objective

is

to

determine

the

trade-off

between

the

number

of

polls

that

are

available

to

each

of

the

tasks

and

the

performance

guarantee

of

the

online

scheduler

or

equivalently

to

determine

the

minimum

number

of

polls

per

task

so

that

a

strictly

constant

competitive

ratio

is

achieved

Additionally

we

consider

the

case

of

unit-size

tasks

that

are

assigned

to

related

machines

Thus

each

task

t

-LSB-

m

-RSB-

has

to

be

assigned

to

a

machine

host

-LRB-

t

-RRB-

-LSB-

n

-RSB-

using

the

following

information

that

is

provided

to

it

the

current

loads

of

d

suitably

chosen

machines

-LRB-

the

kind

of

the

suitable

selection

is

one

of

the

basic

elements

of

a

scheduler

and

will

be

called

the

polling

strategy

from

now

on

-RRB-

and

an

assignment

strategy

that

determines

the

host

of

t

among

these

d

selected

candidates

on

behalf

of

t

In

what

follows

we

shall

consider

homogeneous

schedulers

ie

schedulers

that

apply

exactly

the

same

protocol

on

all

the

tasks

that

are

inserted

into

the

system

This

choice

is

justified

by

the

fact

that

no

task

is

allowed

to

have

access

to

knowledge

concerning

previous

or

forthcoming

choices

of

other

tasks

in

the

system

except

only

for

the

current

loads

of

those

machines

that

have

been

chosen

to

be

its

candidate

hosts

Additionally

we

shall

use

the

terms

-LRB-

capacitated

-RRB-

bins

instead

of

-LRB-

related

-RRB-

machines

and

-LRB-

identical

-RRB-

balls

instead

of

-LRB-

unit-size

-RRB-

tasks

interchangeably

due

to

the

profound

analogy

of

the

problem

under

consideration

with

the

corresponding

Balls

Bins

problem

Polling

Strategies

The

way

a

scheduler

ALG

lets

each

newly

inserted

task

choose

its

d

candidate

hosts

is

called

a

polling

strategy

-LRB-

PS

-RRB-

We

call

the

strategies

that

poll

candidate

machines

homogeneously

for

all

the

inserted

tasks

of

the

same

size

homogeneous

polling

strategies

-LRB-

HPS

-RRB-

In

the

present

work

we

consider

the

tasks

to

be

indistinguishable

Each

task

upon

its

arrival

knows

only

the

loads

of

the

machines

that

it

polls

along

with

the

speed

-LRB-

or

equiv

capacity

wrt

bins

-RRB-

vector

c

of

the

system

This

is

why

we

focus

our

interest

in

schedulers

belonging

to

HPS

Depending

on

the

dependencies

of

the

polls

that

are

taken

on

behalf

of

a

task

we

classify

the

polling

strategies

as

follows

Oblivious

polling

strategies

-LRB-

HOPS

-RRB-

In

this

case

we

consider

that

the

polling

strategy

on

behalf

of

a

newly

inserted

task

t

consists

of

an

independent

-LRB-

from

other

tasks

-RRB-

choice

of

a

d-tuple

from

-LSB-

n

-RSB-

d

according

to

a

fixed

probability

distribution

f

-LSB-

n

-RSB-

d

-LSB-

-RSB-

This

probability

distribution

may

only

depend

on

the

common

offline

information

provided

to

each

of

the

tasks

It

should

be

clear

that

any

kind

of

d

independent

polls

-LRB-

with

or

without

replacement

-RRB-

on

behalf

of

each

task

falls

into

this

family

of

polling

strategies

Thus

the

whole

polling

strategy

is

the

sequence

of

m

d-tuples

chosen

independently

-LRB-

using

the

same

probability

distribution

f

-RRB-

on

behalf

of

the

m

tasks

that

are

to

be

inserted

into

the

system

Clearly

for

any

polling

strategy

belonging

to

HOPS

the

d

random

polls

on

behalf

of

each

of

the

m

tasks

could

have

been

fixed

prior

to

the

start

of

the

assignments

Adaptive

polling

strategies

-LRB-

HAPS

-RRB-

In

this

case

the

i

th

poll

on

behalf

of

ball

t

-LSB-

m

-RSB-

is

allowed

to

exploit

the

information

gained

by

the

i

previous

polls

on

behalf

of

the

same

ball

That

is

unlike

the

case

of

HOPS

where

the

choice

of

d

candidates

of

a

task

was

oblivious

to

the

current

system

state

now

the

polling

strategy

is

allowed

to

direct

the

next

poll

to

specific

machines

of

the

system

according

to

the

outcome

of

the

polls

up

to

this

point

In

this

case

all

the

polls

on

behalf

of

each

task

have

to

be

taken

at

runtime

upon

the

arrival

of

the

task

Remark

It

is

commented

that

this

kind

of

polling

strategies

are

not

actually

helpful

in

the

case

of

identical

machines

where

HOPS

schedulers

achieve

asymptotically

optimal

performance

-LRB-

see

-LSB-

-RSB-

-RRB-

Nevertheless

we

prove

here

that

this

is

not

the

case

for

the

related

machines

It

will

be

shown

that

oblivious

strategies

perform

rather

poorly

in

this

setting

while

HAPS

schedulers

achieve

actually

asymptotically

optimal

performance

Assignment

Strategies

Having

chosen

the

d-size

set

of

candidate

hosts

for

task

t

-LSB-

m

-RSB-

the

next

thing

is

to

assign

this

task

to

one

of

these

machines

given

their

current

loads

and

possibly

exploiting

the

knowledge

on

the

way

that

they

were

selected

We

call

this

procedure

the

assignment

strategy

The

significant

question

that

arises

here

is

the

following

Given

the

polling

strategy

adopted

and

the

knowledge

that

is

acquired

at

runtime

by

the

polled

d-tuple

on

behalf

of

a

task

t

-LSB-

m

-RSB-

which

would

be

the

optimal

assignment

strategy

for

this

task

so

that

the

eventual

maximum

load

in

the

system

be

minimized

In

the

Unit

Size

Tasks-Identical

Machines

case

when

each

of

the

d

polls

is

chosen

iur

-LRB-

with

replacement

-RRB-

from

-LSB-

n

-RSB-

Azar

et

al.

-LRB-

-LSB-

-RSB-

-RRB-

show

that

the

best

assignment

strategy

is

the

minimum

load

rule

and

requires

O

-LRB-

log

n

-RRB-

polls

per

task

for

a

strictly

constant

competitive

ratio

Consequently

V

ocking

-LRB-

-LSB-

-RSB-

-RRB-

has

suggested

the

always

go

left

strategy

which

-LRB-

in

combination

with

a

properly

chosen

oblivious

polling

strategy

-RRB-

only

requires

a

total

number

of

O

-LRB-

loglog

n

-RRB-

in

order

to

achieve

a

constant

competitive

ratio

In

the

same

work

it

was

also

shown

that

one

should

not

expect

much

by

exploiting

possible

dependencies

of

the

polls

in

the

case

of

unit-size

tasks

that

are

placed

into

identical

machines

since

the

load

of

the

fullest

machine

is

roughly

the

same

as

the

one

achieved

in

the

case

of

non-uniform

but

independent

polls

using

the

always

go

left

rule

Nevertheless

things

are

quite

different

in

the

Related

Machines

case

we

show

by

our

lower

bound

-LRB-

section

-RRB-

that

even

if

a

scheduler

ALG

considers

any

oblivious

polling

strategy

and

the

best

possible

assignment

strategy

ALG

has

a

strict

competitive

ratio

of

at

least

n

where

d

is

the

number

of

oblivious

polls

per

task

This

implies

that

in

the

case

of

the

related

machines

there

is

still

much

space

for

the

adaptive

polling

strategies

until

the

lower

bound

of

-LRB-

loglog

n

-RRB-

polls

per

task

is

matched

Related

Work

In

the

case

of

assigning

unit-size

tasks

to

identical

machines

there

has

been

a

lot

of

intensive

research

during

the

last

decade

If

each

task

is

capable

of

viewing

the

whole

status

of

the

system

upon

its

arrival

-LRB-

we

call

this

the

Full

Information

case

-RRB-

then

Graham

's

greedy

algorithm

assures

a

competitive

ratio

that

asymptotically

tends

to

n

-LRB-

-LSB-

-RSB-

-RRB-

Nevertheless

when

the

tasks

are

granted

only

a

limited

number

of

polls

things

are

much

more

complicated

In

the

case

of

unit-size

tasks

and

a

single

poll

per

task

the

result

of

Gonnet

-LSB-

-RSB-

has

proved

that

for

m

n

the

maximum

load

is

-LRB-

o

-LRB-

-RRB-

-RRB-

ln

n

lnln

n

when

the

poll

of

each

task

is

chosen

iur

from

the

n

machines

whp

In

-LSB-

-RSB-

an

explicit

form

for

the

expected

maximum

load

is

given

for

all

combinations

of

n

and

m.

From

this

work

it

easily

seen

that

for

m

n

ln

n

the

maximum

load

is

m

n

-LRB-

m

ln

n/n

-RRB-

which

implies

that

by

means

of

competitive

ratio

m

n

is

actually

the

hardest

instance

In

the

case

of

d

polls

per

task

a

bunch

of

new

techniques

had

to

be

applied

for

the

analysis

of

such

schedulers

The

main

tools

used

in

the

literature

for

this

problem

have

been

the

layered

induction

the

witness

tree

argument

and

the

method

of

fluid

models

-LRB-

a

comprehensive

presentation

of

these

techniques

may

be

found

in

the

very

good

survey

of

Mitzenmacher

et

al.

-LSB-

-RSB-

-RRB-

In

the

seminal

paper

of

Azar

Broder

Karlin

and

Upfal

-LSB-

-RSB-

it

was

proved

that

the

proposed

scheduler

abku

that

chooses

each

of

the

polls

of

a

task

iur

from

-LSB-

n

-RSB-

and

then

assigns

the

task

to

the

candidate

machine

of

minimum

load

achieves

a

maximum

load

that

is

at

most

m

n

lnln

n

ln

d

-LRB-

-RRB-

This

implies

a

strictly

O

lnln

n

ln

d

competitive

ratio

or

equivalently

at

least

O

-LRB-

ln

n

-RRB-

polls

per

task

would

be

necessary

in

order

to

achieve

a

strictly

constant

competitive

ratio

In

-LSB-

-RSB-

the

always

go

left

algorithm

was

proposed

which

assures

a

maximum

load

of

at

most

m

n

lnln

n

d

ln

-LRB-

-RRB-

and

thus

only

needs

an

amount

of

O

-LRB-

loglog

n

-RRB-

polls

per

task

in

order

to

achieve

a

strictly

constant

competitive

ratio

In

addition

it

was

shown

that

this

is

the

best

possible

that

one

may

hope

for

in

the

case

of

assigning

unit-size

tasks

to

identical

machines

with

only

d

-LRB-

either

oblivious

or

adaptive

-RRB-

polls

per

task

The

Online

Assignment

of

Tasks

to

Related

Machines

problem

has

been

thoroughly

studied

during

the

past

years

for

the

Full

Information

case

-LRB-

eg

see

chapter

in

-LSB-

-RSB-

-RRB-

In

particular

it

has

been

shown

that

a

strictly

-LRB-

small

-RRB-

constant

competitive

ratio

can

be

achieved

using

the

slowfit-like

algorithms

that

are

based

on

the

idea

of

exploiting

the

least

favourable

machines

-LRB-

this

idea

first

appeared

in

-LSB-

-RSB-

-RRB-

The

case

of

Limited

Information

has

attracted

little

attention

up

to

this

point

some

recent

works

-LRB-

-LSB-

-RSB-

-RRB-

study

the

case

of

each

task

having

a

single

poll

for

its

assignment

to

one

of

the

-LRB-

possibly

related

-RRB-

machines

when

the

probability

distributions

of

the

tasks

comprise

a

Nash

Equilibrium

For

example

in

-LSB-

-RSB-

it

was

shown

that

in

the

Related

Machines

case

a

coordination

ratio

-LRB-

ie

the

ratio

between

the

worst

possible

expected

maximum

load

among

Nash

Equilibria

over

the

offline

optimum

-RRB-

of

O

log

n

logloglog

n

However

when

all

the

task

weights

are

equal

then

it

was

shown

by

Mavronicolas

and

Spirakis

-LSB-

-RSB-

that

the

coordination

ratio

A

probabilistic

event

A

is

said

to

hold

with

high

probability

-LRB-

whp

-RRB-

if

for

some

arbitrarily

chosen

constant

IP

-LSB-

A

-RSB-

n

is

O

log

n

loglog

n

As

for

the

case

of

d

in

the

Related

Machines

problem

up

to

the

author

's

knowledge

this

is

the

first

time

that

this

problem

is

dealt

with

New

results

In

this

work

we

show

that

any

HOPS

scheduler

requires

at

least

O

log

n

loglog

n

polls

in

order

to

achieve

a

strictly

constant

competitive

ratio

vs.

an

oblivious

adversary

The

key

point

in

this

lower

bound

argument

is

the

construction

of

a

system

of

d

groups

of

machines

running

at

the

same

speed

within

each

group

while

the

machine

speeds

-LRB-

comparing

machines

of

consecutive

groups

-RRB-

fall

by

a

fixed

factor

and

on

the

other

hand

the

cumulative

capacities

of

the

groups

are

raised

by

the

same

factor

Then

it

is

intuitively

clear

that

any

HOPS

scheduler

can

not

keep

track

of

the

current

status

within

each

of

these

d

groups

while

having

only

d

polls

per

new

task

and

thus

it

will

have

to

pay

the

cost

of

misplacing

balls

in

at

least

one

of

these

groups

More

specifically

we

show

the

following

lower

bound

Theorem

d

the

competitive

ratio

of

any

d-hops

scheduler

is

at

least

n

Then

we

propose

a

new

d-hops

scheduler

OBL

which

if

it

is

fortified

with

the

additional

knowledge

of

the

total

number

of

tasks

to

be

inserted

then

it

achieves

the

following

upper-bound

Theorem

Let

lnln

n

lnlnln

n

d

and

suppose

that

the

size

of

the

input

sequence

is

given

as

offline

information.If

OBL

provides

each

task

with

-LRB-

at

most

-RRB-

polls

then

it

has

a

strict

competitive

ratio

that

drops

double-exponentially

with

d.In

particular

the

cost

of

OBL

is

with

high

probability

at

most

OBL

-LRB-

m

-RRB-

-LRB-

o

-LRB-

-RRB-

-RRB-

n

d

-LRB-

d

-RRB-

ADV

-LRB-

m

-RRB-

It

is

commented

that

all

the

schedulers

for

the

Identical

Machines-Limited

Information

case

up

to

now

used

minimum

load

as

the

profound

assignment

rule

On

the

other

hand

OBL

was

inspired

by

the

slowfit

approaches

for

the

Related

Machines-Full

Information

problem

and

the

fact

that

a

greedy

scheduler

behaves

badly

in

that

case

Up

to

the

author

's

knowledge

the

idea

of

using

the

slow-est

machine

possible

first

appeared

in

-LSB-

-RSB-

Additionally

a

layered

induction

argument

is

employed

for

bounding

the

amount

of

tasks

that

flow

from

the

slower

to

the

faster

machines

in

the

system

This

then

allows

the

use

of

relatively

simple

arguments

for

bounding

the

maximum

load

of

the

tasks

that

end

up

in

a

small

fraction

of

the

system

that

consists

of

the

fastest

machines

Clearly

this

upper

bound

is

near-optimal

-LRB-

up

to

a

multiplicative

constant

-RRB-

since

it

matches

the

-LRB-

loglog

n

-RRB-

lower

bound

of

the

Unit

Size

Tasks-Identical

Machines

problem

-LRB-

-LSB-

-RSB-

-RRB-

which

is

a

subcase

of

our

setting

Finally

we

propose

a

haps

scheduler

-LRB-

ADAPT

-RRB-

that

combines

the

previous

hops

scheduler

with

a

classical

guessing

Note

that

this

does

not

imply

preemption

of

tasks

which

is

not

allowed

in

our

setting

but

rather

the

event

that

a

task

hits

slower

machines

that

are

already

overloaded

and

thus

has

to

assign

itself

to

a

faster

machine

argument

for

the

cost

of

ADV

and

assures

a

cost

roughly

times

the

cost

of

OBL

Theorem

For

any

input

sequence

of

identical

tasks

that

have

to

be

assigned

to

n

related

machines

using

at

most

polls

per

task

the

cost

of

ADAPT

is

-LRB-

whp

-RRB-

ADAPT

-LRB-

-RRB-

O

n

d

-LRB-

d

-RRB-

ADV

-LRB-

-RRB-

This

section

contains

a

simple

argument

for

the

claimed

lower

bound

on

online

schedulers

that

devote

a

single

poll

per

new

task

ie

d

Clearly

by

their

nature

these

are

HOPS

schedulers

since

there

is

no

actual

option

for

the

assignment

strategy

The

proof

for

the

lower

bound

of

these

schedulers

is

rather

simple

but

it

will

and

shed

some

light

to

the

essence

of

the

construction

for

the

proof

of

the

general

lower

bound

that

will

follow

in

the

next

section

Let

's

assume

that

there

exists

a

HOPS

scheduler

that

only

uses

poll

per

task

and

claims

to

be

strict

a-competitive

against

any

oblivious

adversary

ADV.

Initially

ADV

chooses

an

arbitrary

real

number

r

n

which

will

be

fixed

in

the

end

so

as

to

maximize

the

lower

bound

on

a.

Let

also

the

variables

C

total

C

max

denote

the

total

capacity

and

the

maximum

possible

polled

capacity

using

one

poll

-LRB-

ie

the

maximum

bin

capacity

in

this

case

-RRB-

in

the

system

Consequently

ADV

uses

the

following

system

of

capacitated

bins

so

that

these

values

are

preserved

C

C

max

C

i

C

max

r

i

C

total

C

max

C

max

r

-LRB-

n

-RRB-

Observe

that

the

capacity

of

bin

i

is

r

times

smaller

than

C

max

while

on

the

other

hand

the

cumulative

capacity

of

the

last

n

machines

is

r

times

larger

than

the

capacity

of

the

largest

bin

in

the

system

Consider

also

the

following

abbreviations

of

probabilities

and

events

that

may

occur

upon

the

arrival

of

a

new

ball

E

i

bin

i

is

hit

by

a

ball

P

IP

-LSB-

E

-RSB-

P

P

Obviously

due

to

the

assumption

of

a-competitiveness

a

C

total

C

max

C

max

-LRB-

-RRB-

C

max

r

C

max

n

r

since

ALG

could

choose

to

assign

all

the

incoming

balls

to

the

largest

bin

in

the

system

The

question

that

arises

is

whether

there

exists

a

scheduler

that

can

do

better

than

that

We

consider

the

following

input

sequences

w

w

ALG

-LRB-

-RRB-

IE

-LSB-

L

max

-LRB-

-RRB-

-RSB-

P

w

C

max

P

w

Cmax

r

ADV

-LRB-

-RRB-

w

C

max

a-comp

a

P

r

P

r

P

-LRB-

r

-RRB-

P

r-a

t

w

t

w

In

this

case

the

loads

of

all

the

bins

will

tend

to

their

expected

values

and

thus

-LRB-

-RRB-

IE

-LSB-

-LRB-

-RRB-

-RSB-

P

w

C

max

ALG

-LRB-

-RRB-

IE

-LSB-

L

max

-LRB-

-RRB-

-RSB-

P

w

C

max

ADV

-LRB-

-RRB-

w

C

total

a-comp

a

w

C

total

P

w

C

max

P

aC

max

C

total

a

r

Combining

the

two

bounds

on

P

we

get

aC

max

C

total

a

r

P

r

a

r

a

r

a

n

r

a

n

r

a

r

n

r

r

n

a

r

n

r

r

r

n

r

r

r

n

which

is

maximized

for

r

n

and

assures

a

lower

bound

on

a

of

n

Remark

It

is

worth

noting

that

the

lower

bound

com-pletely

depends

on

the

number

of

bins

in

the

system

and

on

the

ratio

r

C

max

C

min

and

does

not

depend

at

all

on

the

total

capacity

of

the

system

C

total

In

this

section

we

study

the

behaviour

of

homogeneous

schedulers

that

adopt

an

oblivious

polling

strategy

-LRB-

ie

the

polling

strategy

is

from

HOPS

-RRB-

and

an

arbitrary

assignment

strategy

We

call

these

d-hops

schedulers

since

the

choice

of

the

d

candidates

on

behalf

of

each

ball

is

done

independently

for

each

ball

according

to

a

common

probability

distribution

f

-LSB-

n

-RSB-

d

-LSB-

-RSB-

Recall

that

the

choice

of

the

candidate

bins

for

each

ball

is

oblivious

to

the

current

system

state

and

thus

could

have

been

fixed

prior

to

the

beginning

of

the

assignments

Theorem

d

the

competitive

ratio

of

any

d-hops

scheduler

is

at

least

n

Proof

Let

f

-LSB-

n

-RSB-

d

-LSB-

-RSB-

be

the

adopted

Oblivious

polling

strategy

by

an

arbitrary

d-hops

scheduler

ALG

Assume

also

that

ALG

uses

the

best

possible

assignment

strategy

given

this

polling

strategy

that

is

each

ball

chooses

its

own

candidate

bins

according

to

f

and

then

it

may

assign

itself

to

an

arbitrarily

chosen

host

among

its

candidates

depending

on

the

current

loads

of

the

candidate

bins

Assume

also

that

ALG

claims

a

-LRB-

strict

-RRB-

competitive

ratio

a

against

oblivious

adversaries

As

parameters

of

the

problem

we

consider

again

the

quantities

C

total

n

i

C

i

and

C

max

the

total

capacity

of

a

system

of

n

related

machines

and

the

maximum

capacity

that

may

be

returned

by

a

single

poll

We

shall

describe

an

adversary

ADV

that

initially

chooses

an

arbitrary

real

number

r

n

and

then

considers

the

system

of

-LRB-

d

groups

of

-RRB-

n

capacitated

bins

that

is

described

in

Table

Observe

that

this

construction

preserves

the

following

two

invariants

when

considering

two

successive

groups

of

bins

F

F

d

Group

of

Bins

Number

of

Bins

in

group

Capacity

per

Bin

Cumulative

Group

Capacity

F

C

max

C

max

F

r

-LRB-

r

-RRB-

C

max

r

-LRB-

r

-RRB-

C

max

F

r

-LRB-

r

-RRB-

C

max

r

r

-LRB-

r

-RRB-

C

max

F

r

-LRB-

r

-RRB-

C

max

r

r

-LRB-

r

-RRB-

C

max

F

d

r

-LRB-

r

-RRB-

C

max

r

r

-LRB-

r

-RRB-

C

max

F

d

n

r

r

-LRB-

r

-RRB-

n

r

C

max

r

d

-LRB-

n

r

d

r

-RRB-

C

max

Table

The

system

of

-LRB-

d

groups

of

-RRB-

capacitated

bins

considered

by

ADV

for

the

proof

of

the

Lower

Bound

on

d-HOPS

schedulers

-LRB-

-RRB-

when

going

from

one

group

to

its

successor

the

bin

capacities

decrease

by

a

factor

of

r

and

-LRB-

-RRB-

the

cumulative

capacity

of

the

first

groups

is

larger

than

the

cumulative

capacity

of

the

first

groups

by

a

factor

of

r

We

shall

denote

by

C

-LSB-

F

-RSB-

the

cumulative

capacity

of

any

group

of

bins

F

-LSB-

n

-RSB-

Remark

The

preservation

of

invariant

-LRB-

-RRB-

when

d

implies

that

C

-LSB-

F

d

-RSB-

r

-LRB-

-RRB-

C

max

n

r

d

r

C

max

r

-LRB-

r

-RRB-

C

max

n

r

d

r

r

We

fortify

ALG

by

allowing

a

perfect

balance

of

the

bins

of

a

group

F

whenever

at

least

one

poll

on

behalf

of

a

new

ball

goes

to

a

bin

of

this

group

This

is

actually

in

order

to

capture

the

notion

of

the

perfect

assignment

strategy

given

the

polling

strategy

claim

stated

above

Clearly

this

does

not

cause

any

problem

since

we

are

looking

for

a

lower

bound

Because

ALG

could

lock

its

d

choices

to

the

first

d

groups

of

the

system

it

is

obvious

that

its

competitive

ratio

a

is

at

most

a

C

total

C

-LSB-

d

F

-RSB-

n

r

Consider

now

the

d

events

E

F

is

hit

by

a

ball

-LRB-

d

-RRB-

while

P

IP

-LSB-

E

-RSB-

-LRB-

call

it

the

hitting

probability

of

group

F

-RRB-

is

the

probability

of

at

least

one

bin

from

F

being

hit

by

a

ball

We

shall

charge

ALG

according

to

the

hitting

probabilities

that

its

polling

strategy

determines

Notice

that

these

are

fixed

at

the

beginning

of

the

assignments

since

the

polling

strategy

of

ALG

is

an

oblivious

strategy

Furthermore

the

following

conditional

hitting

probabilities

are

also

determined

uniquely

by

the

polling

strategy

of

ALG

i

j

-LSB-

n

-RSB-

i

j

P

i

j

IP

-LSB-

E

i

E

E

E

j

-RSB-

Q

i

j

IP

-LSB-

E

i

E

E

E

j

-RSB-

Finally

let

B

-LRB-

-RRB-

-LRB-

d

-RRB-

denote

the

maximum

number

of

balls

that

may

be

hosted

by

bins

of

the

set

F

without

violating

the

assumption

of

a-competitiveness

of

ALG

when

the

input

sequence

of

tasks

is

chosen

by

ADV.

The

following

lemma

states

an

inherent

property

of

any

d-hops

scheduler

Lemma

For

any

unless

ALG

admits

a

competitive

ratio

a

-LRB-

-RRB-

-LRB-

-RRB-

d

r

the

following

property

holds

d

P

a

r

Proof

We

prove

this

lemma

by

considering

the

following

input

sequences

of

balls

of

the

same

-LRB-

arbitrarily

chosen

-RRB-

size

w

In

this

case

we

know

that

ADV

-LRB-

-RRB-

w

C

max

The

cost

-LRB-

ie

the

expectation

of

maximum

load

-RRB-

of

ALG

is

ALG

-LRB-

-RRB-

P

w

C

max

-LRB-

P

-RRB-

Q

rw

C

max

-LRB-

Q

-RRB-

Q

r

w

C

max

-LRB-

Q

-RRB-

-LRB-

Q

d

-RRB-

r

d

w

C

max

Due

to

the

demand

for

a-competitiveness

of

ALG

against

ADV

this

then

implies

a

r

P

P

a

r

r

d

In

this

case

ADV

will

use

only

the

bins

of

F

and

thus

he

will

pay

a

cost

of

ADV

-LRB-

-RRB-

r

w

r

C

max

r

w

C

max

As

for

the

cost

of

ALG

we

shall

only

charge

it

for

the

input

subsequence

of

balls

that

definitely

hit

groups

F

F

-LRB-

call

it

-RRB-

Our

purpose

is

from

this

sequence

of

tasks

to

determine

P

ie

the

conditional

hitting

probability

of

group

F

given

that

all

the

previous

groups

are

hit

by

a

ball

Clearly

IE

-LSB-

-RSB-

P

r

P

-LRB-

where

for

symmetry

of

representation

we

let

P

P

-RRB-

Recall

that

B

-LRB-

-RRB-

denotes

the

maximum

number

of

balls

that

may

be

assigned

to

the

bins

of

the

first

groups

given

the

claimed

competitive

ratio

a

by

ALG

and

the

input

sequence

Then

we

have

wB

-LRB-

-RRB-

C

-LSB-

F

-RSB-

a

ADV

-LRB-

-RRB-

B

-LRB-

-RRB-

a

r

Thus

there

is

a

subsequence

of

that

consists

of

those

tasks

which

can

not

be

assigned

to

the

bins

of

the

first

groups

due

to

the

a-competitiveness

constraint

All

these

tasks

have

to

exploit

their

remaining

-LRB-

at

most

-RRB-

d

polls

among

the

bins

of

-LSB-

n

-RSB-

F

It

is

clear

that

ALG

has

no

reason

to

spoil

more

than

one

poll

per

group

due

to

the

optimal

assignment

strategy

that

it

adopts

Thus

we

can

safely

assume

that

there

remain

exactly

d

polls

for

the

remaining

groups

Obviously

IE

-LSB-

-RSB-

IE

-LSB-

-RSB-

B

-LRB-

-RRB-

r

P

ar

r

-LRB-

r

a

-RRB-

where

for

simplification

of

notation

we

use

the

bounding

quence

a

r

a

r

and

This

is

true

because

P

a

r

a

r

while

we

assume

inductively

that

P

By

showing

that

P

a

r

we

shall

also

have

assured

that

P

We

apply

the

Markov

Inequality

-LRB-

on

the

complementary

random

variable

-RRB-

to

find

a

lower

bound

on

the

size

of

IP

-LSB-

r

-LRB-

r

r

r

a

-RRB-

-RSB-

Now

it

is

clear

that

if

ALG

claims

a

competitive

ratio

a

-LRB-

-RRB-

-LRB-

r

-RRB-

d

r

-LRB-

-RRB-

r

r

then

at

least

one

ball

of

will

belong

to

with

probability

at

least

Thus

either

ALG

has

a

-LRB-

-RRB-

-LRB-

-RRB-

dr

or

-LRB-

by

simply

charging

it

only

for

this

very

specific

ball

-RRB-

ALG

-LRB-

-RRB-

-LSB-

P

-LRB-

P

-RRB-

r

-RSB-

r

w

C

max

which

combined

with

the

demand

for

a-competitiveness

and

the

cost

of

ADV

for

implies

that

P

a

r

We

finally

try

the

following

input

sequence

in

case

that

ALG

still

claims

a

competitive

ratio

a

-LRB-

-RRB-

-LRB-

-RRB-

d

r

For

this

input

sequence

it

is

clear

that

ADV

-LRB-

-RRB-

w

C

total

w

r

r

n

r

d

C

max

For

ALG

we

again

consider

the

subsequence

of

balls

that

definitely

hit

the

first

d

groups

of

the

system

Clearly

IE

-LSB-

-RSB-

since

we

now

consider

an

infinite

sequence

of

incoming

balls

As

for

the

upper

bound

on

the

balls

that

the

first

d

groups

can

host

this

is

again

given

by

the

demand

for

a-competitiveness

wB

-LRB-

-RRB-

C

-LSB-

F

-RSB-

wB

-LRB-

-RRB-

r

C

max

a

w

r

r

n

r

d

C

max

B

-LRB-

-RRB-

r

r

r

n

r

d

a

r

The

subsequence

that

has

to

exploit

a

single

poll

among

the

bins

of

-LSB-

n

-RSB-

F

has

size

at

least

IE

-LSB-

-RSB-

IE

-LSB-

-RSB-

B

-LRB-

-RRB-

r

r

r

n

r

d

a

r

-LRB-

d

-RRB-

a

r

r

r

r

n

r

d

a

r

d

a

r

where

for

the

last

inequality

we

consider

that

n

r

Since

we

consider

that

a

-LRB-

-RRB-

-LRB-

-RRB-

dr

we

can

be

sure

that

r

and

thus

the

cost

of

ALG

will

be

lower

bounded

by

the

expected

load

of

the

bins

in

F

d

due

to

the

tasks

of

a

ADV

-LRB-

-RRB-

ALG

-LRB-

-RRB-

P

d

w

-LRB-

r

r

-RRB-

C

max

a

r

r

n

r

d

P

d

-LRB-

-RRB-

-LRB-

-RRB-

dr

r

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

n

r

-LRB-

-RRB-

a

r

r

d

which

is

not

possible

for

any

and

n

r

d

Thus

we

conclude

that

ALG

can

not

avoid

a

competitive

ratio

a

min

-LRB-

d

-RRB-

r

n

r

for

any

and

n

r

d

which

for

and

n

r

d

gives

the

desired

bound

In

this

section

we

prove

that

the

missing

information

for

an

oblivious

scheduler

to

perform

efficiently

is

the

size

of

the

input

sequence

More

specifically

considering

that

the

input

size

is

provided

as

offline

information

to

each

of

the

newly

inserted

tasks

we

construct

an

oblivious

scheduler

that

exploits

this

information

along

with

a

slowfit

assignment

rule

and

a

layered

induction

argument

for

the

flow

of

balls

from

slower

to

faster

bins

in

order

to

achieve

a

strictly

constant

competitive

ratio

with

only

O

-LRB-

loglog

n

-RRB-

polls

per

task

Assume

that

m

unit-size

balls

are

thrown

into

a

system

of

n

capacitated

bins

with

capacities

C

max

C

n

C

C

C

min

Assume

also

that

each

ball

is

allowed

to

poll

up

to

bins

and

then

it

has

to

assign

itself

to

one

of

these

candidates

We

additionally

assume

that

C

max

n

d

As

it

will

become

clear

later

by

the

analysis

if

this

was

not

the

case

then

it

could

only

be

in

favour

of

the

oblivious

scheduler

that

we

propose

because

this

would

allow

the

absorption

of

the

large

additive

constants

in

the

performance

guarantee

of

the

scheduler

We

consider

-LRB-

wlog

-RRB-

that

the

capacity

vector

c

is

nor-malized

by

n

c

so

that

n

i

C

i

n

We

also

assume

that

the

total

size

m

the

input

sequence

is

given

to

every

newly

inserted

ball

This

implies

that

each

ball

can

estimate

the

cost

ADV

-LRB-

m

-RRB-

opt

-LRB-

ie

the

optimum

offline

assignment

of

the

m

unit-size

balls

to

the

n

capacitated

bins

-RRB-

and

thus

it

can

know

a

priori

the

subset

of

bins

that

may

have

been

used

by

ADV

during

the

whole

process

Having

this

in

mind

we

can

assume

that

every

bin

in

the

system

is

legitimate

that

is

it

might

have

been

used

by

the

optimum

solution

otherwise

we

could

have

each

ball

ignore

all

the

illegitimate

bins

in

the

system

Thus

opt

max

C

min

m

C

total

max

C

min

m

n

Finally

we

assume

that

each

of

the

legitimate

bins

of

the

system

gets

at

A

bin

i

-LSB-

n

-RSB-

may

have

been

used

by

ADV

if

and

only

if

i

opt

least

one

ball

in

the

optimum

offline

schedule

This

does

not

affect

the

performance

of

ADV

while

it

may

only

deteriorate

the

performance

of

an

online

scheduler

Nevertheless

it

assures

that

m

n

opt

m

n

meaning

that

the

fractional

load

on

the

bins

is

actually

a

good

estimation

of

opt

Let

the

load

of

bin

i

-LSB-

n

-RSB-

at

time

t

-LRB-

that

is

right

after

the

assignment

of

the

t

th

ball

of

the

input

sequence

-RRB-

be

denoted

as

t

-LRB-

i

-RRB-

q

t

-LRB-

i

-RRB-

C

i

where

q

t

-LRB-

i

-RRB-

is

the

number

of

balls

assigned

to

bin

i

up

to

that

time

The

following

definition

refers

to

the

notion

of

saturated

bins

in

the

system

ie

overloaded

bins

wrt

the

designed

performance

guarantee

of

an

oblivious

scheduler

Definition

A

bin

i

-LSB-

n

-RSB-

is

called

saturated

upon

the

arrival

of

a

new

ball

t

m

if

and

only

if

it

has

-LRB-

i

-RRB-

a

opt

-LRB-

a

-RRB-

where

-LRB-

a

-RRB-

is

called

the

designed

performance

guarantee

of

the

oblivious

scheduler

Let

r

-LSB-

d

-RSB-

i

r

min

-LCB-

i

-LSB-

n

-RSB-

i

j

C

j

r

n

-RCB-

Then

we

consider

the

following

partition

of

the

set

of

bins

-LSB-

n

-RSB-

into

d

groups

of

-LRB-

roughly

geometrically

-RRB-

decreasing

cumulative

capacities

F

-LCB-

i

-RCB-

F

r

-LCB-

i

i

r

-RCB-

r

d

F

d

-LCB-

i

d

n

-RCB-

Although

the

cumulative

capacity

of

group

F

r

may

vary

from

n

r

C

max

to

n

r

C

max

for

ease

of

the

following

computations

we

assume

that

asymptotically

r

-LSB-

d

-RSB-

C

-LSB-

F

r

-RSB-

n

r

and

C

-LSB-

F

d

-RSB-

n

d

We

denote

by

C

min

the

capacity

of

the

smallest

bin

in

F

-LSB-

d

-RSB-

We

now

consider

the

following

ideal

scheduler

that

uses

an

oblivious

polling

strategy

and

an

assignment

strategy

based

on

the

slowfit

rule

This

scheduler

-LRB-

we

call

it

OBL

-RRB-

initially

discards

all

the

illegitimate

bins

in

the

system

using

the

knowledge

of

m

Then

first

it

normalizes

the

capacity

vector

of

the

remaining

bins

and

afterwards

it

considers

the

grouping

mentioned

above

and

adopts

the

following

pair

of

strategies

POLLING

r

d

group

F

r

gets

exactly

poll

which

is

chosen

among

the

bins

of

the

group

proportionally

to

the

bin

capacities

That

is

r

-LSB-

d

-RSB-

i

F

r

IP

-LSB-

bin

i

F

r

is

a

candidate

host

of

a

ball

-RSB-

C

i

C

-LSB-

F

r

-RSB-

The

remaining

d

polls

are

assigned

to

the

bins

of

group

F

d

either

to

the

d

fastest

bins

or

according

to

the

polling

strategy

of

always

go

left

depending

only

on

the

parameters

-LRB-

c

d

-RRB-

of

the

problem

instance

ASSIGNMENT

Upon

the

arrival

of

a

new

ball

t

-LSB-

m

-RSB-

the

smallest

polled

bin

from

d

F

-LRB-

starting

from

F

to

F

a.s.o.

-RRB-

that

is

unsaturated

gets

this

ball

-LRB-

slowfit

rule

-RRB-

In

case

that

all

the

first

d

polls

of

a

ball

are

already

saturated

then

this

ball

has

to

be

assigned

to

a

bin

of

F

d

using

its

remaining

d

candidates

Within

group

F

d

either

the

minimum

post

load

rule

-LRB-

ie

the

bin

of

minimum

load

among

the

d

choices

from

F

d

is

chosen

taking

into

account

also

the

additional

load

of

the

new

ball

-RRB-

or

the

slowfit

rule

Observe

that

this

is

offline

information

and

thus

this

decision

can

be

made

at

the

beginning

of

the

assignment

process

for

all

the

balls

of

the

input

sequence

is

applied

depending

on

the

offline

parameters

-LRB-

c

d

-RRB-

of

the

problem

instance

If

all

the

polled

bins

are

saturated

then

the

minimum

post

load

rule

is

applied

among

them

Ties

are

always

broken

in

favour

of

smaller

bins

-LRB-

ie

slower

machines

-RRB-

The

following

theorem

gives

the

performance

of

OBL

when

the

additional

information

of

the

input

size

is

also

provided

offline

Theorem

For

lnln

n

lnlnln

n

d

when

the

size

of

the

input

sequence

m

n

is

given

as

offline

information

OBL

has

a

strict

competitive

ratio

that

drops

double-exponentially

with

d.In

particular

the

cost

of

OBL

is

-LRB-

whp

-RRB-

at

most

OBL

-LRB-

m

-RRB-

-LRB-

o

-LRB-

-RRB-

-RRB-

n

d

-LRB-

d

-RRB-

ADV

-LRB-

m

-RRB-

Proof

Let

m

be

the

first

ball

in

the

system

that

hits

only

saturated

bins

by

its

polls

Our

purpose

is

to

determine

the

value

of

a

in

the

designed

performance

guarantee

-LRB-

a

-RRB-

so

that

the

probability

of

ball

existing

to

be

polynomially

small

As

stated

before

the

technique

that

we

shall

employ

is

a

layered

induction

argument

on

the

number

of

balls

that

are

passed

to

the

right

of

group

F

r

r

-LSB-

d

-RSB-

For

the

assignment

of

the

balls

that

end

up

in

group

F

d

we

use

a

slightly

modified

version

of

the

always

go

left

scheduler

of

-LSB-

-RSB-

that

gives

an

upper

bound

on

the

maximum

load

in

group

F

d

-LRB-

we

denote

this

by

L

max

-LSB-

F

d

-RSB-

-RRB-

This

upper

bound

on

L

max

-LSB-

F

d

-RSB-

holds

with

high

probability

This

assignment

is

only

used

when

it

produces

a

smaller

maximum

load

than

the

brute

assignment

of

all

the

balls

ending

up

in

F

d

to

the

d

fastest

bins

of

the

group

We

shall

consider

a

notion

of

time

that

corresponds

to

the

assignments

of

newly

arrived

balls

into

the

system

At

time

t

m

the

t

th

ball

of

the

input

sequence

is

thrown

into

the

system

and

it

has

to

be

immediately

assigned

to

one

of

its

candidates

Consider

the

polls

on

behalf

of

a

ball

to

be

ordered

according

to

the

groups

from

which

they

are

taken

Observe

then

that

each

ball

t

is

assigned

to

the

first

unsaturated

bin

that

it

hits

from

the

first

d

groups

or

to

a

bin

in

group

F

d

Thus

-LSB-

d

-RSB-

each

ball

that

has

been

assigned

to

group

F

up

to

-LRB-

and

including

-RRB-

ball

has

definitely

failed

to

hit

an

unsaturated

bin

in

all

the

groups

F

F

For

any

ball

t

m

and

-LSB-

d

-RSB-

let

Q

t

-LRB-

-RRB-

denote

the

number

of

balls

that

have

been

assigned

to

group

F

up

to

time

t

-LRB-

ie

right

after

the

assignment

of

the

t

th

ball

-RRB-

while

Q

t

-LRB-

-RRB-

denotes

the

balls

that

have

been

assigned

to

the

right

of

group

F

that

is

to

bins

of

-LSB-

n

-RSB-

F

Thus

Q

t

-LRB-

-RRB-

d

Q

t

-LRB-

-RRB-

Let

also

S

t

-LRB-

-RRB-

denote

the

set

of

saturated

bins

in

F

at

time

t.

Then

-LSB-

d

-RSB-

t

IP

-LSB-

t

hits

a

saturated

bin

in

F

-RSB-

C

-LSB-

S

-LRB-

-RRB-

-RSB-

C

-LSB-

F

-RSB-

C

-LSB-

S

-LRB-

-RRB-

-RSB-

C

-LSB-

F

-RSB-

Observe

now

that

-LSB-

d

-RSB-

Q

-LRB-

-RRB-

iF

q

-LRB-

i

-RRB-

iS

-LRB-

-RRB-

C

i

q

-LRB-

i

-RRB-

C

i

C

-LSB-

S

-LRB-

-RRB-

-RSB-

-LRB-

a

-RRB-

C

-LSB-

S

-LRB-

-RRB-

-RSB-

C

-LSB-

F

-RSB-

Q

-LRB-

-RRB-

-LRB-

a

-RRB-

C

-LSB-

F

-RSB-

P

-LRB-

-RRB-

Recall

that

up

to

time

we

can

be

sure

that

Q

-LRB-

-RRB-

-LRB-

a

-RRB-

C

-LSB-

F

-RSB-

-LRB-

because

all

these

balls

are

assigned

to

unsaturated

bins

-RRB-

which

in

turn

assures

that

P

Inequality

-LRB-

-RRB-

implies

that

had

we

known

Q

-LRB-

-RRB-

then

the

number

Q

-LRB-

-RRB-

of

balls

before

ball

that

go

to

the

right

of

set

F

would

be

stochastically

dominated

by

the

number

of

successes

in

Q

-LRB-

-RRB-

Bernoulli

trials

with

success

probability

P

We

shall

denote

this

number

by

B

-LRB-

Q

-LRB-

-RRB-

P

-RRB-

In

the

following

lemma

we

apply

the

Chernoff-Hoeffding

bound

on

these

Bernoulli

trials

to

get

an

upper

bound

on

the

amount

of

balls

that

have

to

be

assigned

beyond

group

F

for

-LSB-

d

-RSB-

as

a

function

of

the

number

of

balls

that

have

been

assigned

beyond

group

F

Lemma

-LSB-

d

-RSB-

and

for

an

arbitrary

constant

with

probability

at

least

n

Q

-LRB-

-RRB-

max

-LSB-

Q

-LRB-

-RRB-

-RSB-

-LRB-

a

-RRB-

n

ln

n

Q

-LRB-

-RRB-

Proof

Let

's

assume

that

we

already

know

the

number

Q

-LRB-

-RRB-

of

balls

that

have

already

failed

in

F

F

Then

Q

-LRB-

-RRB-

is

stochastically

dominated

by

the

random

variable

B

-LRB-

Q

-LRB-

-RRB-

P

-RRB-

By

applying

Chernoff-Hoeffding

bounds

-LRB-

-LSB-

-RSB-

p.

-RRB-

on

these

Bernoulli

trials

we

get

that

IP

-LSB-

B

-LRB-

Q

-LRB-

-RRB-

P

-RRB-

Q

-LRB-

-RRB-

-LRB-

P

-RRB-

-RSB-

exp

-LRB-

Q

-LRB-

-RRB-

-RRB-

IP

B

-LRB-

Q

-LRB-

-RRB-

P

-RRB-

Q

-LRB-

-RRB-

P

ln

n

Q

-LRB-

-RRB-

n

where

we

have

set

ln

n

Q

-LRB-

-RRB-

But

recall

that

P

Q

-LRB-

-RRB-

-LRB-

a

-RRB-

C

-LSB-

F

-RSB-

and

also

that

C

-LSB-

F

-RSB-

n

Thus

we

conclude

that

with

probability

at

least

n

Q

-LRB-

-RRB-

B

-LRB-

Q

-LRB-

-RRB-

P

-RRB-

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

-LRB-

a

-RRB-

n

ln

n

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

-LSB-

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

-RSB-

-LRB-

a

-RRB-

n

ln

n

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

-LRB-

a

-RRB-

n

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

-LRB-

a

-RRB-

n

ln

n

Q

-LRB-

-RRB-

Q

-LRB-

-RRB-

-LSB-

Q

-LRB-

-RRB-

-RSB-

-LRB-

a

-RRB-

n

ln

n

Q

-LRB-

-RRB-

-LRB-

a

-RRB-

n

Q

-LRB-

-RRB-

from

which

we

get

the

desired

bound

Consider

now

the

following

finite

sequence

Q

-LRB-

-RRB-

max

-LSB-

Q

-LRB-

-RRB-

-RSB-

-LRB-

a

-RRB-

n

ln

n

Q

-LRB-

-RRB-

-LSB-

d

-RSB-

Q

-LRB-

-RRB-

m

For

the

integrity

of

the

representation

we

set

Q

-LRB-

-RRB-

m

We

then

bound

the

number

of

balls

that

end

up

in

group

F

d

by

the

d

th

term

of

this

sequence

Lemma

With

probability

at

least

dn

at

most

Q

-LRB-

d

-RRB-

balls

end

up

in

group

F

d

Proof

The

proof

of

this

lemma

is

relatively

simple

and

due

to

lack

of

space

it

is

differed

to

the

full

version

of

the

paper

Consequently

we

estimate

a

closed

form

for

the

first

terms

of

the

bounding

sequence

Q

-LRB-

-RRB-

d

that

was

determined

earlier

Lemma

The

first

log

log

m

ln

n

log

-LRB-

-RRB-

terms

of

the

bounding

sequence

Q

-LRB-

-RRB-

d

are

given

by

the

closed

form

Q

-LRB-

-RRB-

m

-LRB-

a

-RRB-

Proof

Assume

that

was

the

first

element

in

the

sequence

for

which

-LSB-

Q

-LRB-

-RRB-

-RSB-

-LRB-

a

-RRB-

n

ln

n

Q

-LRB-

-RRB-

-LRB-

-RRB-

Then

up

to

term

the

sequence

is

dominated

by

the

right-hand

term

of

the

above

inequality

and

thus

r

Q

-LRB-

r

-RRB-

r

-LRB-

a

-RRB-

n

-LSB-

Q

-LRB-

r

-RRB-

-RSB-

r

-LRB-

a

-RRB-

n

r

-LRB-

a

-RRB-

n

-LSB-

Q

-LRB-

r

-RRB-

-RSB-

r

-LRB-

a

-RRB-

n

r

-LRB-

a

-RRB-

n

-LRB-

a

-RRB-

n

-LSB-

Q

-LRB-

r

-RRB-

-RSB-

r

r

-LRB-

a

-RRB-

n

-LSB-

Q

-LRB-

-RRB-

-RSB-

r

r

-LRB-

r

-RRB-

-LRB-

a

-RRB-

n

r

-LSB-

Q

-LRB-

-RRB-

-RSB-

r

r

-LRB-

a

-RRB-

n

m

r

m

m

r

a

r

since

-LRB-

a

-RRB-

n

m

a

We

now

plug

in

this

closed

form

for

Q

-LRB-

-RRB-

in

inequality

-LRB-

-RRB-

to

get

the

following

ln

n

-LRB-

a

-RRB-

n

m

m

lnn

-LRB-

a

-RRB-

n

-LRB-

-RRB-

-LRB-

a

-RRB-

n

m

ln

n

-LRB-

-RRB-

a

From

the

above

and

the

definition

of

it

is

easy

to

see

that

max

r

-LSB-

d

-RSB-

ln

m-ln

lnln

n

ln

r

ln

a

By

setting

A

log

m

lnln

n

ln

ln

and

B

ln

a

we

get

the

solution

A

W

-LSB-

B

ln

exp

-LRB-

A

ln

-RRB-

-RSB-

ln

where

W

-LRB-

x

-RRB-

is

the

Lambert

W

Function

-LRB-

-LSB-

-RSB-

-RRB-

By

approximating

this

function

by

ln

x

lnln

x

-LRB-

since

x

B

ln

exp

-LRB-

A

ln

-RRB-

-RRB-

we

conclude

that

log

log

m

ln

n

log

-LRB-

-RRB-

Lemma

Assume

that

m

n

and

d

lnln

n-lnlnln

n

-LRB-

-RRB-

ln

If

we

set

a

n

d

-LRB-

d

-RRB-

the

cost

of

OBL

is

upper

bounded

-LRB-

whp

-RRB-

by

OBL

-LRB-

m

-RRB-

-LRB-

o

-LRB-

-RRB-

-RRB-

-LRB-

a

-RRB-

ADV

-LRB-

m

-RRB-

Proof

The

cost

of

OBL

up

to

time

is

upper-bounded

by

max

-LCB-

L

max

-LSB-

F

d

-RSB-

-LRB-

a

-RRB-

opt

-RCB-

since

in

the

first

d

groups

no

saturated

bin

ever

gets

another

ball

and

all

the

bins

are

legitimate

We

choose

a

so

that

the

cost

in

the

first

d

groups

is

at

least

as

large

as

L

max

-LSB-

F

d

-RSB-

-LRB-

whp

-RRB-

given

the

upper

bound

Q

-LRB-

d

-RRB-

on

the

number

of

balls

that

end

up

in

the

last

group

Thus

the

probability

of

existing

will

be

then

polynomially

small

because

at

least

one

poll

in

F

d

will

have

to

be

unsaturated

whp

This

then

implies

the

claimed

upper

bound

on

the

performance

of

OBL

Due

to

lack

of

space

the

complete

proof

of

this

lemma

is

presented

in

the

full

version

of

the

paper

Combining

the

statements

of

all

these

technical

lemmas

we

conclude

with

the

desired

bound

on

the

competitive

ratio

of

OBL

In

the

previous

section

we

have

proposed

a

hops

scheduler

that

is

based

on

the

knowledge

of

the

size

of

the

input

sequence

and

then

assures

that

its

own

performance

is

never

worse

than

-LRB-

a

-RRB-

opt

whp

In

this

section

we

propose

a

haps

scheduler

-LRB-

call

it

ADAPT

-RRB-

whose

main

purpose

is

to

guess

the

value

opt

of

the

optimum

offline

cost

by

a

classical

guessing

argument

and

then

let

OBL

do

the

rest

of

the

assignments

This

approach

is

in

complete

analogy

with

the

online

schedulers

of

the

Related

Machines-Full

Information

problem

-LRB-

see

-LSB-

-RSB-

pp.

-RRB-

The

only

difference

is

that

OBL

has

a

performance

which

holds

whp

and

this

is

why

the

final

result

also

holds

whp

A

significant

difficulty

of

ADAPT

is

exactly

this

guessing

mechanism

that

will

have

to

be

based

on

the

limited

information

provided

to

each

of

the

new

tasks

Our

goal

is

not

to

assume

that

any

kind

of

additional

information

-LRB-

eg

global

environment

variables

-RRB-

is

provided

to

the

balls

other

than

the

capacity

vector

and

the

current

loads

of

each

ball

's

candidates

ADAPT

sacrifices

one

of

the

available

polls

per

ball

in

order

to

create

such

a

good

guessing

mechanism

Of

course

a

different

approach

that

would

be

based

on

the

outcome

of

some

of

the

polls

-LRB-

eg

a

constant

fraction

of

the

polls

-RRB-

in

order

to

estimate

a

proper

online

prediction

of

opt

would

be

more

interesting

in

the

sense

that

it

would

not

create

a

communication

bottleneck

for

the

tasks

Nevertheless

the

purpose

of

ADAPT

is

mainly

to

demonstrate

the

possibility

of

constructing

such

an

adaptive

scheduler

whose

performance

is

close

to

that

of

OBL

Let

's

assume

that

the

system

now

has

n

capacitated

bins

-LRB-

C

min

C

C

C

n

C

max

-RRB-

Assume

also

that

each

new

task

is

provided

with

polls

Fix

a

number

r

Then

an

r-guessing

mechanism

proceeds

in

stages

Stage

-LRB-

-RRB-

contains

a

-LRB-

consecutive

-RRB-

subsequence

of

tasks

from

the

input

sequence

that

use

the

same

prediction

r

for

their

assignment

We

set

C

max

The

following

definitions

refer

to

the

notions

of

eligible

and

saturated

machines

upon

the

arrival

of

a

new

task

t

-LSB-

m

-RSB-

into

the

system

that

are

used

in

a

similar

fashion

as

in

-LSB-

-RSB-

where

the

concept

of

eligible-saturated

machines

was

introduced

Definition

Suppose

that

task

t

m

belongs

to

stage

The

set

of

eligible

machines

for

t

is

E

i

-LSB-

n

-RSB-

C

i

r

while

a

machine

i

-LSB-

n

-RSB-

is

considered

to

be

saturated

upon

the

arrival

of

task

t

if

-LRB-

i

-RRB-

a

r

a

where

a

max

E

d

-LRB-

d

-RRB-

Notice

that

the

static

information

of

-LRB-

E

a

-RRB-

only

depends

on

the

capacity

vector

c

and

the

number

d

of

polls

per

task

Thus

it

can

be

easily

computed

by

each

of

the

tasks

or

alternatively

it

can

be

provided

a

priori

to

all

the

tasks

as

additional

offline

information

ADAPT

proceeds

in

phases

and

works

as

follows

Upon

the

arrival

of

a

new

ball

t

-LSB-

m

-RSB-

first

the

fastest

bin

in

the

system

is

polled

and

the

stage

s

-LRB-

t

-RRB-

to

which

this

ball

belongs

is

determined

according

to

the

following

rule

s

-LRB-

t

-RRB-

IN

r

a

q

-LRB-

n

-RRB-

r

a

-LRB-

obviously

for

stage

only

the

second

inequality

must

hold

-RRB-

The

remaining

polls

of

task

t

are

taken

from

group

E

s

-LRB-

t

-RRB-

in

a

fashion

similar

to

that

of

OBL

The

assignment

strategy

of

ADAPT

is

exactly

the

same

with

that

of

OBL

with

the

only

difference

that

whenever

the

first

candidates

of

a

task

are

already

saturated

then

this

task

is

assigned

to

the

fastest

machine

in

the

system

n

If

the

latter

event

causes

machine

n

to

become

also

saturated

then

by

the

definition

of

the

stages

this

task

is

the

last

ball

of

the

current

stage

and

a

new

stage

-LRB-

with

an

r

times

larger

prediction

-RRB-

starts

from

the

next

task

on

Lemma

Suppose

that

ADAPT

uses

r

Let

h

denote

the

stage

at

which

the

prediction

h

of

ADAPT

reaches

opt

for

the

first

time.Then

the

last

stage

of

ADAPT

is

at

most

h

Proof

The

case

of

h

is

easy

since

this

implies

that

opt

max

and

then

ADAPT

can

not

be

worse

than

OBL

which

-LRB-

having

the

right

prediction

-RRB-

succeeds

to

assign

all

the

incoming

tasks

-LRB-

but

for

those

that

might

fit

in

machine

n

without

making

it

saturated

-RRB-

below

-LRB-

a

-RRB-

opt

whp

So

let

's

consider

the

case

where

h

Let

E

denote

the

set

of

legitimate

machines

in

the

system

-LRB-

ie

E

-LCB-

i

-LSB-

n

-RSB-

i

opt

-RCB-

-RRB-

The

amount

of

work

inserted

into

the

system

during

the

whole

assignment

process

is

bounded

by

C

-LSB-

E

-RSB-

opt

By

definition

of

h

r

C

max

opt

r

h

C

max

Stage

h

ends

when

machine

n

becomes

saturated

Let

W

-LRB-

-RRB-

denote

the

total

amount

of

work

assigned

during

stage

while

R

-LRB-

-RRB-

denotes

the

amount

of

remaining

work

at

the

end

of

stage

We

shall

prove

that

the

amount

of

remaining

work

at

the

end

of

stage

h

is

not

enough

to

make

OBL

fail

within

stage

h

Observe

that

at

the

end

of

stage

h

no

machine

has

exceeded

a

load

of

-LRB-

a

h

-RRB-

h

-LRB-

a

h

-RRB-

r

h

C

max

-LRB-

by

definition

of

stages

-RRB-

On

the

other

hand

during

stage

h

each

of

the

eligible

machines

i

E

h

needs

a

load

of

more

than

a

h

h

a

h

r

h

C

max

in

order

to

become

saturated

for

this

stage

We

denote

the

additional

free

space

of

bin

i

E

h

by

f

ree

i

-LRB-

h

-RRB-

a

h

r

h

C

max

-LRB-

a

h

-RRB-

r

h

C

max

C

i

while

F

REE

-LRB-

h

-RRB-

iE

h

f

ree

i

-LRB-

h

-RRB-

a

h

r

h

C

max

-LRB-

a

h

-RRB-

r

h

C

max

C

-LSB-

E

h

-RSB-

a

h

r

h

C

max

a

h

r

a

h

C

-LSB-

E

h

-RSB-

-LRB-

a

h

-RRB-

optC

-LSB-

E

h

-RSB-

ra

h

a

h

a

h

a

h

-LRB-

-RRB-

is

the

cumulative

free

space

granted

to

the

eligible

bins

of

the

system

for

stage

h

It

only

remains

to

prove

that

the

amount

of

work

that

has

to

be

dealt

with

by

OBL

using

only

the

bins

of

-LSB-

n

-RSB-

is

less

than

F

REE

-LRB-

h

-RRB-

-LRB-

a

h

-RRB-

Observe

now

that

OBL

uses

the

bins

of

E

h

and

has

to

deal

with

an

amount

of

work

less

than

W

-LRB-

h

-RRB-

f

ree

n

-LRB-

h

-RRB-

R

-LRB-

h

-RRB-

a

h

r

h

C

max

r

a

h

a

h

C

max

C

-LSB-

E

-RSB-

opt

a

h

-LRB-

-RRB-

opt

C

max

opt

-LRB-

C

-LSB-

E

-RSB-

-LRB-

-RRB-

a

h

C

max

-RRB-

-LRB-

C

-LSB-

E

-RSB-

C

max

-RRB-

opt

if

we

set

r

This

is

because

a

h

a

h

and

at

the

end

of

stage

h

bin

n

must

have

already

become

saturated

Observe

also

that

E

h

E

h

E

-LCB-

n

-RCB-

by

definition

of

h.

Thus

C

-LSB-

E

h

-RSB-

C

-LSB-

E

-RSB-

C

max

Thus

the

amount

of

work

that

has

to

be

served

by

OBL

during

stage

h

is

less

than

opt

C

-LSB-

E

h

-RSB-

Then

setting

r

in

inequality

-LRB-

-RRB-

assures

that

ra

h

a

h

a

h

a

h

-LRB-

recall

that

a

h

a

h

-RRB-

and

thus

the

remaining

work

at

the

end

of

stage

h

is

not

enough

to

make

OBL

fail

The

following

theorem

is

now

a

straightforward

consequence

of

the

previous

lemma

Theorem

For

any

input

sequence

of

identical

tasks

that

have

to

be

assigned

to

n

related

machines

using

at

most

polls

per

task

the

cost

of

ADAPT

is

-LRB-

whp

-RRB-

ADAPT

-LRB-

-RRB-

O

n

d

-LRB-

d

-RRB-

ADV

-LRB-

-RRB-

In

this

work

we

have

studied

the

problem

of

exploiting

limited

online

information

for

the

assignment

of

a

sequence

of

unit-size

tasks

to

related

machines

We

have

shown

that

the

oblivious

schedulers

that

perform

asymptotically

optimally

in

the

case

of

identical

machines

deteriorate

significantly

in

this

case

We

have

then

determined

an

adaptive

scheduler

that

actually

mimics

the

behaviour

of

an

ideal

oblivious

scheduler

in

order

to

achieve

roughly

the

asymptotically

optimal

performance

similar

to

the

case

of

the

identical

machines

As

for

further

research

the

issue

of

providing

only

limited

information

to

online

algorithms

is

critical

in

many

problems

for

which

an

objective

is

also

the

minimization

of

the

communication

cost

In

this

category

fall

most

of

the

network

design

problems

Another

interesting

line

of

research

would

be

the

avoidance

of

communication

bottlenecks

for

such

limited

information

online

algorithms

Additionally

it

would

be

very

interesting

to

study

the

case

of

tasks

of

arbitrary

sizes

being

injected

into

the

system

The

author

wishes

to

thank

Paul

Spirakis

for

valuable

discussions

during

the

write-up

of

the

paper

and

also

for

suggesting

an

appropriate

terminology

on

the

categorization

of

polling

strategies

-LSB-

-RSB-

J.

Aspens

Y.

Azar

A.

Fiat

S.

Plotkin

O.

Waarts

Online

Machine

Scheduling

with

Applications

to

Load

Balancing

and

Virtual

Circuit

Routing

In

Journal

of

ACM

Vol

-LRB-

-RRB-

pp.

-LSB-

-RSB-

Y.

Azar

A.

Broder

A.

Karlin

E.

Upfal

Ballanced

Allocations

In

Proc

of

ACM-STOC

-LRB-

-RRB-

pp.

Also

in

SIAM

J.

Computing

-LRB-

-RRB-

pp.

-LSB-

-RSB-

P.

Berman

M.

Charikar

M.

Karpinski

Online

Load

Balancing

for

Related

Machines

In

Proc

of

Int

Workshop

on

Algorithms

and

Data

Structures

-LRB-

-RRB-

LNCS

Springer-Verlag

pp.

-LSB-

-RSB-

P.

Berenbrink

A.

Czumaj

A.

Steger

and

B.

V

ocking

Balanced

Allocations

The

Heavily

Loaded

Case

In

Proc

of

nd

ACM-STOC

-LRB-

Portland

-RRB-

pp.

-LSB-

-RSB-

A.

Bar-Noy

A.

Freund

J.

Naor

NewAlgorithms

for

Related

Machines

with

Temporary

Jobs

In

Journal

of

Scheduling

Vol

-LRB-

-RRB-

pp.

-LSB-

-RSB-

A.

Borodin

R.

El

Yaniv

Online

Computation

and

Competitive

Analysis

Cambridge

Univ.

Press

-LSB-

-RSB-

R.

Corless

G.

Gonnet

D.

Hare

D.

Jeffrey

D.

Knuth

On

the

Lambert

W

Function

In

Advances

in

Computational

Mathematics

Vol

-LRB-

-RRB-

pp.

-LSB-

-RSB-

A.

Czumaj

and

B.

V

ocking

Tight

Bounds

for

Worst-Case

Equilibria

In

Proc

of

th

ACM-SIAM

SODA

-LRB-

San

Francisco

-RRB-

pp.

-LSB-

-RSB-

Y.

Azar

Online

Load

Balancing

In

Online

Algorithms

The

State

of

the

Art

A.

Fiat

G.

Woeginger

-LRB-

Eds

-RRB-

LNCS

Springer

pp.

-LSB-

-RSB-

G.

Gonnet

Expected

Length

of

the

Longest

Probe

Sequence

in

Hash

Code

Searching

In

J.

of

ACM

Vol

No.

-LRB-

-RRB-

pp.

-LSB-

-RSB-

M.

Habib

C.

McDiarmid

J.

Ramirez-Alfonsin

B.

Reed

-LRB-

Eds

-RRB-

Probabilistic

Methods

for

Algorithmic

Discrete

Mathematics

ISBN

Springer-Velrag

-LSB-

-RSB-

E.

Koutsoupias

C.

Papadimitriou

Worst-case

Equilibria

In

Proc

of

th

Annual

Symposium

on

Theoretical

Aspects

of

Computer

Science

-LRB-

STACS

-RRB-

LNCS

Springer-Verlag

pp.

-LSB-

-RSB-

M.

Mavronicolas

P.

Spirakis

The

Price

of

Selfish

Routing

In

Proc

of

rd

ACM-STOC

-LRB-

Krete

-RRB-

pp.

-LSB-

-RSB-

M.

Mitzenmacher

A.

Richa

R.

Sitaraman

The

Power

of

Two

Random

Choices

A

Survey

of

Techniques

and

Results

In

Handbook

of

Randomized

Algorithms

-LRB-

to

appear

-RRB-

Also

available

through

michaelm

-LSB-

-RSB-

A.

Steger

M.

Raab

Balls

into

Bins

A

Simple

and

Tight

Analysis

In

Proc

of

RANDOM

LNCS

Springer

Verlag

pp.

-LSB-

-RSB-

J.

Sgall

Online

Scheduling

In

Online

Algorithms

The

State

of

the

Art

A.

Fiat

G.

Woeginger

-LRB-

Eds

-RRB-

LNCS

Springer

pp.

-LSB-

-RSB-

D.

Shmoys

J.

Wein

D.

Williamson

Scheduling

Parallel

Machines

Online

In

Proc

of

the

nd

IEEE-FOCS

-LRB-

-RRB-

pp.

-LSB-

-RSB-

B.

V

ocking

HowAsymmetry

Helps

Load

Balancing

In

Proc

of

th

IEEE-FOCS

-LRB-

NewYork

-RRB-

pp.

hops
input sequence
online load balancing
lower bounds
limited information
oblivious scheduler
related machines
scheduling
online information