Broadcast with Partial Knowledge - CiteSeerX

Broadcast

with

Partial

(Preliminary

Baruch

Awerbuch

*

Israel

Cidon

Version)

t

David

Shay Peleg

Abstract This

work

concerns

the problem

message

cessor has partial tial

information

prior held

of date

consequently, conflict

ing

of broadcast-

efficiently

when

knowledge

of the broadcast

be out

Kutten

each proabout

the

The

par-

message.

by the processors

or otherwise different

erroneous,

processors

information.

might

Tight

hold

bounds

Electrical Israel. fIBM

Engineering, T.J.

Yorktown

Watson

Heights,

The Research

NY

Haifa p,o.

center

and applications

Mathematics

are discussed.

1

Introduction

tasks

in object

the topology which

case the view

derlying

of

32000,

Box 704,

network

at the

view

is an inventory

at each site),

and Com-

ture,

and

(e.g.,

a link

here

are subject fails,

necessary

One obvious Permission to copy without fee all or part of this material k granted provided that the copies are not made or distributed for direct comthe ACM

permission

of the Association

otherwise, 01991

or to republish,

ACM

copyright and notice

notice

for Computing

requires

and

the title

is given that copying Machinery.

a fee and/or

0-89791-439-2/91/0007/0153

specific

dated

views

Broadcast

of the is by

on initiating

To copy

the

permission.

object

to the

$1.50

time 153

which

changes

is consumed

is modified).

consistent

This

whenever

It

mecha-

and updated

at the different

a broadcast

The in na-

to have an efficient

algorithm.

held

database.

unit

record

algorithm

complexity

case the

are dynamic

sites.

for maintaining

of a distributed

possibility

resources

the resources

a resource

(in

of the un-

to occasional

a database

of the object

is thus

(in

be

network

or even a general

considered

or released,

Fellowship.

sites

object algorithm

a change

up-

is the Full is based

of the entire

of message

of

may

or certain

listing

of

sites

object

is a description

system

deal

“view”

separate

This

graph),

held

objects

the

of a communication

for maintaining

and its date appear,

prob-

computing

in many

system.

views

advantage,

pro-

computing

maintaining

nism

mercial

condi-

of the broadcast

distributed

concurrently

Award

publication

such

Motivation

The Weizmann Institute, Rehovot 76100, Israel. Supported in part by an Allen Fellowship, by a Walter and Elise Haaa Career Development and by a Bantrell

under

distributed

lems

a common

Laboratory, Harvard Uni02138. Partially supported

of Applied

other

a distributed

NOO014-85-K-0445.

WDepartment puter Science,

broadcast

to

with

10598.

~Aiken Computation versity, Cambridge, MA by ONR

and Faculty

Technion,

for

tocol

1.1

0078, ARO contract DAAL03-86-K-0171, NSF contract CCR861 1442, and a special grant from IBM. T*J. Watson Research center, p-o. Box tIBM 10598,

~

are

of Mathematics and Lab. for Computer Science, M. I. T., Cambridge, MA 02139. Supported by Air Force Contract TNDGAFOSR-86-

NY

Mansour

and

may

“Dept.

Heights,

Yishay

tions,

Many

704, Yorktown

$

~

established

ing a large contents

Knowledge

view

of

occurs.

Due

pipelining,

the

of this algorithm

is relatively

low.

Ontheother

be very

hand,

wasteful

object

may

tively

have

a correct

and

few

broadcast

the

suggested

for

bility

very

makes

of this

of updating minimal

do not

small

such

conditions,

tions

to

the

munication it

amount

of for

and

time

out

of the database

update

Partial

y“

with

problem,

is,

we and

Under

space. efficient to

Broadcast

lem

can be formulated

with

Partial

an

asynchronous

regarding

pi

has

an

m-bit

view

of

processor’s

lo-

in order

to

com-

the

knowledge”

the

inputs

that

prob-

which

of its

the

namely

(besides

its

own

This

[ACK90],

where

of database

protocols.

Even

lem,

none

previously

The

makes

neighbor-knowledge

of the

weaker

neighbors.

in

date

1.3

processors. the

assumption,

is justified

is shown

has no infor-

of other

knows

ver-

each processor

mentioning

problem,

processor

each

strongest

and

inputs

“neighbor

sumption with

input,

it is worth of

the

for

it

comes

and this

topology weaker

known

both in communication

complexity

asfor uP-

prob-

solutions and time.

measures

Consider

ion po,

Our

through-

correct

in which

that input)

prob-

Knowledge

communicant

of n+ 1 processors,

processor

its own

version

can be char-

as follows.

and

communication

we solve

problem,

comset-

paper,

knows

However,

bottleneck

the

the

and

sion of this mation

“communication

and

as a re-

of the network.

to use the

time

they

changes

to each processor

only

solu-

In this

Broadcast

the

are efficient

The

each

w, and

free in context

model

topology

about

given

In this

lem

consisting

network

different

plexities.

the

database,

problem called

have

in the past,

losses,

is to inform

cal inputs

Knowledge.

The

the broadcaster

of message

descripone as a

messages

object

main

simple

may

different

The

In particu-

the

respect

the

object.

processors to

local

the

overheads.

that

as a fairly

complexit

with

in the

due

pi.

the correct

out

coordination

we look

problem

turns

acterized

two

The

from

repre-

is w = Wo,

be no possi-

That

initial

object

nature

a distributed

any

at processor

of the

differ

Wi is stored

all the processors

complex-

input,

the local

the asynchronous

is to study

assumptions.

assume only

every

input

goal

pipelining

local

can be in-

BGJ+85]).

the time

work

object

of changes

got from sult

The

broadcaster.

Wi may

descriptions

update

their

broadcaster’s

pi and describes

of the

minimize

purpose

problem

1.2

topology to

lar,

This

in

of the problem

description

result “nec-

of the

as follows.

by the

tions

algorithms

SG89,

information

which

held

incre-

only

value

formulation

correct

to

while

is the

the

as the

to the prob-

Wo.

sentation

angle,

knowledge

the

seems

write

the

at processor

high.

The

ting,

the

MRR80, there

this

of as having

of

processors

all

terpreted

of

of rela-

transmitted.

handling

to employ

under

from

there

lem

This

of the system.

heart

([ACK90,

method,

“most”

in which

is

the

Unfortunately,

of

partial

extreme

solution

w=

should

is distinguished

p.

In a correct

processors

of the object,

strategy,

is at

problem

allow

view

information

strategy

ity

Viewed

of prior

other

Update

essary”

this

picture

to the processors

mental

the

can be thought the entire

available

strategy

that

processor

broadcaster. output

a success-

need to be informed

advantage

On

fact

changes.

problem

taking

that

maintenance the

the object, the

is clear

to utilize

already

since the

large.

it

consistency

strive

T.oi, and

might

in communication,

be rather

Consequently, ful

thisalgorithm

In order

network,

. . . . pn,

local

to quantify

with

ing local

knowledge,

input

measure

that

154

the possibility

captures

we first

of exploit-

introduce

the level

a new

of “informa-

tion”

of the

knowledge

sor.

Let

held

by processor

which

the

~i,

from

discrepancy

the

the

local

total

Our

&~= goal

tween

A = ~i

8 = A/n, =

rnZLX;

the

these discrepancies

ition

that

tocols

should

most

are

of the protocol

We therefore

express

time

complexity ~.

1.4

the

correct”,

may

require

1.5

Our

first

with

Partial

to

Broadcast

protocol,

munication,

i.e.

other

hand

cast

can be done can terminate

one would

like

respect

aiming be close

to

The in

Incremental

Update

poses

an

strategy

with

“correct”

view

the

a “correction” positions In

where this

propagates

source,

till

nodes

the

algo-

our

algo-

a slightly

better

Note stored

discrepancy

constraints.

upper

codes.

time

bound

that ways,

are

codes

random

network.

[BOGW88] privacy

Rabin

a reliable

codes

to achieve in

BCH

[Rab89]

a colm-

uses codes routing

and

codes

a malicious

fault-tolerant

uses

Goldwasser use

in

in

for so’lv-

[Met84]

protocols

Ben-Or,

lin-

before

algorithms

retransmission

to the

using

used

Metzner

we

proces-

subject

derived

were

distributed problems. and

in all cases,

at the various

bounds Such

Wigderson

to

envircmto achieve with

a low

overhead.

to while

Using theory ory,

proposed

the

a pro-

simple and

we

bounds

approach.

is that

arguments

from

communication

are

able

are almost

average

to

show

tight.

information

complexity that

the-

our

upper

We argue that

discrepancy

is ~,

the

commu-

Q(A

log(~))

to

nication

complexity

which

cent ains

and

time

input

~ log(~)). We also argue that in the cabse that no information is known about the dis-

neighbor’s

is

a “correction

the network are

its

corrected.

the

crepancies,

from

send ft(nrn)

It 155

is at least

when

transmits

list,

algorithm, through

all

A,

to

A is known,

in arbitrary

guarantee

complexity.

alternative

strategy

erroneous. the

time

log ~)

that

the inputs

ment.

complexity

probabil-

log m+n

O(n log $ + m + log ~)

sors to differ

plete

algorithm

It has success

complexity.

Reed-Solomon

and

Partial

cation

efficient

Thus,

random-

+ m + log ~) and the same communi-

On the

time.

discrepancy

of this

neighbor

wave”

total

achieves

in com-

complexity,

this

in the

with

c is a parameter

ing various

the broad-

on this

of

protocol

an efficient

and uses O(A and

constructing

Full

fashion

+ m)

essence

cessor all

since

communication

near-optimal

[ACK90]

The

in O(n

reducing

the

bits.

in a pipelined

to improve to

towards

maintaining

is wasteful

fast,

to

of O(n allow

Broadcast is the

fl(nm)

it is rather

thus with

which

require

even the

Broadcast

problem.

Assuming

rithm

and

are mea-

the

the

l–c,

where

rithm.

Our to

and

network

we provide

solution

time,

then

as a function

problem

appears

results

communication of

model.

solution

Knowledge

assump-

exploitation

algorithm

In this paper,

solutions

obvious

there

efficient

ized

ear The

for

under

Q(J . n) time.

ity at least

be small.

complexities

knowledge,

case of a path

Knowledge

pro-

views

should

complexity

Basic

intu-

the communication

The

in the bit

the

to discrepancy

of our solution

n and

sured

be-

“almost

the overhead

of m,

relationships

if

even

average dis-

i.e.,

that

in this

simple

also

of broadcast

inputs,

of neighbor

pipelining

is the

maximum

be proportional

processors

differs

Define

following

complexity

processors’

be no possibility

and the complexity

algorithms, the

tion

in

{6’i}.

is to study

of broadcast

Wi

bi, the

and the

be stressed

input

w, which

of the object.

should

of bits

at pi,

input

discrepancy

crepancy

6i of the

description

description

discrepancy

by each proces-

pi be the number

the broadcaster’s

correct

of

held

complexity

is at

any deterministic bits,

even if there

least

protocol

fl(n

+

would

are no discrep-

ancies bounds

Organization

1.7

at all.

The comparison

of our protocols

is given

in Figure

and lower

The

1.

rest

lows.

of the

In

1.6

Applications

to

topology

3, we present

One application network

of our work

problem

of Topology

task is at the heart protocols

can each

of its

adjacent

i.e.,

but

each processor

is based

on the

tions

and reconnection

cates

implementation

ert heless,

the

is efficient

in

though

not

tially

status

this global

of

network

of this

tion

of

strategy.

and

protocols for

it is possible

of the problem Knowledge fectively

to those

Namely,

Broadcast

Partial

model,

have since

found

their

hash function

introduc-

[WC79].

X = {h : A ~

many

B}

A fam-

is called

a

if for any al # a2 c A holds:

= 61 and

1 = 62] = — [B[2

h(a2)

the probability

ble choices

essenup-

formly

sig-

sal hash functions.

complexities with

chosen

problem

are many as follows.

B = 2P.

(Note

uni-

~. families

of simple

One example

structed

and

Let

univer-

can be con-

p be a prime

and let

IBI = p.) Then

ef-

to the

solution update

from

over the possi-

is randomly

Partial

Update,

Knowledge

is taken

of h, which

for

is a family

the

In

problem,

the

hash

protocol

of universal above

function

hash

example requires

functions.

the only

encoding two

of a

elements

2P, and also p, therefore

we can describe

and time.

such

a hash

only

pointing

bits.

(Note

overheads out

are presented the

in the

charges

a message

functions

from

whereas

presented

any

a topology

or equal

It is worth ity results

to

is most

the

the former

one can construct munication

where

(al-

observation

of Topology

given

with

lower

that is the

to relate

4.

functions

hash

bl, b2 c 13 the following

There

of Broadcasting

reducing

latter.

Nev-

topology

purposes

on the

procedure

leads

of [ACK90]

our

bounds

in Section

and Wegman

Prob[h(al)

[ACK90].

A consequence

(algorithm

the lower

parti-

communication

time),

bound

upper

applications

by Carter

universal and

compli-

broadcast

terms

our

hash

interesting

link

strategy.

significantly

resulting in

Update

recurring

In Section

Universal

Universal

is

of [ACK90]

communication-optimal

nificant

which

of the

2.1

link

of the protocol

Incremental

theory.

Preliminaries

2

status

each

algorithm

and coding

are established

ily of functions

possibility

with

of the

with

update

The

that

Ini-

information.

The topology

date

as follows. whether

purpose

neces-

universal

network The

is unaware

The

problem

some

This

ACG+90].

is aware

links,

links.

to supply

practical

BGJ+85,

processor

is up or down, of other

Update.

be formulated

tially,

status

of many

[MRR80,

problem

is to the classical

quote

as fol-

use concerning

Finally,

AVERAGE).

paper

is structured

2 we

for later

hash functions

update

paper

section

sary results

of the

that

results

our complex-

in

[ACK90]

complexity

one complexity

of size O(log

corn=

depend

in the bit complexity

message

only

in both

hash

are

units

on A.)

function,

represented

model for

Another

n) bits.

following. 156

function that

the

using encoding

Later,

when

it is assumed with

O(log

way to view We

are

IBI)

O(log

using that

a universal it

can

be

bits.

the parameters

interested

IB 1)

of h does not

is the

is a family

of

— Algorithm Full

Communication

broadcast

(folklore)

Update

[ACK90]

Incr.

Our

nm n+

Our

lower

bound

hash functions

lowing

property:

ments,

the

function

c

%,

probability

e = l/[131, of hash

distinct

them

c.

to

From

hash function

y 1/[ 13\.

we conclude

functions

that

and lower

2.1

1. The check bit trail

there

is a family

encoding

3. The

size is

theoretic

back-

encoding and

order

slightly

ground

basic

later

results

from

code Cm,d : {O, 1}~ that

transforms a codeword

The

codes

ing codeword check bits. the

code

c~,d(~).

c~,d The

attaches lengths bit

by lC@(~)

A code the original word

Cm,d(w)

The

trail

word

following

ties possessed

is a mapping

The

only

w ~ {O, 1 }~

about

is the

words

in

are stanthe

the

BCH

codes,

by the

code

encoding,

are

m.

extend to

our

the

codeword

[,

in the se-

we first

extend

than states

if

decoded

bits

will

After

the

to later

to be check-bit

properties

2~ –

input

m, d are omitted

d are clear

from

(e.g., size.

the

input

bits

and

the decodon in the codes

that

2.1.

The

in Theorem

subscripts

1.

be removed.

Cm,d referred

are meant

satisfy

be d-correcting the

codes

paper

U33PeCtiVelY.

from

the padding

All

7rL).

cc)de

appropriate

ing,

by

The

of length

a word

w by

and that

we need to comment

the encoding.

perform

that

is O(d log

to 2~ – 1 – [C~,d(w)]

differs

theorem

When

of r

to

zeros)

have

codes can

time

trail

that

have

can

will

of BCH

then

are denoted

in no more

simply padding

bit

codes

we

so they

use of arbitrary

BCH

We

time

It is well known

in polynomial

of check property

by

operations

theorem,

w ~ {O, l}m

w can be correctly z that

=

require

of check bits that

of the entire

to

result-

is a “trail”

I and lc~,d(~)

word

any

show

be accomplished

A

to be of the form

to

Cm,d is said

from

decoding

respectively)

and encoding

length

namely,

the trail

le~,d(~)l

in ‘m and d.

modify

the

paper

p c {O, 1}’

Denote

on

O G {O, l}~+r.

in this

ti is assumed

and the check qUt31

word =

based theory.

{O, l}~+r

codes,

where

are

coding

c~,d(w)

considered

“check-bit”

O = wl[p,

s

on

an input

into dard

and

e~’d$

to

the decoding some

is of length

all the above properties.

developed

m/3,

the 1o1-

2. The code 6’m,d is d-correcting.

choosing

polynomial

tools

d