Laboratoire

de

l'Informatique

du

Parall?lisme

Ecole

Normale

Sup?rieure

de

Lyon

Unit

de

recherche

associ?e

au

CNRS

n

Broadcasting

and

Multicasting

by

Shortest

Path

in

Cut-through

Routed

Networks

Johanne

Cohen

Pierre

Fraigniaud

Jean-Claude

K?onig

Andr?e

Raspaud

November

Research

Report

No

Ecole

Normale

Sup?rieure

de

Lyon

Adresse

lectronique

lip@lip.ens-lyon.fr

T?l?phone

-LRB-

-RRB-

T?l?copieur

-LRB-

-RRB-

All?e

d'Italie

Lyon

Cedex

France

Broadcasting

and

Multicasting

by

Shortest

Path

in

Cut-through

Routed

Networks

Johanne

Cohen

Pierre

Fraigniaud

Jean-Claude

K?onig

Andr?e

Raspaud

November

Abstract

This

paper

addresses

the

one-to-all

broadcasting

problem

and

the

one-to-many

broad

casting

problem

Broadcasting

is

the

information

dissemination

problem

which

consists

for

a

node

of

a

network

to

send

a

same

piece

of

information

to

all

the

other

nodes

Mul

ticasting

is

a

partial

broadcasting

in

the

sense

that

only

a

subset

of

nodes

forms

the

destination

set

Both

problems

have

many

applications

in

parallel

and

distributed

com

puting

In

this

paper

we

study

these

problems

under

both

line

model

and

cut-through

model

We

present

a

new

time

optimal

broadcasting

and

multicasting

algorithm

in

both

line

model

and

cut-through

model

We

show

how

the

behavior

of

this

algorithm

can

again

be

improved

in

case

of

tree

networks

We

will

also

show

that

most

of

the

optimization

problems

relative

to

the

maximization

of

the

efficiency

of

broadcasting

or

multicasting

algorithms

in

term

of

switching

times

or

vertex

load

are

NP-complete

R?esum?e

Ce

papier

traite

des

probl?emes

de

diffusion

et

de

multidiffusion

La

diffusion

a

partir

d'un

processeur

u

est

l'op

eration

qui

consiste

a

envoyer

un

message

a

tous

les

pro-

cesseurs

a

partir

de

l'initiateur

unique

u.

Une

multidiffusion

est

une

diffusion

partielle

l'ensemble

de

destinations

est

une

partie

des

noeuds

du

r?eseau

Ces

types

de

prob

l?emes

sont

li?es

aux

communications

globales

structur?ees

telles

que

celles

d?efinies

dans

la

biblioth?eque

MPI

-LRB-

Message

Passing

Interface

-RRB-

Dans

cet

article

nous

consid?erons

deux

modes

de

communication

le

mode

communications

ar

etes-disjointes

et

le

mode

commutation

de

circuits

Nous

nous

concentrons

uniquement

sur

le

nombre

d'

etapes

Nous

montrons

principalement

qu'il

est

non

seulement

possible

de

diffuser

en

ne

etapes

dans

le

mode

commutation

de

circuits

mais

aussi

que

toutes

les

communications

r?ealis?ees

a

chaque

etape

de

la

diffusion

peuvent

s

effectuer

sur

des

plus

courts

chemins

Ce

papier

traite

aussi

de

nombreux

autres

param?etres

comme

la

charge

des

sommets

etc.

Nous

montrons

que

minimiser

ces

param?etres

sont

des

probl?emes

NP-complets

Mots-cl

es

Diffusion

Multidiffusion

mode

communication

ar

etes-disjointes

mode

commutation

de

circuit

Broadcasting

and

Multicasting

by

Shortest

Path

in

Cut-through

Routed

Networks

Johanne

Cohen

LIP

-LCB-

CNRS

Ecole

Normale

Sup?erieure

de

Lyon

Lyon

Cedex

France

Pierre

Fraigniaudy

LIP

-LCB-

CNRS

Ecole

Normale

Sup?erieure

de

Lyon

Lyon

Cedex

France

Jean-Claude

K?onig

D?epartement

d'informatique

Universit?e

d'Evry

Evry

Cedex

France

Andr?e

Raspaud

LaBRI

-LCB-

CNRS

Universit?e

de

Bordeaux

Talence

Cedex

France

November

Abstract

This

paper

addresses

the

one-to-all

broadcasting

problem

and

the

one-to-many

broadcasting

problem

usually

simply

called

broadcasting

and

multicasting

respectively

Broadcasting

is

the

information

dissemination

problem

which

consists

for

a

node

of

a

network

to

send

a

same

piece

of

information

to

all

the

other

nodes

Multicasting

is

a

partial

broadcasting

in

the

sense

that

only

a

subset

of

nodes

forms

the

destination

set

Both

problems

have

many

applications

in

parallel

and

distributed

computing

For

instance

a

broadcasting

procedure

is

included

in

the

collective

communication

routines

of

the

MPI

library

Multicasting

has

also

many

applications

for

the

control

and

the

programming

of

distributed

memory

parallel

computers

and

of

networks

of

processors

In

this

paper

we

study

these

problems

under

both

line

model

and

cut-through

model

The

former

assumes

long

distance

calls

between

non

neighboring

processors

The

latter

completes

the

line

model

by

taking

into

account

the

use

of

a

routing

function

Long

distance

calls

are

possible

in

any

circuit-switched

or

wormhole

routed

machine

and

in

any

networks

supporting

ATM

protocol

or

optical

facilities

It

is

known

that

one

can

find

time

optimal

broadcast

and

multicast

protocols

in

the

line

model

in

polynomial

time

Unfortunately

theses

protocols

are

usually

inefficient

from

a

practical

point

of

view

because

they

do

not

use

the

resources

of

the

network

in

a

balanced

way

We

present

a

new

time

optimal

broadcasting

and

multicasting

algorithm

in

both

line

model

and

cut-through

model

This

algorithm

efficiently

use

the

bandwidth

of

the

network

We

show

how

the

behavior

of

this

algorithm

can

again

be

improved

in

case

of

tree

networks

We

will

also

show

that

most

of

the

optimization

problems

relative

to

the

maximization

of

the

efficiency

of

broadcasting

or

multicasting

algorithms

in

term

of

switching

times

or

vertex

load

are

NP-complete

Additional

support

by

the

DRET

of

the

DGA

yAdditional

support

by

the

research

program

ANM

of

the

CNRS

Introduction

The

broadcast

and

multicast

problem

Given

any

point-to-point

interconnection

network

-LSB-

-RSB-

modeled

as

a

graph

G

-LRB-

V

E

-RRB-

broadcasting

is

the

information

dissemination

problem

which

consists

to

send

a

same

piece

of

information

from

a

unique

source

to

all

the

other

nodes

of

the

network

Such

a

communication

scheme

typically

appears

in

many

applications

on

distributed

memory

parallel

computers

-LSB-

-RSB-

and

is

one

of

the

kernel

of

the

collective

communication

routines

of

the

MPI

library

-LRB-

Message

Passing

Interface

-LSB-

-RSB-

-RRB-

Broadcasting

is

actually

a

particular

case

of

multicasting

in

which

a

source

node

has

a

unique

message

to

send

to

a

subset

of

nodes

Again

multicasting

has

many

applications

for

the

control

of

parallel

computers

like

barrier

synchronization

or

cache

coherence

It

is

also

a

basic

tool

used

to

implement

parallel

data-bases

-LSB-

-RSB-

Another

major

example

of

the

use

of

a

multicasting

procedure

is

when

a

given

user

of

a

parallel

machine

can

only

make

use

of

a

subset

of

processors

-LRB-

a

group

of

processor

in

the

MPI

terminology

-RRB-

and

not

of

the

entire

machine

In

this

case

broadcasting

a

message

in

the

user

domain

at

the

application

level

is

actually

a

multicast

at

the

system

level

The

broadcasting

problem

was

intensively

studied

under

the

store-and-forward

routing

mode

in

which

packets

that

proceed

along

a

path

in

the

network

must

be

entirely

stored

on

each

intermediate

node

before

being

transmitted

to

the

next

one

The

reader

can

consult

the

surveys

-LSB-

-RSB-

The

communication

constraint

that

was

mostly

considered

in

the

literature

is

the

telephone

model

in

which

communications

take

place

in

a

sequence

of

calls

such

that

-LRB-

i

-RRB-

a

call

involves

exactly

two

neighboring

nodes

-LRB-

ii

-RRB-

a

node

can

participate

to

at

most

one

call

at

a

time

and

-LRB-

iii

-RRB-

each

call

takes

a

unit

of

time

A

round

is

then

defined

as

the

set

of

calls

performed

in

parallel

at

the

same

time

Although

hypothesis

-LRB-

iii

-RRB-

might

be

not

realistic

since

it

does

not

take

into

account

the

length

of

the

messages

that

are

exchanged

along

the

communication

links

it

applies

for

short

message

lengths

or

when

startup

times

are

much

larger

than

propagation

times

Moreover

even

if

for

very

long

messages

the

linear

model

-LSB-

-RSB-

gives

a

good

approximation

of

the

real

complexity

of

the

communications

in

message

passing

systems

Best

algorithms

obtained

in

this

model

are

often

derived

from

algorithms

obtained

in

the

telephone

model

by

pipelining

messages

-LSB-

-RSB-

Cut-through

routed

networks

In

most

of

the

modern

distributed

memory

parallel

computers

the

store-and-forward

routing

mode

has

been

replaced

by

various

types

of

virtual

cut-through

routing

-LSB-

-RSB-

including

directconnect

on

several

Intel

's

machines

circuit-switching

and

wormhole

routing

-LSB-

-RSB-

In

the

circuitswitched

mode

when

a

node

x

sends

a

message

to

a

non

neighboring

node

y

a

path

is

created

between

x

and

y

to

directly

connect

these

two

nodes

The

message

from

x

is

then

transmitted

along

this

path

Intermediate

processors

do

not

receive

the

message

that

goes

through

them

because

nodes

generally

consist

of

several

components

the

processing

unit

its

local

memory

and

a

router

which

is

in

charge

of

transmitting

messages

-LRB-

see

Figure

-RRB-

A

router

sends

a

message

to

the

local

memory

of

its

node

only

when

the

message

is

precisely

destined

to

its

node

Wormhole

routing

differs

from

the

circuit-switched

routing

in

message

transmission

In

wormhole

routing

a

message

is

splited

into

small

units

called

flits

The

first

flit

is

used

to

inform

each

intermediate

node

about

the

route

followed

by

the

message

and

the

remaining

flits

follow

in

a

pipeline

fashion

-LRB-

the

last

flit

releases

the

intermediate

connections

-RRB-

Wormhole

routing

does

not

require

a

whole

path

to

be

reserved

between

the

source

and

the

destination

it

makes

use

of

a

number

of

links

proportional

to

the

length

of

the

message

Long

distance

calls

are

also

possible

in

any

networks

supporting

ATM

protocol

-LSB-

-RSB-

and

in

any

all-optical

networks

-LSB-

-RSB-

Like

any

virtual

cut-through

routing

circuit-switched

and

wormhole

routing

are

not

sensitive

to

the

path

length

Indeed

the

time

Tx!y

-LRB-

L

-RRB-

to

send

a

message

of

size

L

from

x

to

y

is

generally

modeled

by

dx!y

ffi

Lb

where

is

the

start-up

time

that

consists

of

local

initialization

processes

-LRB-

hard

or

soft

-RRB-

ffi

is

the

switching

time

of

the

routers

dx!y

is

the

length

of

the

route

from

x

to

y

and

b

is

the

bandwidth

of

the

links

-LRB-

see

-LSB-

-RSB-

-RRB-

In

general

we

have

AE

ffi

and

AE

Therefore

as

for

the

store-and-forward

mode

assuming

Tx!y

-LRB-

L

-RRB-

constant

is

a

good

approximation

when

L

is

not

too

large

The

line

and

cut-through

models

In

this

paper

we

are

interested

in

the

communication

complexity

of

broadcasting

and

multicasting

in

virtual

cut-through

routed

networks

In

the

following

we

will

make

use

in

particular

of

the

so-called

line

model

-LSB-

-RSB-

which

supposes

that

-LRB-

-RRB-

a

call

involves

exactly

two

nodes

-LRB-

these

two

nodes

might

be

at

distance

greater

than

one

-RRB-

-LRB-

-RRB-

a

node

can

participate

to

at

most

one

call

at

a

time

-LRB-

-RRB-

each

call

takes

a

unit

of

time

-LRB-

Tx!y

-LRB-

L

-RRB-

-RRB-

and

-LRB-

-RRB-

any

two

paths

corresponding

to

two

simultaneous

calls

must

be

edge-disjoint

Unlike

the

telephone

model

which

allows

neighbor-to-neighbor

communications

only

the

line

model

allows

long

distance

calls

between

non

neighboring

processors

as

it

is

possible

in

any

virtual

cut-through

implementation

However

the

line

model

suffers

from

a

major

drawback

Indeed

in

virtual

cut-through

routing

the

paths

followed

by

messages

are

in

general

determined

by

a

routing

function

In

a

network

modeled

by

a

graph

G

-LRB-

V

E

-RRB-

a

routing

function

R

is

a

collection

of

local

functions

R

fRx

V

E

x

V

g

such

that

any

message

destined

to

node

y

arriving

or

originated

in

x

is

transmitted

from

x

along

the

edge

-LRB-

or

channel

-RRB-

Rx

-LRB-

y

-RRB-

-LRB-

To

be

well

defined

Rx

must

satisfy

that

Rx

-LRB-

y

-RRB-

is

an

incident

edge

of

x

for

any

y

V

that

is

Rx

V

Ex

where

Ex

f

-LRB-

x

y

-RRB-

E

y

V

g.

-RRB-

Two

typical

examples

of

routing

functions

are

e-cube

routing

in

hypercubes

-LRB-

messages

are

routed

in

dimension

order

-RRB-

and

XY

routing

on

meshes

-LRB-

messages

are

first

routed

horizontally

and

then

vertically

-RRB-

Note

however

that

a

lot

of

recent

papers

on

multicasting

in

meshes

-LRB-

see

the

references

in

-LSB-

-RSB-

-RRB-

make

use

of

more

tricky

routing

functions

which

offer

better

properties

To

take

into

account

the

natural

use

of

a

routing

function

we

will

consider

the

following

simple

additional

hypothesis

to

the

line

model

-LRB-

-RRB-

paths

followed

by

messages

are

constructed

by

application

of

a

routing

function

R

fRx

V

Ex

x

V

g

at

each

intermediate

node

In

this

paper

the

line

model

plus

hypothesis

is

called

the

cut-through

model

Note

that

one

may

be

interested

in

considering

other

types

of

routing

functions

A

typical

example

is

when

Rx

Ex

V

Ex

that

is

when

the

local

decision

depends

of

the

input

port

of

the

message

Such

type

of

routing

functions

called

input-dependent

is

particularly

useful

in

wormhole

routing

to

take

into

account

the

several

virtual

channels

possibly

introduced

to

avoid

deadlocks

-LSB-

-RSB-

However

since

deadlock

avoidance

is

not

the

main

interest

of

this

paper

we

will

restrict

our

study

to

the

simple

hypothesis

although

most

of

the

results

derived

in

this

paper

apply

also

for

input-dependent

routing

functions

Many

results

have

been

also

derived

when

hypothesis

is

replaced

by

the

all-ports

hypothesis

that

is

when

a

node

can

simultaneously

communicate

with

as

many

nodes

as

its

number

of

ports

-LRB-

see

for

instance

the

references

in

-LSB-

-RSB-

-RRB-

Similarly

the

broadcasting

and

the

gossiping

problems

have

also

been

investigated

when

hypothesis

is

replaced

by

a

vertex-disjoint

constraint

-LSB-

-RSB-

or

by

a

pseudovertex-disjoint

constraint

-LSB-

-RSB-

Unfortunately

most

of

these

variations

yield

NP-complete

problems

when

looking

for

time

optimal

broadcasting

or

multicasting

protocols

Farley

's

theorem

In

the

line

model

most

of

the

known

results

concern

particular

network

architectures

as

trees

-LSB-

-RSB-

cycles

-LSB-

-RSB-

meshes

or

tori

-LSB-

-RSB-

A

major

result

which

applies

to

any

topology

is

due

to

Farley

-LSB-

-RSB-

who

has

shown

that

under

the

line

model

for

any

network

G

of

n

nodes

the

broadcasting

time

from

any

node

of

G

is

ne

This

result

will

be

referred

as

Farley

's

Theorem

in

the

following

Note

that

this

theorem

is

optimal

since

hypotheses

and

imply

that

the

number

of

informed

nodes

can

at

most

double

at

each

round

Note

also

the

strong

difference

between

the

line

model

and

the

telephone

model

under

the

telephone

model

knowing

whether

Processor

Memory

RouteroutputlinksInputlinksFrommemory

Tomemory

FromothernodesTo

othernodes

Figure

A

node

of

the

network

the

broadcasting

time

of

an

arbitrary

node

of

an

arbitrary

network

is

less

than

an

arbitrary

constant

is

NP-complete

-LSB-

-RSB-

Also

the

diameter

is

not

a

limitative

constraint

for

broadcasting

in

the

line

model

whereas

it

is

a

lower

bound

for

any

broadcasting

algorithm

under

the

telephone

model

Farley

's

theorem

can

be

easily

extended

to

the

multicast

problem

for

any

network

G

-LRB-

V

E

-RRB-

for

any

source

node

u

V

and

for

any

destinations

set

D

ae

V

u

D

the

multicasting

time

of

u

in

D

is

jDje

As

broadcasting

multicasting

has

been

intensively

studied

in

the

past

under

both

store-and-forward

or

cut-through-like

models

-LRB-

see

the

references

in

-LSB-

-RSB-

-RRB-

In

most

of

the

cases

the

network

is

fixed

-LRB-

generally

a

mesh

-RRB-

and

the

influence

of

the

multicasting

algorithm

on

the

traffic

was

measured

by

simulation

Note

also

that

many

papers

dealing

with

multicasting

make

use

of

an

additional

hypothesis

called

the

path-based

hypothesis

-LSB-

-RSB-

which

specifies

that

a

node

is

able

to

read

any

message

traversing

its

router

In

this

paper

we

will

consider

the

line

model

applied

to

arbitrary

network

topologies

Limitation

of

Farley

's

theorem

Farley

's

theorem

does

not

close

the

discussion

about

the

line

model

-LRB-

for

instance

papers

-LSB-

-RSB-

optimize

other

criteria

related

to

the

topology

of

the

network

-RRB-

Indeed

in

practice

it

is

usually

required

that

the

traffic

generated

by

a

broadcasting

or

a

multicasting

does

not

interfere

with

other

possible

traffic

For

instance

any

broadcasting

initiated

by

a

specific

user

of

a

multiuser

parallel

machine

inside

its

own

processor

domain

must

not

slow

down

communications

of

other

users

especially

when

their

respective

domains

are

disjoint

More

generally

multicastings

are

often

initiated

by

system

processes

which

must

not

reduce

the

ability

of

any

user

processes

to

exchange

messages

at

their

maximum

rate

In

other

words

one

requires

the

number

of

resources

used

to

multicast

or

to

broadcast

to

be

minimal

or

at

least

small

The

ne-time

protocol

of

Farley

's

theorem

is

based

on

transmitting

the

message

along

the

edge

of

a

spanning

tree

and

therefore

does

not

allow

to

minimize

the

resources

used

to

perform

a

broadcast

For

instance

the

routes

followed

by

the

several

copies

of

the

broadcasted

message

can

be

quite

long

compared

to

the

shortest

paths

between

their

sources

and

destinations

The

length

of

the

paths

is

not

a

major

issue

for

cut-through

routing

However

the

simultaneous

use

of

a

lot

of

long

paths

implies

that

many

links

will

be

reserved

in

spite

of

other

possible

communications

Our

main

goal

in

this

paper

is

to

figure

out

whether

it

is

possible

to

derive

optimal

ne-time

broadcast

protocols

achieving

better

use

of

the

available

resources

of

the

network

Among

the

several

possible

parameters

which

measure

the

efficiency

of

a

multicast

or

a

broadcast

protocol

we

will

consider

Number

of

links

We

will

be

interested

in

minimizing

the

total

number

of

communication

links

which

are

used

either

at

each

round

of

the

protocol

or

globally

during

the

whole

protocol

The

less

number

of

links

is

used

the

most

furtive

is

the

protocol

Number

of

transmitters

A

transmitter

is

a

router

which

is

explicitly

used

to

forward

a

message

during

a

broadcast

of

a

multicast

protocol

We

will

be

interested

in

minimizing

the

number

of

transmitters

again

either

at

each

round

of

the

algorithm

or

during

the

whole

protocol

Minimizing

the

number

of

transmitters

allows

to

potentially

decrease

the

number

of

nodes

which

will

be

disturbed

by

the

multicast

protocol

Load

of

the

transmitters

Alternatively

we

will

be

interested

in

minimizing

the

maximum

number

of

communication

paths

that

a

given

transmitter

handles

simultaneously

-LRB-

that

is

the

maximum

number

of

paths

induced

by

a

broadcasting

or

a

multicasting

which

traverse

a

router

simultaneously

-RRB-

We

would

like

to

not

overload

intermediate

routers

so

that

they

keep

their

ability

to

route

other

messages

at

their

maximum

speed

Of

course

there

is

a

tradeoff

between

the

number

of

transmitters

and

the

load

of

these

transmitters

Maximum

path

length

We

will

be

interested

in

minimizing

the

maximum

length

of

the

paths

used

during

a

broadcast

or

a

multicast

protocol

This

parameter

will

allow

us

to

estimate

the

possible

degradation

of

the

time

complexity

of

the

protocol

when

the

switching

time

of

the

routers

can

not

be

neglected

when

a

message

proceed

along

a

long

path

For

all

these

parameters

we

will

give

algorithms

minimizing

their

values

or

we

will

show

that

the

corresponding

decision

problem

is

NP-complete

Contents

and

organization

of

the

paper

In

Section

we

will

present

an

example

which

clearly

shows

the

difference

between

the

line

model

and

the

cut-through

model

In

Section

we

will

derive

a

new

polynomial

algorithm

which

returns

for

any

networks

G

a

time

optimal

multicast

protocol

-LRB-

and

as

a

particular

case

a

time

optimal

broadcast

protocol

-RRB-

in

G

in

the

line

model

This

protocol

minimizes

the

total

number

of

edges

used

at

each

round

-LRB-

Theorem

-RRB-

A

major

point

is

that

our

new

algorithm

also

applies

to

the

cut-through

model

under

the

simple

condition

that

the

routing

function

generates

shortest

paths

only

-LRB-

Theorem

-RRB-

whereas

all

time

optimal

protocols

derived

in

the

literature

up

to

now

required

to

be

applied

to

the

cut-through

model

a

routing

function

using

the

edges

of

a

spanning

tree

-LRB-

such

a

routing

function

can

not

be

used

in

practice

since

it

creates

a

lot

of

contentions

at

the

root

of

the

tree

and

the

lengths

of

the

routes

could

be

much

too

large

compared

to

the

shortest

paths

between

the

sources

and

the

destinations

-RRB-

Of

course

there

is

no

reason

why

our

algorithm

which

minimizes

the

number

of

edges

used

at

each

round

should

also

minimize

the

sum

taken

over

all

the

rounds

of

the

total

number

of

edges

used

at

every

round

In

fact

we

will

show

that

the

decision

problem

corresponding

to

this

minimization

problem

is

NP-complete

-LRB-

Theorem

-RRB-

However

we

will

show

that

our

algorithm

is

optimal

up

to

a

logarithmic

factor

-LRB-

Theorem

-RRB-

In

section

we

will

derive

specific

lower

and

upper

bounds

for

trees

-LRB-

Theorem

-RRB-

The

rest

of

the

paper

-LRB-

Sections

and

-RRB-

is

dedicated

to

the

NP-completeness

of

minimizing

the

number

-LRB-

Theorem

-RRB-

or

the

load

-LRB-

Theorem

-RRB-

of

the

transmitters

or

minimizing

the

total

switching

time

i.e.

the

maximum

length

of

the

routing

paths

used

to

broadcast

or

multicast

-LRB-

Theorem

-RRB-

These

results

imply

in

particular

that

routers

must

be

sophisticated

enough

to

support

simultaneous

routing

of

many

paths

without

degradation

of

their

switching

time

Similarly

even

if

it

is

not

a

major

issue

to

minimize

the

length

of

the

routes

these

results

show

that

broadcasting

or

multicasting

messages

of

small

size

is

difficult

to

optimize

when

the

switching

time

can

not

be

neglected

Finally

Section

contains

some

concluding

remarks

Line

model

versus

cut-through

model

Before

starting

our

study

let

us

point

out

the

difference

between

the

line

model

and

the

cutthrough

model

to

make

clear

these

different

models

Let

us

consider

the

network

on

Figure

It

is

a

double-star

-LRB-

V

E

-RRB-

with

two

centers

u

and

v

and

n

rays

First

let

us

show

uv

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Figure

A

double

star

of

vertices

how

to

broadcast

in

ne

rounds

from

any

source

of

in

the

line

model

-LRB-

Farley

's

theorem

insures

that

such

a

broadcast

protocol

exists

-RRB-

Nodes

at

the

extremities

of

the

rays

are

labeled

from

to

n

node

u

is

labeled

and

node

v

is

labeled

n

Note

that

nodes

u

and

v

play

the

same

role

There

are

only

two

cases

to

be

considered

either

the

source

of

the

broadcasting

is

a

center

-LRB-

say

node

-RRB-

or

it

is

one

of

the

extremities

of

the

n

rays

-LRB-

say

node

-RRB-

In

both

cases

one

can

always

proceed

so

that

both

nodes

and

are

aware

of

the

broadcasted

message

after

one

round

Now

if

one

consider

the

subgraph

of

the

double

star

obtained

by

removing

all

the

edges

between

v

and

the

other

nodes

but

u

we

obtain

a

usual

star

of

center

u.

On

this

star

the

ne

next

rounds

are

given

by

at

round

i

i

if

the

node

labeled

k

is

aware

of

the

message

then

it

informs

the

node

labeled

k

There

is

no

edge

contention

and

after

round

ne

all

nodes

are

aware

of

the

message

Let

us

now

consider

the

routing

function

R

that

routes

the

messages

on

the

double

star

as

follows

V

w

u

w

v

V

Rw

-LRB-

x

-RRB-

u

V

Ru

-LRB-

x

-RRB-

v

V

Rv

-LRB-

x

-RRB-

x

Assume

n

It

is

not

possible

to

broadcast

in

rounds

from

any

source

in

the

double

star

of

vertices

under

the

cut-through

model

Indeed

to

broadcast

in

rounds

the

last

round

would

consists

of

calls

performed

at

the

same

time

Therefore

at

least

two

of

these

calls

would

use

the

arc

from

u

to

v

which

is

impossible

-LRB-

Recall

that

both

line

model

and

cut-through

model

require

edge-disjoint

paths

-RRB-

This

example

shows

that

the

cut-through

model

is

more

restrictive

than

the

line

model

In

particular

it

is

not

true

that

for

any

network

G

and

for

any

routing

function

R

on

G

there

exists

a

broadcast

protocol

from

any

node

of

G

that

performs

in

ne

rounds

However

one

can

state

the

following

result

Property

-LRB-

cut-through

version

of

Farley

's

theorem

-RRB-

Under

the

cut-through

model

for

any

network

G

of

n

nodes

there

exists

a

routing

function

R

such

that

the

broadcasting

time

from

any

node

of

G

is

ne

Proof

There

exist

many

proofs

of

Farley

's

theorem

and

similar

results

-LRB-

see

-LSB-

-RSB-

-RRB-

All

of

them

use

an

arbitrary

spanning

tree

of

the

considered

network

and

all

the

calls

are

performed

along

the

edges

of

this

spanning

tree

Any

spanning

tree

induces

a

routing

function

R

since

there

is

a

unique

path

between

any

two

nodes

in

a

tree

Therefore

all

paths

used

by

the

broadcast

protocols

derived

in

-LSB-

-RSB-

can

be

generated

by

a

routing

function

The

proof

of

Property

produces

a

routing

function

which

follows

the

edges

of

a

spanning

tree

of

the

network

Such

a

routing

function

induces

a

lot

of

contentions

and

hot-spots

when

it

is

used

for

other

communication

problems

Indeed

the

traffic

is

not

balanced

and

the

root

of

the

tree

is

clearly

overloaded

Moreover

though

the

use

of

shortest

paths

is

not

necessarily

important

in

network

using

circuit-switched

or

wormhole

routing

such

a

routing

function

may

route

messages

exchanged

between

neighbors

along

a

path

of

length

twice

the

diameter

of

the

network

In

the

next

section

we

will

show

that

it

is

possible

do

much

better

Using

the

minimum

number

of

links

In

this

section

we

will

focus

on

the

total

number

of

links

that

are

used

by

a

broadcasting

or

a

multicasting

algorithm

at

each

round

or

during

the

whole

protocol

As

we

saw

this

parameter

can

be

considered

as

a

measure

of

the

furtiveness

of

the

protocol

Note

that

by

total

number

of

links

used

during

the

whole

protocol

we

mean

the

sum

over

all

the

rounds

of

the

number

of

links

used

at

each

round

At

each

round

of

a

broadcasting

algorithm

nodes

aware

of

the

message

are

matched

with

other

nodes

that

did

not

have

received

the

message

yet

Let

us

formalize

this

fact

Definition

Let

U

be

a

subset

of

vertices

of

a

graph

G

-LRB-

V

E

-RRB-

A

pseudo-matching

in

U

is

a

set

P

of

b

c

pairwise

edge-disjoint

paths

in

G

such

that

every

vertex

of

U

-LRB-

but

one

if

jU

j

is

odd

-RRB-

is

an

extremity

of

a

path

in

P

The

following

result

shows

in

particular

that

a

pseudo-matching

in

U

exists

for

any

choice

of

U

ae

V

We

denote

by

d

-LRB-

x

y

-RRB-

the

distance

between

two

nodes

x

and

y

of

a

graph

G.

Lemma

Let

U

be

a

subset

of

vertices

of

a

graph

G

-LRB-

V

E

-RRB-

One

can

group

the

vertices

of

U

in

pairs

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

xk

yk

-RRB-

where

k

b

c

xi

U

yj

U

xi

xj

yi

yj

and

xi

yj

for

all

i

j

kg

such

that

any

shortest

path

between

xi

and

yi

is

edge-disjoint

with

any

shortest

path

between

xj

and

yj

i

j

and

such

that

Pki

d

-LRB-

xi

yi

-RRB-

is

minimum

among

all

the

possible

choices

of

the

pairs

-LRB-

xi

yi

-RRB-

i

k.

Proof

Let

m

jU

j

and

let

us

consider

the

complete

graph

Km

of

m

vertices

where

each

vertex

of

Km

is

identified

to

a

vertex

of

U

We

add

weights

on

the

edges

of

Km

the

edge

fx

yg

has

weight

d

-LRB-

x

y

-RRB-

the

distance

between

x

and

y

in

G.

Consider

a

perfect

matching

of

minimum

weight

in

Km

This

matching

induces

a

set

P

of

b

c

shortest

paths

in

G

such

that

every

vertex

of

U

-LRB-

but

one

if

jU

j

is

odd

-RRB-

is

an

extremity

of

a

path

in

P.

-LRB-

In

case

of

multiple

shortest

paths

between

two

matched

vertices

choose

one

arbitrarily

-RRB-

We

claim

that

the

paths

of

P

are

pairwise

edge-disjoint

Indeed

assume

that

two

paths

Pfx

yg

and

of

P

are

not

edge-disjoint

It

means

that

there

exist

two

vertices

z

and

such

that

Pfx

yg

Pfx

zg

-LSB-

Pfz

-LSB-

yg

and

zg

-LSB-

P

fz

-LSB-

or

-LSB-

P

zg

-LSB-

Pfz

where

Pfz

and

P

fz

have

at

least

one

edge

in

common

Assume

without

loss

of

generality

that

zg

-LSB-

P

fz

-LSB-

It

implies

that

the

two

matching

fx

yg

and

can

be

replaced

by

two

other

matching

fx

and

fy

The

former

matching

has

a

weight

of

d

-LRB-

x

z

-RRB-

d

-LRB-

y

-RRB-

d

-LRB-

z

-RRB-

d

-LRB-

-RRB-

d

-LRB-

z

-RRB-

whereas

the

later

has

a

weight

of

d

-LRB-

x

-RRB-

d

-LRB-

y

-RRB-

which

is

less

or

equal

than

d

-LRB-

x

z

-RRB-

d

-LRB-

y

-RRB-

d

-LRB-

z

-RRB-

d

-LRB-

-RRB-

Since

d

-LRB-

z

-RRB-

we

obtain

a

contradiction

with

the

fact

that

the

original

matching

is

of

minimum

weight

and

therefore

the

paths

of

P

are

pairwise

edge-disjoint

We

can

define

the

weight

of

a

pseudo-matching

P

in

U

as

the

sum

of

the

lengths

of

all

the

paths

in

P

We

are

interested

in

minimizing

the

weights

of

the

several

pseudo-matching

generated

at

any

round

of

a

multicasting

or

a

broadcasting

algorithm

A

pseudo-matching

of

small

weight

requires

low

use

of

the

bandwidth

to

performs

exchanges

between

the

extremities

of

its

paths

The

following

theorem

shows

that

such

this

minimization

is

possible

in

polynomial

time

It

strongly

improves

Farley

's

theorem

Theorem

For

any

network

G

-LRB-

V

E

-RRB-

of

n

nodes

and

any

node

u

of

G

one

can

compute

in

polynomial

time

a

multicast

protocol

from

u

to

any

set

D

in

G

u

D

which

performs

in

jDje

rounds

in

the

line

model

and

such

that

-LRB-

i

-RRB-

all

the

calls

are

performed

along

shortest

paths

-LRB-

ii

-RRB-

at

any

of

the

jDje

rounds

the

weight

of

the

corresponding

pseudo-matching

is

minimum

Proof

The

jDje

pseudo-matching

are

constructed

backward

Start

with

D

and

by

Lemma

compute

a

pseudo-matching

in

of

minimum

weight

and

containing

shortest

paths

only

Then

choose

one

of

the

two

extremities

of

each

path

in

This

choice

is

arbitrary

but

for

the

source

u

that

must

be

selected

Keep

also

the

unique

isolated

unmatched

vertex

in

if

exists

All

these

nodes

form

a

set

Then

compute

a

pseudo-matching

in

of

minimumweight

and

containing

shortest

paths

only

Again

this

is

possible

by

Lemma

Then

we

extract

a

set

from

as

was

extracted

from

and

repeat

the

process

until

a

set

Ui

is

reduced

to

u.

Clearly

i

satisfies

i

jDje

This

protocol

can

be

computed

in

polynomial

time

because

one

can

compute

a

perfect

matching

of

minimum

weight

in

the

complete

graph

in

polynomial

time

-LSB-

-RSB-

and

thus

we

can

apply

the

construction

of

Lemma

in

polynomial

time

Now

this

result

also

holds

in

the

cut-through

model

because

in

the

proof

of

Lemma

the

way

the

shortest

paths

are

selected

does

not

matter

In

particular

they

can

have

been

constructed

by

any

shortest

path

routing

function

Therefore

we

get

the

following

result

which

strongly

improves

Property

Theorem

For

any

network

G

-LRB-

V

E

-RRB-

of

n

nodes

any

node

u

of

G

and

any

shortest

path

routing

function

on

G

one

can

compute

in

polynomial

time

a

multicast

protocol

from

u

to

any

set

D

in

G

u

D

which

performs

in

jDje

rounds

in

the

cut-through

model

and

such

that

at

any

of

the

jDje

rounds

the

weight

of

the

corresponding

pseudo-matching

is

minimum

Theorem

says

that

whatever

is

the

network

and

the

routing

function

on

this

network

if

this

routing

function

routes

messages

along

shortest

paths

it

will

be

possible

to

broadcast

and

multicast

optimally

in

logarithmic

time

Most

of

the

routing

functions

used

in

the

usual

topologies

-LRB-

meshes

multi-dimensional

tori

-RRB-

routes

on

shortest

paths

-LRB-

XY

routing

e-cube

routing

-RRB-

If

wormhole

routing

is

used

it

is

interesting

to

point

out

that

there

is

no

contradiction

between

the

search

for

a

deadlock

free

routing

function

and

the

search

for

a

routing

function

that

insures

optimal

broadcasting

Indeed

most

of

the

classical

deadlock

free

routing

functions

generate

shortest

paths

only

Theorems

and

both

say

that

one

can

easily

minimize

the

annoyance

of

each

round

of

any

multicasting

algorithm

and

therefore

one

can

easily

insure

that

the

additional

traffic

due

to

this

process

will

not

perturb

too

much

other

user

traffic

Unfortunately

it

does

not

say

that

the

sum

of

the

lengths

of

all

the

paths

used

during

the

whole

multicasting

is

minimum

The

next

result

shows

that

minimizing

this

parameter

is

NP-complete

Theorem

The

following

problem

is

NP-complete

Minimum-Total-Path-Length

-LRB-

MTPL

-RRB-

Instance

A

graph

G

-LRB-

V

E

-RRB-

a

vertex

u

of

G

a

subset

D

ae

V

u

D

and

an

integer

k.

Question

Does

there

exist

a

multicast

protocol

from

u

to

D

in

G

performing

in

jDje

rounds

in

the

line

model

and

such

that

the

sum

of

all

the

lengths

of

all

the

communication

paths

generated

by

this

protocol

is

at

most

k

The

proof

of

Theorem

is

based

on

the

following

lemma

Lemma

The

following

problem

is

NP-complete

Log-Broadcast

-LRB-

LogB

-RRB-

Instance

A

graph

G

-LRB-

V

E

-RRB-

of

n

vertices

and

a

vertex

u

of

G.

Question

Does

there

exist

a

broadcast

protocol

from

u

in

G

performing

in

at

most

ne

rounds

in

the

telephone

model

Proof

It

is

well

known

that

the

following

problem

has

been

shown

to

be

NP-complete

by

Johnson

-LRB-

see

-LSB-

-RSB-

for

his

proof

-RRB-

Broadcast

Instance

A

graph

G

-LRB-

V

E

-RRB-

a

vertex

u

of

G

and

an

integer

k.

Question

Does

there

exist

a

broadcast

protocol

from

u

in

G

performing

in

at

most

k

rounds

in

the

telephone

model

Lemma

can

be

proved

using

slightly

the

same

technic

as

the

one

used

in

-LSB-

-RSB-

to

show

that

the

problem

Broadcast

is

NP-complete

The

only

difference

is

that

one

have

to

add

vertices

until

k

is

equal

to

ne

Proof

of

Theorem

MTPL

is

clearly

in

NP

MTPL

is

NP-complete

by

transformation

from

Problem

LogB

Let

G

u

be

an

instance

of

the

LogB

problem

We

transform

this

instance

in

an

instance

G

D

u

k

of

the

MTPL

problem

by

setting

D

V

and

k

n

Clearly

if

there

exists

a

broadcast

protocol

from

u

in

G

performing

in

at

most

ne

rounds

in

the

telephone

model

then

there

exists

a

broadcast

protocol

from

u

in

G

in

ne

rounds

in

the

line

model

and

such

that

the

sum

of

all

the

lengths

of

the

communication

paths

is

at

most

n

Reciprocally

if

there

exists

a

broadcast

protocol

from

u

in

G

performing

in

ne

rounds

in

the

line

model

and

such

that

the

sum

of

all

the

lengths

of

the

communication

paths

is

at

most

n

then

all

calls

are

performed

between

neighboring

vertices

and

therefore

there

exists

a

broadcast

protocol

from

u

in

G

performing

in

at

most

ne

rounds

in

the

telephone

model

Theorem

implies

that

it

is

difficult

to

optimize

the

total

edge-load

of

a

multicasting

or

a

broadcasting

algorithm

However

one

can

approximate

this

value

up

to

a

logarithmic

factor

More

precisely

for

any

network

G

-LRB-

V

E

-RRB-

any

node

u

V

and

any

set

D

ae

V

u

D

let

us

denote

by

S

-LRB-

G

D

u

-RRB-

the

minimum

taken

over

all

the

multicast

protocols

from

u

to

D

in

G

performing

in

jDje

rounds

of

the

sum

of

all

the

lengths

of

all

the

paths

generated

by

the

protocol

Theorem

shows

that

minimizing

S

-LRB-

G

D

u

-RRB-

is

an

NP-complete

problem

However

we

have

in

the

case

of

the

broadcasting

problem

Theorem

Let

G

-LRB-

V

E

-RRB-

be

any

graph

and

let

u

V

There

exists

a

broadcast

protocol

from

u

in

G

which

performs

in

ne

rounds

in

the

line

model

and

such

that

the

total

sum

of

all

the

paths

generated

by

the

algorithm

is

at

most

neS

-LRB-

G

V

u

-RRB-

Proof

S

-LRB-

G

V

u

-RRB-

n

because

jV

j

vertices

must

receive

the

message

At

any

round

of

the

protocol

of

Theorem

the

pseudo-matching

P

which

is

constructed

forms

a

forest

-LRB-

i.e.

a

set

of

disjoint

trees

-RRB-

Indeed

if

it

would

not

be

the

case

we

could

construct

another

pseudomatching

with

fewer

edges

as

follows

Assume

for

the

purpose

of

contradiction

that

there

is

a

cycle

in

P

and

let

C

xk

k

be

this

cycle

C

is

actually

composed

of

union

of

sub-paths

of

p

paths

of

P

We

identify

particular

vertices

of

C

called

doors

A

door

is

a

vertex

xi

of

C

such

that

-LRB-

xi

mod

k

xi

-RRB-

and

-LRB-

xi

xi

mod

k

-RRB-

do

not

belong

to

the

same

path

of

P.

Clearly

the

number

of

doors

is

If

the

number

of

doors

is

then

C

is

composed

of

parts

of

two

paths

yr

xs

yr

y

g

and

xs

xs

z

These

paths

could

be

easily

replaced

by

P

yr

z

and

P

xs

yr

y

g

This

would

yield

a

pseudo-matching

of

smaller

weight

A

contradiction

If

the

number

of

doors

is

let

P

be

one

of

the

paths

of

P

using

edges

of

C.

P

y

g.

Let

i

be

the

smallest

index

such

that

-LRB-

yi

yi

-RRB-

C

We

have

yi

xj

for

some

j

j

k

xj

is

a

door

There

is

another

path

of

P

say

P

zmg

such

that

-LRB-

xj

mod

k

xj

-RRB-

P

We

replace

the

two

pairs

of

matched

vertices

y

and

zm

by

y

and

zm

The

number

of

edges

used

by

the

new

pseudo-matching

is

the

same

but

the

number

of

doors

of

C

has

been

decreased

by

one

We

iterate

this

process

until

the

number

of

doors

is

yielding

a

contradiction

Therefore

P

is

a

forest

and

hence

it

uses

at

most

n

edges

Thus

our

protocol

generates

paths

of

total

length

at

most

-LRB-

n

-RRB-

ne

S

-LRB-

G

V

u

-RRB-

ne

Broadcasting

in

trees

In

this

section

we

will

derive

upper

and

lower

bounds

on

S

-LRB-

G

D

u

-RRB-

for

trees

Let

us

start

with

few

definitions

Notation

Let

be

any

non

negative

integer

If

then

the

binary

expression

of

is

denoted

by

k

such

that

Pki

We

denote

by

g

-LRB-

-RRB-

the

number

of

's

in

the

binary

representation

of

that

is

g

-LRB-

-RRB-

Pki

i

We

also

define

the

function

f

by

induction

f

-LRB-

-RRB-

f

-LRB-

-RRB-

and

for

f

-LRB-

-RRB-

min

f

-LRB-

d

e

-RRB-

f

-LRB-

b

c

-RRB-

mod

In

term

of

binary

expression

the

reader

can

check

that

for

any

non

empty

binary

string

M

f

-LRB-

-RRB-

f

-LRB-

M

-RRB-

f

-LRB-

-RRB-

f

-LRB-

M

-RRB-

and

for

any

string

N

of

length

at

least

f

-LRB-

N

-RRB-

and

f

-LRB-

-RRB-

f

-LRB-

-RRB-

For

instance

f

-LRB-

-RRB-

f

-LRB-

-RRB-

f

-LRB-

-RRB-

f

-LRB-

-RRB-

f

-LRB-

-RRB-

Let

T

-LRB-

V

E

-RRB-

be

a

tree

with

n

vertices

Let

u

be

the

source

of

a

multicast

of

destination

set

D

u

D

We

consider

T

as

rooted

in

u.

Any

set

U

u

U

then

induces

a

weight

function

wU

on

the

edges

of

T

as

follows

For

any

edge

e

-LRB-

x

y

-RRB-

E

the

removing

of

e

from

T

decomposes

T

is

two

trees

Tx

and

Ty

We

denote

by

Te

the

tree

which

does

not

contain

u

We

define

wU

-LRB-

e

-RRB-

jTe

U

j

We

have

the

following

result

Theorem

In

any

tree

T

-LRB-

V

E

-RRB-

there

exists

a

multicast

protocol

from

its

root

u

to

any

subset

D

of

vertices

u

D

which

performs

in

jDje

rounds

in

the

line

model

and

such

that

P

f

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

S

-LRB-

T

D

u

-RRB-

P

g

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

Proof

As

we

already

did

all

along

this

paper

we

will

consider

the

dual

problem

of

multicasting

that

is

the

accumulation

problem

-LRB-

reversing

the

communication

scheme

for

accumulation

results

in

a

communication

scheme

for

multicasting

-RRB-

In

the

accumulation

problem

each

node

of

D

has

some

piece

of

information

which

must

be

collected

by

u

An

accumulation

algorithm

will

be

described

by

a

sequence

jDje

of

subsets

of

nodes

with

D

jDje

u

and

Ut

ae

Ut

For

t

jDje

Ut

actually

denotes

a

sets

of

nodes

such

that

the

union

of

all

the

pieces

of

information

known

by

them

at

round

t

is

exactly

the

union

of

all

the

pieces

of

information

originally

known

by

the

vertices

of

D

The

set

Ut

is

constructed

by

using

the

information

of

Ut

only

Ut

is

called

the

active

set

at

time

t

A

vertex

belonging

to

an

active

set

is

called

an

active

node

In

the

considered

tree

given

a

set

U

of

active

vertices

an

edge

e

is

said

to

be

even

relatively

to

U

if

the

removal

of

this

edge

decomposes

the

tree

in

two

subtrees

each

rooted

at

one

extremity

of

e

which

have

both

an

even

number

of

active

vertices

of

U

The

edge

e

is

said

to

be

odd

relatively

to

U

otherwise

Upper

bound

Let

us

introduce

another

notation

given

two

vertices

x

and

y

of

the

tree

we

denote

by

st

-LRB-

x

y

-RRB-

the

number

of

the

zero

second

rightmost

-LRB-

penultimate

-RRB-

bits

of

the

binary

expression

of

the

weights

wUt

-LRB-

e

-RRB-

of

the

edges

e

on

the

shortest

path

from

x

to

y

in

T

In

other

words

if

they

are

q

edges

on

the

shortest

path

from

x

to

y

in

T

and

if

the

weights

of

these

q

edges

are

denoted

by

w

-LRB-

-RRB-

w

-LRB-

q

-RRB-

then

st

-LRB-

x

y

-RRB-

q

Pqi

w

-LRB-

i

-RRB-

Finally

for

the

sake

of

simplicity

let

us

first

assume

that

D

Our

accumulation

algorithm

proceeds

as

in

the

proof

of

Theorem

by

a

successive

construction

of

pseudo-matchings

of

the

active

sets

Ut

t

jDje

The

only

thing

that

we

have

to

specify

carefully

is

the

way

to

choose

which

extremity

of

each

path

of

the

pseudo-matching

we

keep

for

the

next

round

More

precisely

round

t

of

our

algorithm

can

be

decomposed

in

two

phases

Construct

a

pseudo-matching

in

Ut

using

only

odd

edges

relatively

to

Ut

For

each

pair

of

vertices

x

and

y

matched

in

the

pseudo-matching

in

Ut

let

z

be

the

first

common

ancestor

of

x

and

y

If

st

-LRB-

z

x

-RRB-

st

-LRB-

z

y

-RRB-

then

choose

the

vertex

which

is

the

closest

to

the

root

u

and

put

it

in

the

set

Ut

-LRB-

choose

arbitrarily

any

of

these

two

vertices

if

d

-LRB-

x

u

-RRB-

d

-LRB-

y

u

-RRB-

-RRB-

If

st

-LRB-

z

x

-RRB-

st

-LRB-

z

y

-RRB-

then

put

x

in

Ut

and

if

st

-LRB-

z

x

-RRB-

st

-LRB-

z

y

-RRB-

then

put

y

in

Ut

We

refer

the

reader

to

-LSB-

-RSB-

to

figure

out

how

phase

can

be

performed

in

polynomial

time

Let

us

show

that

this

jDje-time

algorithm

satisfies

that

the

sum

over

its

jDje

rounds

of

the

number

of

edges

used

at

each

round

is

upper

bounded

by

P

g

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

For

this

purpose

let

us

prove

the

following

Claim

If

there

are

odd

edges

before

the

round

t

-LRB-

that

is

relatively

to

Ut

-RRB-

then

there

are

at

most

ff

odd

edges

after

round

t

-LRB-

that

is

relatively

to

Ut

-RRB-

where

is

the

sum

of

the

penultimate

bits

of

the

weights

wUt

-LRB-

e

-RRB-

of

all

the

edges

e

E.

Indeed

since

jDj

the

number

of

active

nodes

is

divided

by

after

each

round

Clearly

for

any

even

edge

e

at

round

t

wUt

-LRB-

e

-RRB-

wUt

-LRB-

e

-RRB-

and

its

parity

at

round

t

depends

on

the

penultimate

bit

of

wUt

-LRB-

e

-RRB-

an

even

edge

turns

odd

if

the

penultimate

bit

of

its

previous

weight

was

a

The

case

of

odd

edges

is

more

complicated

Any

odd

edge

e

relatively

to

Ut

will

be

used

during

round

t

say

for

a

communication

between

x

and

y.

Let

z

be

the

first

common

ancestor

of

x

and

y

let

wUt

-LRB-

e

-RRB-

and

let

us

consider

the

several

possibility

for

the

choice

of

the

node

v

fx

yg

put

in

Ut

-LCB-

If

v

V

n

Te

then

wUt

-LRB-

e

-RRB-

k

and

hence

e

stays

odd

if

and

only

if

the

penultimate

bit

of

its

previous

weight

was

a

This

case

is

therefore

taken

into

account

in

the

term

of

Claim

-LCB-

If

v

Te

then

wUt

-LRB-

e

-RRB-

k

and

hence

e

stays

odd

if

and

only

if

the

penultimate

bit

of

its

previous

weight

was

a

Since

v

minimizes

the

number

of

zeros

at

the

penultimate

position

of

the

weights

of

the

edges

between

x

y

and

their

common

ancestor

z

this

case

is

therefore

taken

into

account

in

the

term

ff

of

Claim

Therefore

as

claimed

if

there

are

odd

edges

before

the

round

t

then

there

are

at

most

ff

odd

edges

after

round

t

where

is

the

sum

of

the

penultimate

bits

of

the

weight

wUt

-LRB-

e

-RRB-

of

all

the

edges

e

E.

Now

let

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

i

be

the

ith

bit

of

the

binary

representation

of

wD

-LRB-

e

-RRB-

and

let

Si

P

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

i

for

i

ne

By

definition

we

have

X

g

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

ne

X

i

Si

-LRB-

-RRB-

At

the

first

round

at

most

edges

are

used

since

there

are

only

odd

edges

Following

Claim

at

most

edges

are

used

at

the

second

round

More

generally

we

have

Claim

At

any

round

t

t

jDje

at

most

Pti

Si

i

edges

are

used

in

the

pseudomatching

We

prove

this

claim

using

Claim

in

showing

that

the

sum

of

the

penultimate

bits

of

the

weights

wUt

-LRB-

e

-RRB-

of

all

the

edges

e

E

is

equal

to

St.

Indeed

we

have

seen

that

if

wUt

-LRB-

e

-RRB-

is

even

then

wUt

-LRB-

e

-RRB-

wUt

-LRB-

e

-RRB-

and

therefore

the

most

significant

bits

of

wD

-LRB-

e

-RRB-

are

not

modified

If

wUt

-LRB-

e

-RRB-

two

cases

can

be

considered

wUt

-LRB-

e

-RRB-

k

or

wUt

-LRB-

e

-RRB-

k

In

the

former

case

the

most

significant

bits

of

wD

-LRB-

e

-RRB-

are

not

modified

The

latter

case

occurs

only

when

the

penultimate

bits

of

wUt

-LRB-

e

-RRB-

is

a

and

therefore

the

most

significant

bits

of

wD

-LRB-

e

-RRB-

are

not

modified

This

shows

that

the

sum

of

the

penultimate

bits

of

the

weights

wUt

-LRB-

e

-RRB-

of

all

the

edges

e

E

is

equal

to

St.

Now

it

is

easy

to

check

that

jDje

X

t

tX

i

Si

i

jDje

X

i

Si

that

is

using

Equation

jDje

X

t

tX

i

Si

i

X

g

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

This

completes

the

poof

of

the

upper

bound

when

jDj

If

jDj

is

not

a

power

of

two

say

jDj

a

where

a

one

can

consider

multicasting

in

a

tree

T

of

n

a

vertices

consisting

in

T

plus

a

new

vertices

directly

connected

to

the

root

u

and

forming

a

stable

set

S.

Let

D

-LSB-

S.

has

a

power

of

two

vertices

One

can

apply

the

previously

described

algorithm

to

T

and

so

that

a

vertex

of

S

communicates

either

with

another

vertex

in

S

or

with

u.

Let

us

call

this

algorithm

Thus

if

one

does

not

consider

communications

involving

vertices

in

S

induces

an

algorithm

A

which

accumulates

the

information

of

D

in

the

root

u

of

the

tree

T

Note

that

u

is

the

only

possible

idle

vertex

of

any

active

set

during

the

execution

of

A.

Let

us

denote

by

SA

-LRB-

respectively

-RRB-

the

global

sum

of

the

lengths

of

the

paths

generated

by

the

execution

of

A

-LRB-

respectively

-RRB-

Since

is

a

power

of

we

have

P

-LRB-

T

-RRB-

g

-LRB-

-LRB-

e

-RRB-

-RRB-

Actually

we

can

strength

a

bit

this

upper

bound

P

-LRB-

T

-RRB-

g

-LRB-

-LRB-

e

-RRB-

-RRB-

S

-LRB-

a

a

u

-RRB-

where

a

is

the

star

of

a

vertices

rooted

in

u

and

the

function

S

is

defined

as

S

excepted

that

the

minimization

is

taken

over

all

the

multicast

protocols

in

which

the

only

possible

unmatched

active

vertex

is

the

source

of

the

multicast

Now

SA

S

-LRB-

a

a

u

-RRB-

Therefore

S

-LRB-

T

D

u

-RRB-

P

-LRB-

T

-RRB-

g

-LRB-

-LRB-

e

-RRB-

Finally

since

g

-LRB-

-LRB-

e

-RRB-

-RRB-

g

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

we

get

S

-LRB-

T

D

u

-RRB-

P

-LRB-

T

-RRB-

g

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

which

concludes

the

proof

of

the

upper

bound

Lower

bound

Assume

first

that

D

We

prove

by

induction

on

k

that

any

edge

e

is

used

at

most

f

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

times

during

any

multicasting

protocol

from

u

to

D

This

result

clearly

holds

for

k

Let

k

and

assume

that

any

edge

e

is

used

at

most

f

-LRB-

w

-LRB-

e

-RRB-

-RRB-

times

during

any

multicasting

protocol

from

u

to

any

set

D

such

that

jDj

i

k

At

any

round

of

any

multicast

algorithm

performing

in

an

optimal

number

of

rounds

all

the

active

nodes

take

part

of

the

communications

and

as

proved

in

-LSB-

-RSB-

only

the

odd

edges

can

be

used

in

the

pseudo-matching

Let

be

the

set

of

active

nodes

after

the

first

round

of

an

arbitrary

accumulation

algorithm

-LCB-

If

the

edge

e

is

even

before

the

first

round

then

no

call

made

use

this

edge

and

the

weight

of

e

is

simply

divided

by

By

induction

hypothesis

the

edge

e

will

be

used

at

least

f

-LRB-

-LRB-

e

-RRB-

-RRB-

times

during

any

multicast

algorithm

that

is

at

least

f

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

times

by

definition

of

f

since

wD

-LRB-

e

-RRB-

is

even

-LCB-

If

the

edge

e

is

odd

before

the

first

round

then

a

call

between

two

vertices

makes

use

of

this

edge

at

the

first

round

of

any

accumulation

algorithm

According

to

the

choice

of

the

active

node

for

the

next

round

the

weight

of

edge

e

will

be

equal

to

dwD

-LRB-

e

-RRB-

or

bwD

-LRB-

e

-RRB-

Therefore

the

edge

e

is

used

at

least

min

-LRB-

f

-LRB-

dwD

-LRB-

e

-RRB-

-RRB-

f

-LRB-

bwD

-LRB-

e

-RRB-

-RRB-

-RRB-

that

is

at

least

f

-LRB-

-LRB-

wD

-LRB-

e

-RRB-

-RRB-

by

definition

of

the

function

f

The

case

where

jDj

is

not

a

power

of

can

be

solved

using

the

same

argument

as

in

the

proof

of

the

upper

bound

This

concludes

the

proof

Although

we

think

that

the

two

bounds

of

Theorem

differ

in

average

by

a

small

factor

-LRB-

possibly

a

constant

-RRB-

we

did

not

succeed

to

prove

this

fact

There

are

trees

for

which

the

two

bounds

differ

by

more

than

a

constant

However

in

many

examples

like

the

path

the

star

the

complete

binary

tree

and

the

binomial

tree

these

two

bounds

differ

by

at

most

a

factor

of

Also

the

expected

value

of

the

ratio

-LRB-

-RRB-

f

-LRB-

-RRB-

when

is

uniformly

randomly

chosen

is

bounded

by

a

constant

Number

and

load

of

the

routers

In

this

section

we

address

the

ability

of

a

multicasting

algorithm

not

to

disturb

other

applications

by

traversing

nodes

which

are

not

directly

concerned

by

the

multicasted

message

Moreover

we

also

study

the

load

of

the

routers

during

a

multicasting

or

a

broadcasting

algorithm

Indeed

a

router

is

usually

able

to

route

any

permutation

of

its

input

channels

on

its

output

channels

However

the

output

channel

allocation

process

is

sequential

in

general

and

thus

could

be

time

consuming

when

many

messages

traverse

a

same

router

Therefore

it

could

be

an

issue

to

minimize

the

load

of

the

routers

Number

of

routers

We

show

below

that

minimizing

the

number

of

routers

involved

in

a

time

optimal

multicast

algorithm

is

an

NP-complete

problem

Definition

Let

P

be

a

pseudo-matching

in

a

subset

U

of

vertices

of

a

graph

G

A

transmitter

is

a

vertex

of

G

which

is

traversed

by

a

path

of

P

The

following

theorem

shows

that

minimizing

the

number

transmitters

at

any

round

of

a

broadcasting

or

multicasting

algorithm

is

difficult

Theorem

The

following

problem

is

NP-complete

Minimum-Number-Of-Transmitters

-LRB-

MNOT

-RRB-

Instance

A

graph

G

-LRB-

V

E

-RRB-

a

subset

U

ae

V

of

vertices

and

an

integer

k.

Question

Does

there

exist

a

pseudo-matching

in

U

which

has

at

most

k

transmitters

Proof

MNOT

is

clearly

in

NP

MNOT

is

NP-complete

by

transformation

from

Let

C

cmg

be

an

instance

of

-LRB-

where

each

clause

ci

has

exactly

three

literals

ci

-LSB-

-RSB-

ci

-LSB-

-RSB-

and

ci

-LSB-

-RSB-

-RRB-

on

the

boolean

variables

B

bng

We

construct

an

instance

of

MNOT

as

follows

We

associate

two

copies

of

to

each

clause

ci

of

C

In

each

there

is

a

vertex

for

each

literal

They

are

denoted

by

ci

ci

and

ci

We

also

associate

one

copy

of

to

each

boolean

variable

bi

in

B

the

first

partition

is

composed

of

two

vertices

denoted

bi

and

bi

and

the

second

partition

is

composed

of

two

vertices

denoted

xi

and

yi

We

connect

the

literals

of

each

clause

to

the

corresponding

boolean

vertices

as

follows

-LRB-

see

Figure

-RRB-

if

ci

-LSB-

-RSB-

bj

-LRB-

resp

ci

-LSB-

-RSB-

bj

-RRB-

then

add

an

edge

between

the

two

vertices

of

the

form

ci

j

and

the

vertex

bj

-LRB-

resp

bj

-RRB-

We

obtain

a

graph

G

with

n

vertices

Let

U

be

the

subset

of

vertices

composed

of

the

vertices

of

the

complete

graphs

plus

the

vertices

xi

yi

i

n.

Set

k

n

-LRB-

note

that

there

are

at

least

n

transmitters

in

any

pseudo-matching

in

U

because

of

the

vertices

xi

and

yi

in

each

-RRB-

We

claim

that

C

is

satisfiable

if

and

only

if

there

exists

a

pseudo-matching

in

U

with

at

most

n

transmitters

-LRB-

i.e.

with

exactly

n

transmitters

-RRB-

Assume

C

is

satisfiable

We

construct

a

pseudo-matching

in

U

with

exactly

n

transmitters

as

follows

For

i

n

connect

xi

to

yi

with

a

path

traversing

the

vertex

bi

if

the

solution

provides

bi

true

or

traversing

the

vertex

bi

if

the

solution

provides

bi

false

For

i

m

at

least

one

of

the

literals

in

ci

is

true

Choose

such

a

true

literal

ci

-LSB-

j

-RSB-

and

connect

the

two

vertices

ci

j

by

a

path

traversing

the

vertex

ci

-LSB-

j

-RSB-

Then

in

each

copy

of

corresponding

to

ci

connect

the

two

vertices

ci

and

ci

j

by

a

path

of

length

We

get

a

pseudomatching

in

U

with

exactly

n

transmitters

Conversely

assume

the

exists

a

pseudo-matching

in

U

with

exactly

n

transmitters

Due

to

the

structure

of

U

and

G

there

is

a

transmitter

in

each

copy

of

These

transmitters

induce

xybbxybbxybbjjjjj

j'

j'

j'

j'

j'

j'

j'

ci

ci

ci

ci

ci

ci

Figure

Construction

in

the

proof

of

Theorem

The

clause

ci

satisfies

ci

-LSB-

-RSB-

bj

ci

-LSB-

-RSB-

and

ci

-LSB-

-RSB-

a

assignation

bi

true

or

bi

false

for

all

the

boolean

variables

bi

i

n

In

each

one

vertex

must

be

connected

by

a

path

of

length

at

least

to

a

vertex

of

another

This

path

traverses

one

of

the

transmitters

Therefore

there

is

at

least

a

true

literal

in

each

clause

and

C

is

satisfiable

Remark

From

the

proof

of

Theorem

one

can

check

that

the

problem

MNOT

is

solvable

in

polynomial

time

for

trees

-LRB-

indeed

in

the

terminology

of

this

proof

only

odd

edges

take

part

of

the

pseudo-matching

and

therefore

one

can

check

in

polynomial

time

whether

a

vertex

is

or

is

not

a

transmitter

according

to

the

number

of

its

incident

odd

edges

-RRB-

The

following

result

shows

that

minimizing

the

number

of

involved

router

is

not

only

difficult

when

one

considers

a

single

round

but

also

when

one

considers

the

whole

protocol

Property

The

following

problem

is

NP-complete

Minimum-Number-of-Routers

-LRB-

MNOR

-RRB-

Instance

A

graph

G

a

vertex

u

of

G

and

an

integer

k.

Question

Does

there

exist

a

time

optimal

broadcast

protocol

from

u

in

the

line

model

which

involves

at

most

k

routers

during

its

whole

execution

Proof

The

problem

MNOR

with

k

is

equivalent

to

the

NP-complete

problem

LogB

Indeed

if

there

exists

a

ne-time

broadcasting

algorithm

B

such

that

no

vertex

is

a

transmitter

then

all

the

communications

are

neighbor-to-neighbor

and

therefore

only

use

one

edge

So

the

algorithm

B

satisfies

the

telephone

model

constraints

Conversely

if

B

is

a

ne-time

broadcasting

algorithm

in

the

telephone

model

then

the

corresponding

number

of

transmitters

is

reduced

to

zero

Matching

index

and

load

of

the

routers

Let

P

be

a

pseudo-matching

of

a

subset

of

vertices

U

of

a

graph

G.

Let

us

define

the

load

of

a

given

transmitter

x

to

be

the

number

of

paths

which

are

traversing

x

-LRB-

x

is

not

an

end

vertex

-RRB-

This

parameter

is

denoted

by

-LRB-

U

P

x

-RRB-

The

load

of

the

pseudo-matching

P

in

U

is

denoted

by

-LRB-

U

P

-RRB-

and

is

defined

by

-LRB-

U

P

-RRB-

maxx

-LRB-

U

P

x

-RRB-

Definition

The

matching

index

of

subset

of

vertices

U

of

a

graph

G

-LRB-

V

E

-RRB-

is

the

minimum

taken

over

all

the

pseudo

matching

P

in

U

of

-LRB-

U

P

-RRB-

It

is

denoted

by

-LRB-

U

-RRB-

The

matching

index

of

a

graph

G

is

-LRB-

V

-RRB-

When

the

minimization

is

restricted

to

pseudo-matching

containing

shortest

paths

only

this

parameter

is

denoted

by

m

-LRB-

U

-RRB-

Clearly

if

one

wants

to

minimize

the

load

of

the

routers

one

will

try

to

find

the

pseudo

matching

P

for

which

-LRB-

U

P

-RRB-

-LRB-

U

-RRB-

at

each

round

of

the

multicast

or

the

broadcast

algorithm

Unfortunately

we

get

the

following

result

Theorem

The

following

problem

is

NP-complete

Matching-Index-of-Shortest-Paths

-LRB-

MISP

-RRB-

Instance

A

graph

G

-LRB-

V

E

-RRB-

a

vertex

u

of

G

a

subset

U

ae

V

and

an

integer

k.

Question

m

-LRB-

U

P

-RRB-

k

Proof

Clearly

MISP

NP

because

one

can

check

in

polynomial

time

that

a

pseudo

matching

P

in

U

contains

shortest

path

only

and

satisfies

-LRB-

U

P

-RRB-

k.

To

show

that

MISP

is

NP-complete

let

consider

the

following

problem

Matching-Index-of-Shortest-Path

-LRB-

-RRB-

-LRB-

MISP

-LRB-

-RRB-

-RRB-

Instance

A

graph

G

a

vertex

u

of

G

a

subset

U

ae

V

Question

m

-LRB-

U

P

-RRB-

Let

us

prove

that

MISP

-LRB-

-RRB-

is

NP-complete

by

transformation

from

the

Vertex

Cover

problem

Vertex

Cover

-LRB-

VC

-RRB-

Instance

A

graph

G

-LRB-

V

E

-RRB-

and

a

positive

integer

b.

Question

Is

there

a

vertex

cover

of

size

b

or

less

i.e

a

subset

S

ae

V

such

that

jSj

b

and

for

each

edge

-LRB-

u

v

-RRB-

E

at

least

one

of

u

and

v

belongs

to

S

Let

an

arbitrary

instance

of

VC

be

given

by

the

graph

G

-LRB-

V

E

-RRB-

and

the

positive

integer

b

We

construct

a

graph

-LRB-

V

-RRB-

and

a

subset

of

vertices

U

such

that

in

the

graph

the

subset

U

satisfies

m

-LRB-

U

-RRB-

if

and

only

if

G

has

a

vertex

cover

of

size

b

or

less

The

graph

consists

of

paths

Pi

i

of

three

vertices

-LRB-

xi

yi

zi

-RRB-

plus

other

vertices

Connections

between

these

other

vertices

are

functions

of

the

connections

of

the

vertices

of

G

each

edge

e

of

G

corresponds

to

a

subgraph

Ge

-LRB-

Ve

Ee

-RRB-

of

Such

a

subgraph

is

called

the

gadget

of

e

The

gadget

of

an

edge

e

-LRB-

u

v

-RRB-

E

is

illustrated

on

Figure

It

is

composed

of

vertices

Only

of

them

are

explicitly

drawn

on

Figure

The

others

are

virtually

presented

in

the

form

of

dotted

lines

each

dotted

line

C

represents

a

cross

of

five

vertices

The

five

vertices

at

the

top

-LRB-

resp

bottom

-RRB-

of

the

gadget

of

edge

e

-LRB-

u

v

-RRB-

as

drawn

on

Figure

are

denoted

by

-LRB-

e

ui

-RRB-

i

-LRB-

resp

-LRB-

e

vi

-RRB-

i

-RRB-

We

also

identify

two

particular

vertices

denoted

by

e

-LRB-

-RRB-

and

e

-LRB-

-RRB-

on

Figure

For

each

vertex

v

V

let

ed

be

its

incident

edges

-LRB-

d

is

the

degree

of

v

-RRB-

For

i

d

the

gadget

of

the

edge

ei

is

joined

to

the

gadget

of

edge

ei

by

an

edge

-LRB-

-LRB-

ei

-RRB-

-LRB-

ei

-RRB-

-RRB-

-LRB-

see

Figure

-RRB-

All

these

connections

form

a

path

Qv

f

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

ed

-RRB-

-LRB-

ed

-RRB-

g

Finally

the

first

and

last

vertices

of

Qv

-LRB-

-RRB-

and

-LRB-

ed

-RRB-

are

connected

to

vertices

by

a

complete

bipartite

graph

-LRB-

see

Figure

-RRB-

The

set

U

is

then

defined

by

f

-LRB-

e

-RRB-

-LRB-

e

-RRB-

-LRB-

e

-RRB-

-LRB-

e

-RRB-

e

-LRB-

-RRB-

e

-LRB-

-RRB-

e

Eg

-LSB-

for

each

cross

Cg

-LSB-

fxi

yi

zi

i

All

these

nodes

are

marked

in

grey

on

Figures

and

Of

course

and

U

can

be

constructed

in

polynomial

time

from

G

and

b.

Let

us

list

some

simple

properties

which

are

satisfied

by

the

gadget

of

an

edge

e

-LRB-

u

v

-RRB-

For

any

shortest

path

pseudo-matching

P

in

U

satisfying

-LRB-

U

P

-RRB-

we

have

the

following

properties

In

any

cross

C

vertices

and

must

be

connected

together

otherwise

-LRB-

U

P

-RRB-

Moreover

no

other

matching

can

pass

through

cross

C

otherwise

the

load

of

would

be

increased

-LRB-

e

u

-RRB-

-LRB-

e

u

-RRB-

-LRB-

e

u

-RRB-

-LRB-

e

u

-RRB-

-LRB-

e

u

-RRB-

-LRB-

e

v

-RRB-

-LRB-

e

v

-RRB-

-LRB-

e

v

-RRB-

-LRB-

e

v

-RRB-

-LRB-

e

v

-RRB-

Figure

The

gadget

of

an

edge

e

-LRB-

u

v

-RRB-

used

in

the

proof

of

the

Theorem

-LRB-

doted

lines

represent

a

cross

of

five

vertices

-RRB-

Vertices

in

U

are

marked

in

grey

Vertices

e

-LRB-

-RRB-

and

e

-LRB-

-RRB-

must

be

connected

together

Indeed

connecting

e

-LRB-

-RRB-

by

a

shortest

path

to

any

node

of

U

distinct

from

e

-LRB-

-RRB-

requires

to

pass

through

a

cross

C

and

we

have

seen

that

this

is

not

possible

Vertex

-LRB-

e

-RRB-

can

only

be

connected

to

vertex

-LRB-

e

-RRB-

or

to

a

vertex

which

does

not

belong

to

the

same

gadget

otherwise

the

matching

would

require

to

pass

through

a

cross

C.

If

vertex

-LRB-

e

-RRB-

is

matched

to

vertex

-LRB-

e

-RRB-

then

vertex

-LRB-

e

-RRB-

is

not

matched

to

vertex

-LRB-

e

-RRB-

otherwise

the

load

of

-LRB-

e

-RRB-

or

-LRB-

e

-RRB-

would

be

greater

than

because

of

the

matching

e

-LRB-

-RRB-

-LRB-

-RRB-

In

other

words

at

least

two

of

the

four

vertices

of

the

form

-LRB-

e

-RRB-

-LRB-

e

-RRB-

-LRB-

e

-RRB-

or

-LRB-

e

-RRB-

must

be

matched

to

vertices

which

do

not

belong

to

the

same

gadget

Now

let

us

show

that

there

exists

a

shortest

path

pseudo-matching

in

U

of

maximum

load

if

and

only

if

G

has

a

vertex

cover

of

size

b

or

less

-LRB-

Suppose

that

there

exists

a

shortest

path

pseudo-matching

P

in

U

such

that

-LRB-

U

P

-RRB-

From

properties

to

stated

before

we

already

know

how

matchings

involving

nodes

inside

a

gadget

look

like

For

each

path

Pi

only

one

vertex

of

Pi

can

be

matched

to

a

vertex

not

in

Pi

otherwise

the

load

of

yi

would

be

strictly

greater

than

For

the

same

reason

no

path

connecting

two

vertices

not

belonging

to

Pi

can

pass

through

the

vertex

yi

Without

loss

of

generality

we

can

assume

that

vertex

yi

is

the

vertex

which

must

be

matched

to

a

vertex

not

in

the

path

Pi

Let

u

be

a

vertex

of

G

and

let

ed

be

its

incident

edges

If

the

vertex

-LRB-

-RRB-

is

matched

to

a

vertex

yi

then

from

Property

vertex

-LRB-

-RRB-

will

be

matched

to

vertex

-LRB-

-RRB-

Then

-LRB-

-RRB-

will

be

matched

to

-LRB-

-RRB-

and

so

on

until

vertex

-LRB-

ed

-RRB-

is

matched

to

some

vertex

yj

Let

S

be

the

set

of

vertices

of

G

such

that

u

S

if

and

only

if

-LRB-

-RRB-

and

-LRB-

ed

-RRB-

are

both

matched

to

vertices

of

type

yi

i

in

the

pseudo-matching

defined

above

Note

that

jSj

b.

Assume

that

S

is

not

a

vertex

cover

of

G

Then

let

e

be

an

edge

of

G

such

that

of

its

two

endpoints

belong

to

S.

From

Property

at

least

two

vertices

of

the

gadget

corresponding

to

-LRB-

e

a

-RRB-

-LRB-

e

a

-RRB-

-LRB-

e

a

-RRB-

-LRB-

e

a

-RRB-

-LRB-

e

a

-RRB-

-LRB-

e

a

-RRB-

-LRB-

e

c

-RRB-

-LRB-

e

c

-RRB-

-LRB-

e

c

-RRB-

-LRB-

e

c

-RRB-

-LRB-

e

d

-RRB-

-LRB-

e

d

-RRB-

-LRB-

e

d

-RRB-

-LRB-

e

d

-RRB-

-LRB-

e

b

-RRB-

-LRB-

e

b

-RRB-

Figure

Example

of

the

construction

of

the

proof

of

Theorem

when

b

The

diamonds

are

symbol

of

the

gadget

of

the

edges

drawn

on

Figure

e

must

be

connected

to

a

vertex

not

in

this

gadget

Let

u

be

an

extremity

of

e

such

that

-LRB-

e

-RRB-

and

-LRB-

e

-RRB-

are

both

matched

to

vertices

not

in

the

gadget

of

e.

Vertex

-LRB-

e

-RRB-

is

matched

to

a

vertex

-LRB-

-RRB-

vertex

-LRB-

-RRB-

is

matched

to

a

vertex

-LRB-

-RRB-

and

so

on

Finally

if

the

degree

of

u

is

d

and

if

ed

denote

the

d

incident

edges

of

u

then

vertex

-LRB-

ed

-RRB-

is

matched

to

vertex

-LRB-

ed

-RRB-

of

path

Qu

and

vertex

-LRB-

ed

-RRB-

must

be

matched

to

a

vertex

belonging

to

fyi

i

For

the

same

reason

-LRB-

-RRB-

must

be

matched

to

a

vertex

belonging

to

fyi

i

Thus

u

S

contradiction

Therefore

S

is

a

vertex

cover

of

G

size

b

or

less

-RRB-

Conversely

suppose

that

S

V

is

a

vertex

cover

of

G

of

size

b

or

less

If

jSj

b

then

we

can

add

some

vertices

to

S

in

order

to

obtain

a

vertex

cover

of

exactly

b

vertices

So

we

can

assume

without

loss

of

generality

that

jSj

b.

For

any

vertex

s

of

G

let

d

be

its

degree

and

edbe

its

d

incident

edges

Vertices

of

S

are

labeled

from

to

b

We

construct

the

pseudo-matching

P

in

U

as

follows

ffl

if

s

S

then

let

i

be

its

label

in

S

is

matched

to

-LRB-

-RRB-

-LRB-

-RRB-

is

matched

to

-LRB-

-RRB-

-LRB-

ed

-RRB-

is

matched

to

ffl

if

s

S

then

for

any

i

dg

vertex

-LRB-

ei

-RRB-

is

matched

to

-LRB-

ei

-RRB-

and

the

path

from

e

-LRB-

-RRB-

i

to

e

-LRB-

-RRB-

i

passes

through

the

vertex

-LRB-

ei

-RRB-

where

ei

-LRB-

s

v

-RRB-

One

can

check

that

this

pseudo-matching

P

is

such

that

the

-LRB-

U

P

-RRB-

and

that

all

paths

in

P

are

shortest

paths

Theorem

shows

that

it

is

difficult

to

minimize

the

load

of

the

routers

at

each

round

Unfortunately

this

is

also

true

for

the

whole

protocol

Given

a

graph

G

and

a

vertex

u

of

G

we

denote

by

M

-LRB-

G

u

-RRB-

the

minimum

taken

over

all

the

broadcast

protocols

B

from

u

in

G

which

perform

in

ne

rounds

in

the

line

model

of

maxi

ne

-LRB-

Ui

Pi

-RRB-

where

Pi

is

the

set

of

paths

corresponding

to

the

pseudo-matching

in

Ui

generated

at

round

i

by

the

broadcasting

algorithm

B

We

have

Property

The

following

problem

is

NP-complete

Minimum

Broadcast

Load

-LRB-

MBL

-RRB-

Instance

A

graph

G

a

vertex

u

of

G

and

an

integer

k.

Question

M

-LRB-

G

u

-RRB-

k

Proof

If

there

exists

a

broadcasting

algorithm

B

such

that

M

-LRB-

G

u

-RRB-

then

no

vertex

is

a

transmitter

at

any

round

So

algorithm

B

is

a

broadcasting

algorithm

from

u

in

ne

rounds

in

telephone

model

Clearly

the

reciprocal

holds

Therefore

the

problem

MBL

with

k

is

equivalent

to

the

NP-complete

problem

LogB

Global

switching

time

In

this

section

we

focus

on

the

switching

times

induced

by

long

paths

between

two

vertices

Given

a

pseudo-matching

P

of

a

subset

U

of

vertices

of

a

graph

G

we

denote

by

L

-LRB-

U

P

-RRB-

the

maximum

length

of

the

paths

of

P.

Let

L

-LRB-

U

-RRB-

minP

L

-LRB-

U

P

-RRB-

where

P

is

a

pseudo-matching

of

U

If

the

switching

time

of

the

routers

can

not

be

neglected

L

-LRB-

U

-RRB-

is

an

important

parameter

to

minimize

at

each

round

of

a

multicast

algorithm

Unfortunately

we

have

Theorem

The

following

problem

is

NP-complete

Maximum-Length-of-a-Pseudo-Matching

-LRB-

MLPM

-RRB-

Instance

A

graph

G

-LRB-

V

E

-RRB-

a

vertex

u

of

G

a

subset

U

ae

V

and

an

integer

k.

Question

L

-LRB-

U

-RRB-

k

Proof

MLPM

is

clearly

in

NP

To

prove

that

it

is

NP-complete

we

transform

the

Vertex

Cover

problem

to

MLPM

with

k

Let

an

arbitrary

instance

of

VC

be

given

by

a

graph

G

-LRB-

V

E

-RRB-

and

a

positive

integer

b

We

construct

an

instance

of

MLPM

that

is

a

graph

-LRB-

V

-RRB-

and

a

subset

U

of

vertices

of

such

that

U

has

a

pseudo-matching

P

satisfying

L

-LRB-

U

P

-RRB-

-RRB-

if

and

only

if

G

has

a

vertex

cover

of

size

b

or

less

The

construction

of

is

quite

similar

to

the

one

of

the

proof

of

Theorem

The

only

differences

are

each

of

the

paths

Pi

of

length

is

transformed

into

a

single

vertex

yi

i

the

gadgets

of

the

edges

of

G

are

as

shown

on

Figure

each

bipartite

graph

connecting

the

two

endpoints

of

a

path

Qu

to

the

vertices

yi

i

is

slightly

transformed

by

replacing

each

edge

by

a

path

of

length

The

set

U

is

defined

exactly

as

in

the

proof

of

Theorem

The

reader

can

check

that

the

gadgets

presently

defined

have

the

same

properties

as

the

gadgets

defined

in

the

proof

of

Theorem

In

fact

the

remainder

of

the

proof

is

identical

to

the

proof

of

Theorem

Concerning

the

whole

protocol

we

have

a

similar

bad

news

for

both

the

maximum

length

and

the

sum

of

the

maximum

length

of

the

paths

Given

a

graph

G

and

a

vertex

u

of

G

we

denote

by

-LRB-

e

u

-RRB-

-LRB-

e

u

-RRB-

-LRB-

e

u

-RRB-

-LRB-

e

v

-RRB-

-LRB-

e

v

-RRB-

-LRB-

e

v

-RRB-

-LRB-

e

v

-RRB-

-LRB-

e

v

-RRB-

-LRB-

e

v

-RRB-

-LRB-

e

u

-RRB-

-LRB-

e

u

-RRB-

-LRB-

e

u

-RRB-

Figure

The

gadget

of

edge

e

-LRB-

u

v

-RRB-

used

in

the

proof

of

the

Theorem

Vertices

in

U

are

marked

in

grey

Lsum

-LRB-

G

u

-RRB-

the

minimum

taken

over

all

the

broadcast

protocols

B

from

u

performing

in

ne

rounds

in

the

line

model

of

the

sum

of

all

the

maximum

length

of

the

ne

pseudo-matching

in

U

generated

by

this

protocol

More

formally

Lsum

-LRB-

G

u

-RRB-

minB

ne

i

L

-LRB-

Ui

Pi

-RRB-

where

Pi

is

the

pseudo

matching

of

round

i

of

B

i

ne

Property

The

following

problem

is

NP-complete

Sum-Max

Instance

A

graph

G

a

vertex

u

of

G

and

an

integer

k.

Question

Lsum

-LRB-

G

u

-RRB-

k

Proof

If

there

exists

a

broadcasting

algorithm

B

such

that

Lsum

-LRB-

G

u

-RRB-

ne

then

every

calls

satisfy

the

telephone

model

constraints

Thus

algorithm

B

is

a

broadcasting

algorithm

from

u

in

ne

rounds

in

telephone

model

The

reciprocal

clearly

holds

So

the

Sum-Max

problem

with

k

ne

is

equivalent

to

the

NP-complete

problem

LogB

Given

a

graph

G

and

a

vertex

u

of

G

we

denote

by

Lmax

-LRB-

G

u

-RRB-

the

minimum

taken

over

all

the

broadcast

protocols

B

from

u

performing

in

ne

rounds

in

the

line

model

of

the

maximum

length

of

all

the

paths

generated

at

each

round

in

the

ne

pseudo-matching

generated

by

this

protocol

More

formally

Lmax

-LRB-

G

u

-RRB-

minB

maxi

ne

L

-LRB-

Ui

Pi

-RRB-

where

Pi

is

the

pseudo

matching

of

Ui

at

round

i

of

B

i

ne

Property

The

following

problem

is

NP-complete

Max-Max

Instance

A

graph

G

a

vertex

u

of

G

and

an

integer

k.

Question

Lmax

-LRB-

G

u

-RRB-

k

Proof

Similar

to

Property

the

Max-Max

problem

with

k

is

equivalent

to

the

NP

complete

problem

LogB

Conclusion

and

open

problems

Table

summarizes

the

complexities

of

the

minimization

problems

related

to

the

multicast

and

the

broadcast

problems

Even

if

the

situation

seems

despairing

at

the

first

glance

-LRB-

most

of

the

problems

are

NP-complete

-RRB-

the

reader

must

keep

in

mind

that

the

main

parameter

to

optimize

in

cut-through

networks

is

the

use

of

the

bandwidth

that

is

the

first

property

that

must

satisfy

a

communication

protocol

in

this

model

is

to

be

free

of

link

contention

From

these

points

of

view

we

have

provided

important

results

It

is

possible

to

multicast

or

broadcast

in

an

optimal

number

of

rounds

such

that

all

calls

are

performed

on

shortest

paths

and

the

sum

of

the

paths

lengths

at

any

round

is

minimum

In

other

words

such

that

the

total

bandwidth

required

at

each

round

is

minimum

Even

if

minimizing

the

use

of

the

bandwidth

for

the

whole

protocol

is

NP-complete

we

have

described

a

protocol

which

reaches

the

lower

bound

up

to

a

logarithmic

factor

The

minimization

of

many

second

order

parameters

as

the

number

of

transmitters

or

theirs

load

the

maximum

length

of

the

paths

or

the

sum

of

the

maximum

paths

used

at

each

round

of

the

multicast

or

a

broadcast

protocol

yield

NP-complete

problems

Since

these

problems

might

be

not

so

critical

for

cut-through

routing

we

did

not

try

to

derive

approximated

solutions

However

if

minimizing

these

parameters

turn

to

be

a

major

issue

for

some

specific

reasons

such

algorithms

should

be

given

Broadcasting/multicasting

Pseudo-matching

Sum

of

the

paths

lengths

NP-complete

-LRB-

Theorem

-RRB-

P

-LRB-

Theorem

-RRB-

Number

of

transmitters

NP-complete

-LRB-

Property

-RRB-

NP-complete

-LRB-

Theorem

-RRB-

Load

of

the

transmitters

NP-complete

-LRB-

Property

-RRB-

NP-complete

-LRB-

Theorem

-RRB-

Maximum

path

length

NP-complete

-LRB-

Property

-RRB-

NP-complete

-LRB-

Theorem

-RRB-

Sum

of

the

maximum

paths

lengths

NP-complete

-LRB-

Property

-RRB-

Table

Complexities

of

minimization

problems

related

to

the

pseudo-matching

problem

and

to

the

broadcast

and

multicast

problems

Other

problems

seem

to

be

more

important

to

solve

We

indicate

below

two

directions

that

we

are

currently

investigating

ffl

the

all-port

model

As

we

pointed

out

in

the

introduction

one

can

consider

a

model

in

which

any

node

u

are

able

to

generate

u

messages

which

are

simultaneously

sent

to

the

router

where

u

is

the

degree

of

the

router

associated

to

node

u

If

the

network

is

regular

of

the

degree

we

have

b

-LRB-

G

-RRB-

dlog

ne

ffl

The

gossiping

problem

This

problem

is

also

called

all-to-all

broadcasting

it

consists

of

a

simultaneous

broadcasting

from

all

the

nodes

of

the

network

In

the

line

model

we

have

b

-LRB-

G

-RRB-

g

-LRB-

G

-RRB-

b

-LRB-

G

u

-RRB-

in

performing

first

an

accumulation

of

all

the

messages

in

same

node

and

then

a

broadcasting

from

this

node

Although

such

an

algorithm

is

quite

efficient

in

term

of

rounds

it

does

not

balance

the

traffic

and

create

hot

spot

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