Multiple

Multicast

with

Minimized

Node

Contention

on

Wormhole

k-ary

n-cube

Networks

Ram

Kesavan

and

Dhabaleswar

K.

Panda

Technical

Report

A

preliminary

version

of

this

paper

has

been

presented

in

International

Conference

on

Parallel

Processing

Aug.

Manuscript

is

under

review

by

IEEE

Transactions

on

Parallel

and

Distributed

Systems

Multiple

Multicast

with

Minimized

Node

Contention

on

Wormhole

k-ary

n-cube

Networks

Ram

Kesavan

and

Dhabaleswar

K.

Panda

Dept.

of

Computer

and

Information

Science

The

Ohio

State

University

Columbus

OH

Tel

-LRB-

-RRB-

Fax

-LRB-

-RRB-

E-mail

fkesavan,pandag@cis.ohio-state.edu

Contact

Author

Dhabaleswar

K.

Panda

Abstract

This

paper

presents

a

new

approach

to

minimize

node

contention

while

performing

multiple

multicast/broadcast

on

wormhole

k-ary

n-cube

networks

with

overlapped

destination

sets

The

existing

multicast

algorithms

in

the

literature

deliver

poor

performance

under

multiple

multicast

because

these

algorithms

have

been

designed

with

only

single

multicast

in

mind

The

new

algorithms

introduced

in

this

paper

do

not

use

any

global

knowledge

about

the

respective

destination

sets

of

the

concurrent

multicasts

Instead

only

local

information

and

source-specific

partitioning

approach

are

used

For

systems

supporting

unicast

message-passing

a

new

SPUmesh

-LRB-

SourcePartitioned

Umesh

-RRB-

algorithm

is

proposed

and

shown

to

be

superior

than

the

conventional

Umesh

algorithm

-LSB-

-RSB-

for

multiple

multicast

Two

different

algorithms

SQHL

-LRB-

Source-Quadrant

Hierarchical

Leader

-RRB-

and

SCHL

-LRB-

Source-Centered

Hierarchical

Leader

-RRB-

are

proposed

for

systems

with

multidestination

message-passing

and

shown

to

be

superior

than

the

HL

scheme

-LSB-

-RSB-

All

these

algorithms

perform

-RRB-

times

faster

than

the

existing

algorithms

under

multiple

multicast

and

-RRB-

as

fast

as

existing

algorithms

under

single

multicast

Furthermore

the

SCHL

scheme

demonstrates

that

the

latency

of

multiple

multicast

can

in

fact

be

reduced

as

the

degree

of

multicast

increases

beyond

a

certain

number

Such

results

related

to

multiple

multicast

are

the

first

of

their

kind

in

the

wormhole

literature

Thus

these

algorithms

demonstrate

significant

potential

to

be

used

for

designing

fast

and

scalable

collective

communication

libraries

on

current

and

future

generation

wormhole

systems

preliminary

version

of

the

paper

-LSB-

-RSB-

has

been

presented

in

International

Conference

on

Parallel

Processing

Aug

This

research

is

supported

in

part

by

NSF

Grant

and

NSF

Career

Award

Contents

Introduction

Contention

in

Existing

Multicast

Algorithms

Node

Contention

Contention

in

Existing

Algorithms

The

Umesh

Algorithm

The

HL

Scheme

Estimating

Contention

in

Existing

Algorithms

The

Umesh

Algorithm

The

HL

Scheme

Link

Contention

Improvement

using

Source

Based

Information

Source-Partitioned-Umesh

-LRB-

SPUmesh

-RRB-

Algorithm

using

Unicast

Messages

The

Algorithm

Contention

Free

Property

for

Single

Multicast

Reduced

Contention

for

Multiple

Multicast

Improved

Algorithms

using

Multidestination

Messages

The

Source

Quadrant-based

Hierarchical

Leader

-LRB-

SQHL

-RRB-

Scheme

The

Algorithm

Reduced

Contention

for

Multiple

Multicast

Potential

for

Improvement

The

Source

Centered

Hierarchical

Leader

-LRB-

SCHL

-RRB-

Scheme

The

Algorithm

Reduced

Contention

for

Multiple

Multicast

Simulation

Experiments

and

Results

Multicast

Algorithms

using

Unicast

Messages

Meshes

Meshes

Multicast

Algorithms

using

Multidestination

Messages

Meshes

Meshes

Comparing

Unicast-based

and

Multidestination-based

Schemes

Meshes

Meshes

Scalability

with

Increasing

System

Size

Conclusions

Introduction

The

wormhole-routing

switching

technique

is

becoming

the

trend

in

building

future

parallel

systems

due

to

its

inherent

advantages

like

low-latency

communication

and

reduced

communication

hardware

overhead

-LSB-

-RSB-

IBM

Intel

Paragon

Cray

Ncube

J-Machine

and

Stanford

DASH

are

representative

systems

falling

into

this

category

Gradually

variations

of

this

technique

knows

as

cut-through

routing

are

being

used

in

networks

of

workstations

like

Myrinet

-LSB-

-RSB-

as

well

Distributed-memory

or

distributed-shared

memory

programming

paradigms

are

supported

on

these

systems

For

efficient

support

of

the

above

paradigms

these

systems

require

fast

implementation

of

collective

communication

operations

-LRB-

broadcast

multicast

global

combine

and

barrier

synchronization

-RRB-

-LSB-

-RSB-

Multicast/broadcast

is

a

common

collective

communication

operation

in

parallel

systems

-LSB-

-RSB-

Many

multicast/broadcast

algorithms

have

been

proposed

in

the

literature

in

recent

years

-LSB-

-RSB-

These

algorithms

have

been

primarily

developed

for

single

multicast

However

multiple

multicast

operations

-LRB-

i.e.

two

or

more

multicasts

executing

simultaneously

-RRB-

occur

frequently

in

parallel

systems

Examples

include

cache-invalidation

in

distributed-memory

systems

multiple

broadcast

in

numerical

and

scientific

applications

-LRB-

LU

decomposition

for

example

-RRB-

multiple

broadcast/multicast

operations

during

concurrent

barrier

and

reduction

operations

etc.

In

these

operations

destination

sets

of

different

multicasts

often

overlap

leading

to

nodes

participating

concurrently

in

multiple

multicasts

Efficient

multicast

algorithms

are

typically

hierarchical

in

nature

i.e.

some

destinations

serve

as

intermediate

sources

and

forward

copies

of

the

message

to

other

destinations

when

they

receive

it

Typically

tree-based

algorithms

are

used

to

minimize

the

number

of

communication

start-ups

-LRB-

steps

-RRB-

required

for

multicast

-LSB-

-RSB-

These

algorithms

are

also

generally

contention-free

for

single

multicast

In

order

to

implement

multiple

multicast

in

current

parallel

systems

each

source

node

uses

the

same

algorithm

designed

for

single

multicast

and

constructs

its

multicast

tree

independently

With

overlapped

destination

sets

such

construction

of

trees

may

result

in

node

contention

This

may

significantly

affect

the

overall

latency

of

the

multiple

multicast

operation

Providing

global

knowledge

of

different

multicasts

-LRB-

the

source

and

destinations

-RRB-

to

every

source

is

a

possible

solution

Thus

every

source

can

construct

its

tree

optimally

in

order

to

minimize

node-contention

However

exchange

of

such

global

knowledge

is

not

realistic

for

dynamic

multicast

operations

like

cachecoherency

Therefore

the

challenge

in

solving

the

problem

of

multiple

multicast

is

to

design

a

multicast

algorithm

while

keeping

multiple

multicast

in

mind

Such

an

algorithm

must

-RRB-

provide

an

optimal

solution

for

single

multicast

and

-RRB-

minimize

node

contention

under

multiple

multicast

when

every

source

uses

this

algorithm

to

construct

its

tree

in

the

absence

of

any

global

knowledge

In

this

paper

we

take

on

these

challenges

We

develop

efficient

multicast

algorithms

on

kary

n-cube

wormhole

networks

which

satisfy

the

above

objectives

We

consider

systems

with

unicast

as

well

as

multidestination

message-passing

-LSB-

-RSB-

First

we

identify

and

analyze

the

severity

of

node

contention

when

algorithms

designed

for

single

multicast

are

applied

to

multiple

multicast

We

develop

analytical

models

for

estimating

such

node

contention

for

completely

and

randomly

overlapped

destination

sets

Through

these

analyses

we

conclude

that

node

contention

along

with

link

contention

can

be

reduced

if

some

source-specific

information

can

be

used

to

construct

multicast

trees

for

different

sources

Next

we

develop

algorithms

for

multiple

multicast

For

unicast-based

systems

we

propose

a

new

SPUmesh

-LRB-

Source-Partitioned

Umesh

-RRB-

algorithm

which

uses

source-position

information

to

construct

the

multicast

tree

This

new

algorithm

is

contention

free

and

is

as

efficient

as

the

Umesh

algorithm

-LSB-

-RSB-

for

single

multicast

but

clearly

superior

for

multiple

multicast

For

systems

supporting

multidestination

message-passing

we

propose

two

different

algorithms

SQHL

-LRB-

Source-Quadrant-based

Hierarchical

Leader

-RRB-

and

SCHL

-LRB-

Source-Centered

Hierarchical

Leader

-RRB-

The

SCHL

algorithm

uses

source-position

information

as

well

as

load-balancing

techniques

on

network

links

along

different

directions

to

minimize

multiple

multicast

latency

Both

these

algorithms

are

significantly

better

than

the

HL

scheme

-LSB-

-RSB-

for

multiple

multicast

We

also

compare

the

best

of

both

kinds

of

algorithms

-LRB-

SPUmesh

and

SCHL

-RRB-

to

study

the

benefits

of

multidestination

message-passing

over

unicast

message-passing

for

multiple

multicast

It

is

demonstrated

that

similar

to

the

HL

scheme

-LSB-

-RSB-

for

single

multicast

the

SCHL

scheme

is

also

capable

of

reducing

multiple

multicast

latency

as

the

degree

of

multicast

increases

The

rest

of

the

paper

is

organized

as

follows

Section

discusses

how

contention

leads

to

poor

performance

when

existing

multicast

algorithms

designed

for

single

multicast

are

used

for

multiple

multicast

Section

presents

the

SPUmesh

algorithm

and

evaluates

its

node-contention

property

Section

proposes

the

SQHL

and

the

SCHL

schemes

for

systems

supporting

multidestination

message-passing

Simulation

experiments

and

results

comparing

the

relative

merits

of

the

existing

and

new

algorithms

are

presented

in

Section

Concluding

remarks

are

made

in

Section

Contention

in

Existing

Multicast

Algorithms

In

this

section

we

describe

how

node

and

link

contention

arises

when

existing

multicast

algorithms

are

used

to

implement

multiple

multicast

The

Umesh

algorithm

-LSB-

-RSB-

using

unicast

message

passing

and

the

HL

scheme

-LSB-

-RSB-

using

multidestination

message

passing

are

examined

Analytical

models

to

estimate

node

contention

under

multiple

multicast

are

developed

An

approach

to

reduce

contention

using

source-based

information

is

outlined

Node

Contention

Let

us

consider

multiple

multicasts

occuring

simultaneously

in

a

system

with

each

multicast

having

a

unique

source

and

a

set

of

destinations

The

destination

sets

of

various

concurrent

multicasts

may

overlap

Efficient

multicast

algorithms

are

typically

hierarchical

in

nature

-LSB-

-RSB-

where

some

intermediate

destinations

are

used

to

forward

the

message

to

other

destinations

in

the

set

Therefore

a

node

participating

in

multiple

multicasts

may

need

to

forward

multiple

messages

during

a

given

period

of

time

This

leads

to

serialization

of

the

associated

communication

startups

which

is

defined

as

node

contention

Depending

on

the

number

of

overlapped

nodes

and

the

degree

of

the

overlapping

one

or

more

nodes

may

fall

into

the

critical

path

The

overall

latency

of

the

multiple

multicast

pattern

will

depend

on

these

nodes

and

the

degree

of

node

contention

they

experience

Let

us

analyze

how

node

contention

affects

the

latency

of

multiple

multicasts

while

using

the

Umesh

algorithm

-LSB-

-RSB-

and

the

HL

scheme

-LSB-

-RSB-

Contention

in

Existing

Algorithms

The

Umesh

Algorithm

The

Umesh

algorithm

-LSB-

-RSB-

uses

the

concept

of

a

dimension-ordered

chain

-LRB-

-RRB-

on

meshes

that

use

e-cube

routing

-LSB-

-RSB-

The

chain

is

a

total

ordering

of

the

elements

of

the

set

which

consists

of

the

source

and

all

the

destination

nodes

of

the

multicast

The

ordering

ensures

zero

contention

for

links

between

messages

of

a

multicast

However

this

does

not

prevent

messages

belonging

to

different

concurrent

multicasts

from

contending

for

some

nodes

or

links

Figure

-LRB-

a

-RRB-

shows

how

a

multicast

message

propagates

within

the

dimension-ordered

chain

The

node

to

receive

the

first

message

is

positioned

half-way

in

the

chain

and

is

called

the

half-node

The

algorithm

recursively

identifies

quarter-nodes

one-eighth-nodes

and

so

on

-LRB-

a

-RRB-

source

half-node

quarter-node

one-eighth-node

-LRB-

b

-RRB-

source

half-node

quarter-node

one-eighth-node

Figure

Multicast

message

pattern

using

the

Umesh

algorithm

for

-LRB-

a

-RRB-

multicast

A

and

-LRB-

b

-RRB-

multicast

B

The

sources

half-nodes

and

quarter-nodes

are

highlighted

Let

us

consider

two

multicasts

A

and

B

with

identical

source-destination

sets

and

the

same

dimension-ordered

chain

Also

let

us

assume

that

both

their

sources

lie

in

the

lower

half

of

Figures

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

show

that

both

multicasts

to

a

large

extent

share

the

same

half-node

quarter-nodes

one-eighth-nodes

and

so

on

The

common

half-node

for

A

and

B

has

to

sequentialize

the

four

message

start-ups

that

it

undergoes

Similarly

if

several

multicasts

have

-LRB-

nearly

-RRB-

identical

's

these

multicasts

need

to

share

the

same

nodes

at

the

key

positions

along

the

chain

leading

to

hot

spots

In

the

worst

case

of

multiple

multicast

many-to-all

broadcast

each

broadcast

has

the

same

Therefore

all

the

sources

choose

the

node

half-way

in

the

chain

to

send

their

first

messages

to

the

node

quarter-way

in

the

chain

to

send

their

second

messages

to

and

so

on

This

leads

to

severe

node

contention

and

high

latency

for

the

multiple

multicast

pattern

Figure

-LRB-

a

-RRB-

shows

two

multicasts

A

and

B

with

identical

Their

sources

are

-LRB-

-RRB-

and

-LRB-

-RRB-

respectively

The

dimension-ordered

chain

is

shown

in

Fig.

-LRB-

b

-RRB-

Figures

-LRB-

c

-RRB-

and

-LRB-

d

-RRB-

show

the

multicast

trees

of

A

and

B

respectively

It

can

be

observed

that

nodes

-LRB-

-RRB-

-LRB-

-RRB-

and

the

nodes

in

the

subtrees

rooted

at

these

nodes

experience

contention

from

both

multicasts

These

nodes

sequentialize

the

multicasts

which

results

in

larger

multiple

multicast

latency

For

a

large

number

of

such

concurrent

multicasts

such

sequentialization

can

significantly

affect

their

latencies

A

detailed

analysis

of

the

node

contention

while

using

the

Umesh

algorithm

follows

in

Section

Destination

Node

Source

Destination

Node

-LRB-

a

-RRB-

-LRB-

b

-RRB-

F

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

Node

with

contention

-LRB-

c

-RRB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

Node

with

contention

-LRB-

d

-RRB-

Figure

Node

contention

between

two

concurrent

multicasts

using

the

Umesh

algorithm

-LRB-

a

-RRB-

two

multicasts

A

and

B

on

a

mesh

-LRB-

b

-RRB-

both

multicasts

sharing

the

same

dimension-ordered

chain

-LRB-

c

-RRB-

the

multicast

tree

for

A

with

source

at

-LRB-

-RRB-

and

-LRB-

d

-RRB-

the

multicast

tree

for

B

with

source

at

-LRB-

-RRB-

The

nodes

with

contention

are

highlighted

The

HL

Scheme

Multicasting

with

multidestination

message

passing

while

using

the

HL

scheme

-LSB-

-RSB-

also

has

node

contention

problem

for

multiple

multicast

Let

us

consider

two

concurrent

multicasts

A

and

B

with

identical

source-destination

sets

on

a

mesh

Let

their

respective

sources

be

at

-LRB-

-RRB-

and

-LRB-

-RRB-

as

shown

in

Fig.

It

can

be

observed

that

with

multidestination

message

passing

A

and

B

have

the

same

level

and

level

leader

sets

-LRB-

and

-RRB-

-LSB-

-RSB-

Sequentialization

of

worms

-LRB-

unicast

as

well

as

multidestination

-RRB-

corresponding

to

multicasts

A

and

B

at

the

and

nodes

leads

to

high

latency

for

the

overall

multiple

multicast

operation

With

this

HL

scheme

a

large

number

of

concurrent

multicasts

with

overlapping

leader

sets

leads

to

higher

node

contention

Consider

the

worst

case

of

S

concurrent

broadcasts

The

sets

for

all

the

broadcasts

is

the

same

singleton

set

consisting

of

a

corner

node

This

node

undergoes

two

start-ups

per

broadcast

This

this

will

result

in

the

sequentialization

of

message

start-ups

at

the

corner

node

leading

to

a

significant

increases

in

the

multiple

multicast

latency

A

detailed

analysis

of

the

node

contention

while

using

the

HL

scheme

follows

in

Section

destinationsource

BRCP

worm

Unicast

worm

-LRB-

a

-RRB-

level

leader

level

leader

destinationsource

BRCP

worm

Unicast

worm

-LRB-

b

-RRB-

level

leader

level

leader

Figure

Node

contention

between

two

concurrent

multicasts

with

identical

source-destination

sets

using

the

HL

scheme

-LRB-

a

-RRB-

communication

pattern

for

multicast

A

with

source

at

-LRB-

-RRB-

and

-LRB-

b

-RRB-

communication

pattern

for

multicast

B

with

source

at

-LRB-

-RRB-

Estimating

Contention

in

Existing

Algorithms

We

develop

analytical

models

to

estimate

the

contention

experienced

by

nodes

while

using

the

existing

algorithms

in

a

multiple

multicast

pattern

The

contention

directly

affects

the

latency

of

the

multiple

multicast

pattern

The

Umesh

Algorithm

Let

us

analyze

an

estimate

of

the

node

contention

when

using

the

Umesh

algorithm

for

multiple

multicast

We

consider

two

cases

-LRB-

a

-RRB-

worst

case

scenario

of

total

overlap

of

source-destination

sets

and

-LRB-

b

-RRB-

random

overlap

of

source-destination

sets

In

reality

concurrent

multiple

multicasts

of

an

application

could

lie

anywhere

in

the

range

of

random

to

complete

overlap

of

source-destination

sets

In

current

generation

wormhole

systems

a

communication

step

is

dominated

by

message

start-up

cost

Hence

in

this

section

and

in

analytical

estimates

done

is

later

sections

we

evaluate

node

contention

and

multipe

multicast

latency

in

terms

of

steps

-LRB-

message

start-ups

-RRB-

However

in

Section

we

consider

both

message

start-up

and

propagation

costs

when

presenting

simulation

results

Complete

Overlap

of

Source-Destination

Sets

Consider

a

k-ary

n-mesh

with

S

multicasts

occuring

simultaneously

Let

all

the

multicasts

have

the

same

source-destination

set

and

unique

sources

Let

j?j

D

It

can

be

observed

that

such

complete

overlap

of

source-destination

sets

is

possible

when

S

D.

Under

such

overlapping

the

node

undergoing

the

most

number

of

start-ups

decides

the

lower

bound

on

the

latency

of

a

multiple

multicast

pattern

This

leads

to

Theorem

The

latency

of

a

multiple

multicast

pattern

with

complete

overlap

of

source-destination

sets

-LRB-

S

sources

and

-LRB-

D

-RRB-

destinations

for

each

source

with

identical

S

D

-RRB-

while

using

the

Umesh

algorithm

is

given

by

-LRB-

-LRB-

D

-RRB-

e

-RRB-

steps

for

odd

D

and

is

bounded

below

by

-LRB-

am

bm

-RRB-

steps

for

even

D

where

ai

D

e

and

bi

-LRB-

if

S

D

i

if

S

D

and

m

is

such

that

am

ai

Proof

Let

dD

Let

ds

be

a

source

node

in

one

of

the

multicasts

For

odd

D

ds

chooses

as

the

half-node

independent

of

its

relative

position

in

except

when

s

If

is

one

of

the

sources

it

undergoes

e

start-ups

as

the

half-node

for

each

of

the

other

S

multicasts

and

an

additional

e

start-ups

as

a

source

If

is

not

one

of

the

sources

it

undergoes

e

start-ups

for

each

of

the

S

multicasts

and

an

additional

start-up

is

needed

for

the

first

message

to

arrive

at

Therefore

for

odd

D

the

latency

of

a

multiple

multicast

pattern

is

given

by

-LRB-

-LRB-

D

-RRB-

e

-RRB-

steps

-LRB-

start-ups

-RRB-

For

even

D

let

us

assume

the

best

case

of

uniform

distribution

of

sources

in

The

node

dbD

c

is

chosen

as

the

half-node

in

of

the

multicasts

dbD

c

is

chosen

as

a

quarter-node

in

of

the

multicasts

and

so

on

Depending

on

the

value

of

D

one

of

these

nodes

say

dx

undergoes

the

maximum

number

of

start-ups

in

the

multiple

multicast

pattern

If

S

D

for

estimating

the

lower

bound

let

us

assume

the

best

case

of

dx

not

being

one

of

the

sources

Therefore

if

dx

dbD

c

there

is

an

extra

start-up

before

the

first

message

arrives

at

dx

following

which

dx

undergoes

e

start-ups

Similarily

if

dx

dbD

c

there

are

i

extra

start-ups

before

the

first

message

reaches

dx

following

which

dx

undergoes

-LRB-

-RRB-

S

e

start-ups

The

lower

bound

is

given

by

that

expression

in

which

dx

undergoes

the

maximum

number

of

start-ups

If

S

D

then

dx

undergoes

an

additional

start-ups

as

a

source

In

this

case

the

lower

bound

is

given

by

plus

the

maximum

number

of

start-ups

that

dx

undergoes

Therefore

for

even

D

the

latency

of

a

multiple

multicast

pattern

is

bounded

below

by

-LRB-

am

bm

-RRB-

steps

where

m

is

such

that

am

ai

A

simple

experiment

was

conducted

to

estimate

the

latency

of

multiple

multicast

-LRB-

in

terms

of

steps

-RRB-

using

the

Umesh

algorithm

with

complete

overlap

of

source-destination

sets

on

an

mesh

In

this

experiment

all

system

parameters

-LRB-

propagation

time

router

delay

switching

time

etc

-RRB-

were

ignored

and

only

a

high

start-up

time

was

considered

The

experimental

values

were

averaged

over

runs

and

rounded

off

to

the

nearest

integer

The

third

and

seventh

rows

of

Table

show

the

results

of

this

experiment

for

odd

and

even

D

respectively

The

analytical

values

calculated

by

using

Theorem

are

shown

in

the

fourth

and

eighth

rows

for

odd

and

even

D

respectively

It

can

be

observed

that

the

analytical

values

match

the

experimental

values

very

closely

validating

the

results

of

Theorem

These

results

indicate

that

node

contention

becomes

severe

for

a

multiple

multicast

pattern

as

the

number

of

sources

and

destinations

increase

Table

Experimental

and

analytical

latency

of

multiple

multicast

-LRB-

in

number

of

steps

-RRB-

using

Umesh

on

an

mesh

for

different

values

of

S

and

-LRB-

odd

and

even

-RRB-

D.

Complete

overlap

of

source-destination

sets

is

considered

-LRB-

S

D

-RRB-

S

odd

D

j?j

Experimental

61 76 125 156

Analytical

61 76 125 156

S

even

D

j?j

Experimental

57 77 105 154

Analytical

77 98 146 198

Random

Overlap

of

Source-Destination

Sets

Let

us

now

consider

the

case

of

random

overlap

of

source-destination

sets

Since

it

is

difficult

to

identify

the

node

which

undergoes

the

maximum

node

contention

we

construct

a

lower

bound

on

the

node

contention

by

calculating

the

expected

number

of

startups

sequentialized

at

a

node

using

probabilistic

analysis

Let

us

assume

each

node

has

equal

probability

of

being

in

any

position

in

the

dimension-ordered

chain

-LRB-

-RRB-

of

a

multicast

In

reality

the

nodes

which

lie

in

the

middle

of

the

mesh

have

a

greater

probability

of

being

the

half-node

for

a

multicast

with

large

D.

Similarily

the

nodes

in

other

specific

parts

of

the

mesh

have

a

higher

probability

of

being

specific

nodes

in

a

multicast

with

large

D.

Therefore

the

values

generated

by

using

the

equal

probability

assumption

can

serve

as

a

lower

bound

to

the

multiple

multicast

latency

Again

let

us

consider

S

multicasts

occuring

simultaneously

each

with

j?j

D

An

arbitrary

node

s

which

is

a

source

in

one

of

the

multicasts

-LRB-

and

hence

undergoes

De

startups

-RRB-

can

be

a

half-node

quarter-node

and

so

on

in

the

other

S

multicasts

Let

the

terms

and

so

on

denote

the

events

of

s

being

the

half-node

quarter-node

one-eighth-node

respectively

in

a

multicast

Let

denote

the

event

of

s

not

participating

in

a

multicast

and

let

A

B

denote

the

events

A

and

B

occuring

simultaneously

Now

each

event

has

a

fixed

number

of

start-up

cost

associated

with

it

For

example

has

e

steps

associated

with

it

has

e

and

so

on

Under

the

equal

probability

assumption

as

stated

above

we

have

Theorem

The

latency

of

a

multiple

multicast

pattern

with

random

ovelap

of

source-destination

sets

-LRB-

S

sources

and

-LRB-

D

-RRB-

destinations

for

each

source

with

random

-RRB-

while

using

the

Umesh

algorithm

is

bounded

below

by

De

steps

plus

the

sum

of

the

products

of

the

probabilities

of

each

term

in

the

expression

-LRB-

ED

-RRB-

S

occuring

and

the

number

of

steps

associated

with

each

term

Proof

The

terms

in

the

expression

-LRB-

ED

-RRB-

S

define

all

possible

events

for

a

source

s.

For

example

the

term

represents

the

following

event

s

does

not

participate

in

multicasts

is

the

half-node

of

multicasts

the

quarter-node

of

multicasts

and

the

one-eighth-node

of

multicast

If

-RCB-

-LRB-

A

-RRB-

represents

the

probability

of

the

event

A

occuring

then

the

expected

number

of

start-ups

incurred

by

s

due

to

the

event

occuring

is

given

by

the

expression

-LRB-

e

e

e

-RRB-

-RCB-

-LRB-

-RRB-

Therefore

the

expected

number

of

total

start-ups

incurred

by

s

is

De

-LRB-

as

a

source

-RRB-

plus

the

sum

of

the

products

of

the

probabilities

of

each

term

-LRB-

in

the

above

polynomial

-RRB-

occuring

and

the

number

of

start-ups

s

incurs

for

the

term

This

number

acts

as

a

lower

bound

because

of

the

assumption

of

each

node

having

equal

probability

of

being

in

any

position

in

the

of

a

multicast

For

example

for

values

of

S

and

D

the

expression

is

-LRB-

-RRB-

The

node

s

incurs

start-ups

as

a

source

as

a

half-node

-LRB-

-RRB-

as

a

quarter-node

-LRB-

-RRB-

and

as

a

one-eighth-node

-LRB-

-RRB-

If

s

does

not

participate

in

a

multicast

-LRB-

-RRB-

it

incurs

no

start-ups

Let

us

consider

the

term

The

number

of

start-ups

associated

with

the

term

is

The

number

of

times

the

term

appears

in

the

above

expression

is

The

product

of

the

probability

of

the

term

and

the

associated

start-ups

is

-RCB-

-LRB-

-RRB-

Similarily

the

products

of

the

probabilities

of

other

terms

and

their

respective

startups

are

-RCB-

-LRB-

-RRB-

-RCB-

-LRB-

-RRB-

-RCB-

-LRB-

-RRB-

-RCB-

-LRB-

-RRB-

-RCB-

-LRB-

-RRB-

and

so

on

A

small

experiment

was

conducted

to

estimate

the

number

of

sequentialized

steps

-LRB-

start-ups

-RRB-

that

multiple

multicast

incurs

for

the

Umesh

algorithm

with

random

overlap

of

source-destination

sets

on

a

mesh

The

smaller

size

was

chosen

to

make

the

calculation

of

all

the

terms

of

the

multinomial

expression

feasible

Like

the

previous

experiment

all

system

parameters

were

ignored

and

only

a

high

startup

time

was

considered

The

experiment

values

were

averaged

over

runs

and

rounded

off

to

the

nearest

integer

Table

shows

that

the

analytical

values

calculated

by

using

the

above

theorem

serve

as

good

lower

bounds

to

the

experiment

values

Table

Experimental

and

analytical

latency

of

multiple

multicast

-LRB-

in

number

of

steps

-RRB-

using

Umesh

on

an

mesh

for

different

values

of

S

and

D.

Random

overlap

of

source-destination

sets

is

considered

-LRB-

S

D

-RRB-

S

D

j?j

Experimental

Analytical

The

HL

Scheme

Let

us

analyze

an

estimate

of

the

node

contention

when

using

the

HL

scheme

for

multiple

multicast

on

a

k-ary

n-mesh

We

only

consider

the

case

of

total

overlap

with

identical

source-destination

sets

in

this

section

The

random

overlap

case

is

difficult

to

model

and

analyze

However

we

study

the

impact

of

both

random

as

well

as

total

overlap

on

multiple

multicast

latency

using

simulations

in

Section

Let

us

consider

S

multicasts

with

all

multicasts

having

identical

source-destination

sets

of

size

D

Such

identical

sets

lead

to

identical

leader

sets

This

leads

to

Theorem

The

latency

of

a

multiple

multicast

pattern

with

complete

overlap

of

source-destination

sets

-LRB-

S

multicasts

and

-LRB-

D

-RRB-

destinations

for

each

source

with

identical

source-destination

sets

S

D

-RRB-

on

a

k-ary

n-mesh

using

the

HL

scheme

is

-LRB-

e

Sn

-RRB-

steps

and

is

Sn

steps

Proof

Based

on

a

given

source-destination

set

let

us

consider

a

node

x

from

the

highest

level

leader

set

Lg

g

n

-LSB-

-RSB-

This

node

undergoes

some

start-ups

when

every

source

uses

the

Umesh

algorithm

to

multicast

to

nodes

of

Lg

In

the

worst

case

if

x

is

the

half-node

for

the

Umesh

phase

of

each

multicast

x

undergoes

e

start-ups

for

each

multicast

In

the

best

case

x

incurs

no

start-ups

in

the

Umesh

phase

of

each

multicast

This

occurs

when

jLgj

for

high

values

of

D.

Following

the

Umesh

phase

x

undergoes

a

maximum

of

g

start-ups

for

each

multicast

This

is

to

send

out

g

multidestination

worms

to

cover

each

set

of

which

it

is

a

leader

Therefore

the

number

of

start-ups

sequentialized

at

x

lies

between

-LRB-

e

Sg

-RRB-

and

Sg

As

D

increases

the

value

of

jLgj

reduces

from

k

to

and

the

value

of

g

increases

from

to

n

The

maximum

possible

value

of

jLgj

is

k

and

that

of

g

is

n

Since

it

is

very

complex

to

analytically

predict

the

values

of

jLgj

and

g

for

different

values

of

S

and

D

we

consider

upper

and

lower

bounds

for

this

expression

For

the

upper

bound

occuring

for

low

values

of

D

we

can

take

jLgj

k

and

g

n

The

lower

bound

case

occurs

for

high

values

of

D

where

jLgj

ss

and

g

n.

Let

us

now

estimate

the

latency

of

the

multiple

multicast

pattern

If

the

node

x

is

not

one

of

the

sources

in

the

pattern

an

extra

start-up

is

added

to

account

for

the

startup

of

the

first

message

to

arrive

from

a

source

to

x

This

gives

us

an

upper

bound

of

-LRB-

e

Sn

-RRB-

and

a

lower

bound

of

-LRB-

Sn

-RRB-

for

the

multiple

multicast

latency

On

the

other

hand

if

x

is

a

source

for

one

of

the

multicasts

and

a

half-node

for

the

remaining

S

multicasts

we

get

a

upper

bound

of

-LRB-

-LRB-

S

-RRB-

e

Sn

-RRB-

which

is

equal

to

-LRB-

e

Sn

-RRB-

If

x

is

a

source

and

also

the

sole

member

of

all

the

Lg

sets

it

sends

out

Sg

messages

to

give

us

a

lower

bound

of

Sn

Thefore

the

upper

bound

for

the

the

multiple

multicast

pattern

is

-LRB-

e

Sn

-RRB-

and

the

lower

bound

is

Sn

A

small

experiment

was

conducted

to

estimate

the

number

of

sequentialized

steps

that

multiple

multicast

incurs

for

the

HL

scheme

with

complete

overlap

of

source-destination

sets

on

a

mesh

Like

previous

experiments

all

system

parameters

were

ignored

and

only

a

high

startup

time

was

considered

The

experimental

values

were

averaged

over

runs

and

rounded

off

to

the

nearest

integer

The

fourth

row

of

Table

shows

the

results

of

this

experiment

In

this

mesh

n

and

k

giving

an

upper

bound

of

and

a

lower

bound

of

The

upper

bounds

in

Table

represent

the

worst

case

condition

of

jLgj

k

and

g

n

and

hence

act

like

a

loose

upper

bound

Nevertheless

the

experimental

values

fall

between

the

two

bounds

and

reflect

the

severity

of

node

contention

in

multiple

multicast

Table

Experimental

and

analytical

latency

of

multiple

multicast

-LRB-

in

number

of

steps

-RRB-

using

the

HL

scheme

on

an

mesh

for

different

values

of

S

and

D.

Complete

overlap

of

source-destination

sets

is

considered

-LRB-

S

D

-RRB-

S

D

Upper

Bound

65 65 129 129

Experimental

Lower

Bound

Link

Contention

As

observed

above

when

each

source

node

uses

the

same

single

multicast

algorithm

to

implement

multiple

multicast

the

intermediate

nodes

of

the

multicasts

might

be

identical

leading

to

node

contention

Even

if

they

are

not

identical

they

may

lie

in

close

proximity

with

each

other

When

these

closely

bunched

intermediate

nodes

send

out

messages

as

part

of

their

respective

multicasts

it

leads

to

link

contention

This

may

also

affect

the

overall

latency

of

multiple

multicast

operation

The

effect

of

link

contention

obviously

increases

as

link

propagation

time

becomes

more

of

a

dominant

factor

compared

to

start-up

time

It

also

becomes

dominant

for

long

messages

Thus

a

good

multicast

algorithm

should

make

sure

that

the

intermediate

nodes

for

various

concurrent

multicasts

are

different

and

well

spread

out

over

the

network

The

former

condition

reduces

the

node

contention

experienced

while

the

latter

reduces

link

contention

In

this

paper

we

concentrate

on

reducing

the

node

contention

by

making

sure

that

the

intermediate

nodes

for

different

multicasts

are

as

different

as

possible

A

side

effect

of

the

intermediate

nodes

being

well

spread

out

over

the

mesh

is

that

it

provides

reduced

link

contention

as

discussed

in

Section

Improvement

using

Source

Based

Information

A

method

to

reduce

node

contention

is

to

make

each

multicast

choose

unique

intermediate

nodes

as

different

as

possible

from

the

rest

With

dynamic

multicast

patterns

all

concurrent

multicasts

are

unaware

of

one

another

This

means

that

a

multicast

has

no

information

whatsoever

about

the

source

and

destinations

of

the

other

multicasts

Then

how

can

a

multicast

choose

unique

intermediate

nodes

different

from

those

of

other

multicasts

A

good

multicast

algorithm

should

use

some

local

information

to

make

its

tree

as

unique

as

possible

The

local

information

that

our

new

algorithms

use

is

the

position

of

the

source

in

the

system

This

is

unique

for

each

multicast

If

each

multicast

constructs

its

tree

based

on

the

position

of

its

corresponding

source

then

all

the

multicasts

will

end

up

with

trees

as

unique

as

possible

In

current

-LRB-

Umesh

and

HL

-RRB-

algorithms

the

multicast

trees

are

generated

using

primarily

the

destination

-LRB-

distribution

-RRB-

information

They

depend

to

a

very

small

degree

on

the

position

of

the

source

In

our

new

unicast-based

algorithm

we

use

the

same

dimension-ordered

chain

concept

as

the

Umesh

algorithm

but

use

the

position

of

the

source

in

the

chain

to

decide

the

multicast

tree

In

our

multidestination-based

algorithms

we

choose

the

leader

sets

for

a

multicast

depending

on

the

position

of

its

source

We

present

these

new

algorithms

in

the

following

sections

Source-Partitioned-Umesh

-LRB-

SPUmesh

-RRB-

Algorithm

using

Unicast

Messages

In

this

section

we

propose

a

new

Source

Partitioned

Umesh

-LRB-

SPUmesh

-RRB-

algorithm

to

implement

efficient

multiple

multicast

using

unicast

messages

For

single

multicast

the

algorithm

performs

as

well

as

the

Umesh

algorithm

by

preserving

link

contention-free

property

For

multiple

multicast

the

new

algorithm

performs

far

better

than

the

Umesh

algorithm

by

reducing

node

contention

First

the

algorithm

is

presented

formally

and

its

link

contention-free

property

is

proved

for

single

multicast

Then

its

reduced

node

contention

property

for

multiple

multicast

is

demonstrated

The

Algorithm

As

the

name

suggests

the

Source

Partitioned

Umesh

algorithm

partitions

the

dimension

ordered

chain

according

to

the

position

of

the

source

in

the

chain

Let

the

source

and

destination

addresses

be

sorted

into

a

dimension

ordered

chain

A

new

chain

is

obtained

by

a

rotate-left

operation

on

till

the

source

s

shifts

to

the

beginning

of

Now

the

Umesh

algorithm

is

performed

on

The

algorithm

is

formally

presented

in

Fig.

Figure

-LRB-

a

-RRB-

shows

an

example

of

a

multicast

tree

generated

when

the

SPUmesh

algorithm

is

used

for

multicast

A

from

Fig.

-LRB-

a

-RRB-

SPUmesh

Algorithm

Input

Dn

the

dimension

ordered

chain

Dsource

the

source

node

Output

Send

the

message

to

all

destinations

Procedure

Let

Di

Dsource

where

Di

Create

Di

Di

Dn

Di

Apply

the

Umesh

algorithm

on

with

Di

as

source

Figure

Outline

of

the

SPUmesh

algorithm

The

SPUmesh

algorithm

differs

from

the

Umesh

algorithm

only

with

respect

to

the

dimension

ordered

chain

Changing

to

causes

the

multicast

pattern

to

be

dependent

on

the

position

of

the

source

This

effect

lowers

node

contention

and

latency

for

multiple

multicast

as

compared

to

the

Umesh

algorithm

It

is

to

be

noted

that

a

similar

reordering

of

the

dimension-order

chain

has

been

used

in

-LSB-

-RSB-

to

develop

multicast

algorithms

for

the

torus

network

and

in

-LSB-

-RSB-

for

multipacket

multicast

However

here

we

use

such

reordering

on

meshes

to

reduce

node

contention

for

multiple

multicast

Since

the

multicast

tree

is

generated

in

exactly

the

same

way

as

in

the

Umesh

algorithm

the

SPUmesh

algorithm

takes

no

more

than

De

start-ups

for

a

single

multicast

We

prove

below

that

the

SPUmesh

algorithm

like

the

Umesh

algorithm

guarantees

zero

link

contention

among

the

messages

for

a

single

multicast

Contention

Free

Property

for

Single

Multicast

The

Umesh

algorithm

guarantees

zero

link

contention

among

messages

of

a

single

multicast

-LSB-

-RSB-

for

wormhole

systems

with

dimension

ordering

Let

us

use

the

binary

relation

dimension

order

denoted

as

d

as

defined

in

-LSB-

-RSB-

According

to

the

definition

if

u

d

v

d

x

d

y

then

the

path

taken

by

a

message

from

u

to

v

and

the

path

taken

by

a

message

from

x

to

y

are

arc

disjoint

Link

contention

occurs

when

two

or

more

messages

need

to

use

the

same

physical

link

at

the

same

time

We

assume

meshes

with

dimension-order

X-Y

routing

for

this

proof

Let

us

consider

s

to

be

the

source

and

m

to

be

the

half-node

in

the

SPUmesh

algorithm

Without

loss

of

generality

let

us

assume

that

s

d

m.

Figure

-LRB-

a

-RRB-

shows

the

positions

of

s

and

m

in

the

new

dimension-ordered

chain

and

their

positional

relationaships

with

the

original

dimension-ordered

chain

Figure

-LRB-

b

-RRB-

shows

how

the

positions

of

s

and

m

can

be

used

to

divide

the

mesh

into

three

partitions

According

to

the

new

algorithm

s

sends

the

first

message

to

m.

Subsequently

s

covers

destinations

in

partition

and

m

covers

destinations

in

partitions

and

Let

us

define

the

messages

for

a

single

multicast

on

this

system

as

follows

Definition

A

message

is

defined

as

an

intra-partition

or

inter-partition

message

depending

on

whether

the

message

traverses

within

or

across

a

partition

-LRB-

a

-RRB-

s

m

s

m

F

F

rotate-left

s

m

Partition

Partition

Partition

-LRB-

b

-RRB-

y

x

Figure

Positional

relationships

between

a

typical

source

node

and

the

half-node

-LRB-

a

-RRB-

transformation

from

the

original

dimension-ordered

chain

to

the

new

dimension-ordered

chain

and

-LRB-

b

-RRB-

division

of

the

mesh

into

three

partitions

based

on

their

positions

It

can

be

easily

observed

from

the

algorithm

that

after

s

sends

the

first

message

to

m

all

further

messages

are

intra-partition

messages

except

for

those

inter-partition

messages

sent

from

nodes

in

to

nodes

in

With

respect

to

intra-partition

messages

we

have

Lemma

After

s

sends

the

first

message

to

m

there

is

no

link

contention

between

any

two

intrapartition

messages

Proof

Consider

any

of

the

three

partitions

All

intra-partition

messages

are

sent

strictly

using

the

Umesh

algorithm

according

to

the

dimension

ordering

Therefore

intra-partition

messages

of

a

partition

do

not

contend

for

links

with

each

other

Obviously

intra-partition

messages

of

two

non-adjacent

partitions

can

not

contend

for

links

with

each

other

Let

us

consider

intra-partition

messages

of

two

adjacent

partitions

An

intra-partition

message

can

take

no

link

in

the

x

dimension

which

is

completely

contained

in

its

adjacent

partition

It

can

only

take

one

x

dimension

link

which

straddles

the

inter-partition

boundary

and

this

link

can

not

be

taken

by

an

intra-partition

message

of

the

adjacent

partition

Also

an

intra-partition

message

can

take

the

y

dimension

links

-LRB-

of

the

common

column

-RRB-

in

the

adjacent

partition

only

in

the

negative

direction

while

intra-partition

messages

of

the

adjacent

partition

can

take

those

y

dimension

links

only

in

the

positive

direction

This

proves

that

there

exists

no

link

contention

between

any

two

intra-partition

messages

This

leaves

us

to

concentrate

only

on

the

inter-partition

messages

which

are

sent

from

nodes

in

to

nodes

in

Let

us

define

these

messages

as

follows

Definition

An

inter-partition

message

from

partition

to

partition

is

called

a

back

message

This

leads

to

Lemma

At

any

point

of

time

there

is

atmost

only

one

back

message

in

the

system

Proof

Let

there

be

two

or

more

concurrent

back

messages

Let

us

take

any

two

of

these

messages

and

Let

be

from

node

x

to

node

y

and

be

from

node

u

to

node

v.

Therefore

nodes

x

and

u

are

in

partition

and

nodes

y

and

v

are

in

partition

Without

loss

of

generality

let

us

assume

x

d

u

This

leads

to

only

two

possible

situations

as

shown

in

Figure

-LRB-

a

-RRB-

x

d

u

d

v

d

y

and

-LRB-

b

-RRB-

x

d

u

d

y

d

v

It

can

be

proved

that

both

cases

never

occur

If

x

is

sending

a

message

to

y

in

a

given

step

then

the

set

of

nodes

fij

-LRB-

x

d

i

-RRB-

-LRB-

i

d

y

-RRB-

g

will

receive

the

message

only

during

subsequent

steps

The

node

u

lies

in

this

set

and

thus

can

not

be

sending

a

message

to

v

in

the

same

step

Therefore

there

can

be

atmost

only

one

back

message

being

sent

at

any

point

of

time

Lemma

There

is

no

link

contention

between

a

back

message

and

an

intra-partition

message

belonging

to

partition

or

x

u

v

y

-LRB-

a

-RRB-

x

u

y

v

-LRB-

b

-RRB-

Figure

Possible

cases

for

two

simultaneous

messages

going

from

partition

to

partition

Proof

Let

us

first

consider

an

intra-partition

message

belonging

to

partition

For

dimensional

X-Y

routing

a

back

message

takes

X

dimension

links

only

in

the

negative

direction

An

intrapartition

message

going

from

node

a

to

node

b

always

has

a

d

b

and

hence

it

takes

X

dimension

links

only

in

the

positive

direction

There

is

no

contention

in

the

Y

dimension

because

Y

dimension

links

are

taken

by

the

back

message

only

after

it

reaches

partition

Using

the

same

reasoning

as

above

there

is

no

contention

in

the

X

dimension

betwen

a

back

message

and

an

intra-partition

message

belonging

to

partion

A

back

message

takes

Y

dimension

links

only

along

a

column

that

has

atleast

one

node

in

partition

An

intra-partition

message

of

partition

takes

Y

dimension

links

only

along

the

column

that

has

atleast

one

node

in

partition

The

only

column

that

satisfies

both

these

conditions

is

one

which

has

the

source

node

s

In

this

column

an

intra-partition

message

of

partition

can

only

take

those

Y

dimension

links

which

lie

in

partition

and

are

in

the

positive

direction

On

the

other

hand

a

back

message

can

only

take

those

Y

dimension

links

in

the

positive

direction

which

lie

in

partition

Therefore

there

can

not

be

contention

between

an

intra-partition

message

of

partition

and

a

back

message

Using

the

same

reasoning

mentioned

in

the

beginning

of

this

proof

there

is

no

contention

in

the

X

dimension

between

a

back

message

and

an

intra-partition

message

belonging

to

partition

Let

us

consider

contention

in

the

Y

dimension

for

a

back

message

from

node

u

to

node

v

such

that

v

d

u

There

are

three

cases

the

back

message

takes

-LRB-

a

-RRB-

atleast

one

Y

dimension

link

in

the

positive

direction

or

-LRB-

b

-RRB-

atleast

one

Y

dimension

link

in

the

negative

direction

or

-LRB-

c

-RRB-

no

Y

dimension

links

In

case

-LRB-

c

-RRB-

there

is

no

possible

contention

with

an

intra-partition

message

of

partition

Let

us

take

cases

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

Consider

an

intra-partition

message

of

partition

from

node

a

to

node

b

such

that

a

d

b

d

u

If

this

message

shares

even

one

Y

dimension

link

with

the

back

message

then

a

d

v

According

to

the

Umesh

algorithm

if

u

sends

a

message

to

v

the

set

of

nodes

fij

-LRB-

i

d

v

-RRB-

g

will

only

receive

the

message

in

subsequent

steps

Since

the

node

a

lies

in

this

set

it

is

yet

to

receive

the

message

and

thus

can

not

be

sending

a

message

to

b

in

the

same

step

Therefore

there

can

not

be

contention

between

an

intra-partition

message

belonging

to

partition

and

a

back

message

Lemmas

directly

lead

to

Theorem

The

SPUmesh

algorithm

is

link

contention

free

for

a

single

multicast

The

above

theorem

has

been

proved

assuming

that

s

d

m

The

case

of

m

d

s

can

be

easily

proved

with

similar

arguments

Although

this

theorem

has

been

proved

for

a

mesh

it

is

quite

general

and

can

be

easily

extended

for

higher

dimensional

meshes

Reduced

Contention

for

Multiple

Multicast

Using

the

SPUmesh

algorithm

each

multicast

chooses

a

different

half-node

depending

on

the

position

of

its

corresponding

source

node

This

reduces

the

contention

on

the

centrally

positioned

nodes

of

When

is

divided

recursively

at

each

stage

of

the

algorithm

the

above

effect

carries

over

Let

us

take

the

example

from

Fig.

to

see

the

effect

of

the

SPUmesh

algorithm

Figures

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

illustrate

the

results

for

multicasts

A

and

B

respectively

Now

multicasts

A

and

B

choose

-LRB-

-RRB-

and

-LRB-

-RRB-

as

their

respective

center

nodes

and

experience

zero

node

contention

In

general

the

SPUmesh

algorithm

affects

the

choice

of

the

center

node

at

each

of

the

successive

steps

of

the

algorithm

resulting

in

reduced

node

contention

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LRB-

a

-RRB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LRB-

b

-RRB-

Figure

Zero

node

contention

for

the

example

multiple

multicasts

in

Fig.

with

the

SPUmesh

algorithm

-LRB-

a

-RRB-

the

new

multicast

tree

for

A

and

-LRB-

b

-RRB-

the

new

multicast

tree

for

B.

Analytical

Estimate

of

Node

Contention

Let

us

analyze

an

estimate

of

the

node

contention

when

using

the

SPUmesh

algorithm

for

multiple

multicast

We

only

consider

the

worst

case

of

total

overlap

of

source-destination

sets

here

Theorem

The

latency

of

a

multiple

multicast

pattern

with

complete

overlap

of

source-destination

sets

-LRB-

S

sources

and

-LRB-

D

-RRB-

destinations

for

each

source

with

identical

S

D

-RRB-

while

using

the

SPUmesh

algorithm

is

bounded

above

by

-LRB-

De

Pni

D

e

D

e

-RRB-

steps

where

S

Pni

p

and

p

Proof

Let

s

be

one

of

the

sources

In

the

worst

case

s

can

be

the

half-node

for

atmost

one

other

multicast

the

quarter-node

for

atmost

two

other

multicasts

the

one-eighth

node

for

atmost

four

other

multicasts

and

so

on

The

latency

of

the

multiple

multicast

pattern

is

bounded

above

by

the

maximum

number

of

steps

-LRB-

start-ups

-RRB-

that

s

can

undergo

Let

S

be

written

as

Pni

p

where

p

The

node

s

undergoes

atmost

De

start-ups

as

a

source

e

start-ups

as

a

half-node

e

start-ups

as

a

quarter-node

Therefore

the

multiple

multicast

pattern

latency

is

bounded

above

by

-LRB-

De

Pni

D

e

D

e

-RRB-

steps

A

small

experiment

was

conducted

to

estimate

the

number

of

steps

incurred

by

multiple

multicast

pattern

using

the

SPUmesh

algorithm

with

complete

overlap

of

source-destination

sets

on

an

mesh

Like

previous

experiments

all

system

parameters

were

ignored

and

only

a

high

start-up

time

was

considered

The

experimental

values

were

averaged

over

runs

and

rounded

off

to

the

nearest

integer

The

third

and

seventh

rows

of

Table

show

the

results

of

this

experiment

for

odd

and

even

D

respectively

The

analytical

values

calculated

by

using

Theorem

are

shown

in

the

fourth

and

eighth

rows

for

odd

and

even

D

repectively

It

can

be

observed

that

the

analytical

values

act

as

an

upper

bound

for

the

experimental

values

validating

Theorem

It

can

also

be

seen

that

unlike

the

Umesh

case

D

being

odd

or

even

does

not

affect

the

number

of

steps

The

experimental

results

in

Table

are

significantly

less

when

compared

to

those

in

Table

The

analytical

results

of

Table

when

compared

with

those

of

Table

lead

to

the

following

observation

Observation

The

upper

bound

on

the

latency

of

a

multiple

multicast

pattern

while

using

the

SPUmesh

algorithm

is

less

than

the

lower

bound

on

the

latency

of

the

same

pattern

while

using

the

Umesh

algorithm

Therefore

the

SPUmesh

algorithm

promises

much

better

performance

than

the

Umesh

algo

rithm

for

multiple

multicast

Table

Experimental

and

analytical

latency

of

multiple

multicast

-LRB-

in

number

of

steps

-RRB-

using

SPUmesh

on

an

mesh

for

different

values

of

S

and

odd

and

even

D.

Complete

overlap

of

source-destination

sets

is

considered

-LRB-

S

D

-RRB-

S

odd

D

j?j

Experimental

Analytical

S

even

D

j?j

Experimental

Analytical

Improved

Algorithms

using

Multidestination

Messages

In

this

section

two

multicast

algorithms

are

proposed

for

systems

supporting

multidestination

message-passing

They

are

shown

to

be

better

than

the

HL

scheme

-LSB-

-RSB-

for

multiple

multicast

The

Source

Quadrant-based

Hierarchical

Leader

-LRB-

SQHL

-RRB-

Scheme

The

Algorithm

Let

us

consider

a

multicast

pattern

from

source

s

to

a

destination

set

D.

Let

D

-LSB-

fsg

In

the

ith

step

i

n

the

algorithm

partitions

set

Li

into

disjoint

subsets

each

with

a

leader

node

The

leader

node

can

forward

a

message

to

the

members

of

its

set

using

a

single

multidestination

worm

under

the

dimension-order

routing

model

The

groupings

are

done

exactly

like

the

HL

scheme

However

the

leader

nodes

are

chosen

according

to

the

algorithm

in

Fig.

In

effect

when

dealing

with

the

rth

dimension

r

n

an

-LRB-

n

-RRB-

dimension

plane

Pr

is

drawn

perpendicular

to

the

rth

dimension

such

that

it

divides

the

nD

mesh

into

halves

The

leaders

are

chosen

as

close

to

that

end

of

the

grouping

as

the

half

in

which

the

source

lies

The

remaining

steps

are

the

same

as

the

original

HL

scheme

The

number

of

startups

for

a

single

multicast

using

the

SQHL

scheme

is

the

same

as

when

using

the

HL

scheme

i.e.

-LRB-

-LRB-

jLmj

-RRB-

e

m

-RRB-

The

SQHL

Scheme

Input

Source

s

-LRB-

sn

-RRB-

si

is

the

coordinate

of

s

in

the

ith

dimension

si

ki

ki

is

the

radix

in

the

ith

dimension

A

given

grouping

of

w

destinations

in

the

rth

dimension

g

dwg

where

ti

j

is

the

coordinate

of

di

in

the

jth

dimension

-LRB-

ti

j

kj

kj

is

the

radix

in

the

jth

dimension

-RRB-

Output

Leader

node

dl

g

such

that

dl

can

cover

all

the

members

of

g

with

a

single

multidestination

worm

Procedure

if

sr

c

then

Choose

dl

-LRB-

dl

g

-RRB-

as

the

leader

such

that

di

g

-RRB-

tl

r

ti

r

else

Choose

dl

-LRB-

dl

g

-RRB-

as

the

leader

such

that

di

g

-RRB-

tl

r

ti

r

endif

Figure

Leader

node

choice

in

the

SQHL

Scheme

Figure

shows

the

message

pattern

for

multicast

B

-LRB-

from

Fig.

-LRB-

b

-RRB-

-RRB-

using

the

SQHL

scheme

on

a

mesh

Since

the

source

for

the

multicast

-LRB-

-RRB-

is

in

the

right

upper

quadrant

the

leaders

are

chosen

on

the

rows

to

the

right

end

of

the

mesh

and

the

leaders

are

chosen

on

the

columns

to

the

upper

end

of

the

mesh

Figure

shows

the

patterns

of

multidestination

worms

that

get

generated

for

different

positions

of

the

source

in

a

mesh

with

large

destination

sets

It

is

assumed

that

grouping

is

done

first

along

the

x

dimension

Depending

on

the

position

of

the

source

the

set

tends

towards

one

of

the

edges

and

the

set

towards

one

of

the

corners

of

the

mesh

The

first

set

of

multidestination

worms

-LRB-

arrows

marked

as

in

Fig

-RRB-

go

from

the

nodes

to

the

nodes

and

the

next

set

of

worms

-LRB-

arrows

marked

as

-RRB-

go

from

the

nodes

to

the

remaining

destinations

The

Umesh

stage

of

the

multicasts

is

not

shown

for

clarity

This

scheme

has

potential

to

reduce

node

contention

compared

to

the

HL

scheme

destinationsource

BRCP

worm

Unicast

worm

level

leader

level

leader

Figure

Message

pattern

for

multicast

B

-LRB-

from

Fig.

-LRB-

b

-RRB-

-RRB-

using

the

SQHL

scheme

-LRB-

a

-RRB-

-LRB-

b

-RRB-

-LRB-

c

-RRB-

-LRB-

d

-RRB-

Figure

Multidestination

worm

patterns

generated

by

the

SQHL

scheme

when

the

source

is

in

the

-LRB-

a

-RRB-

left

upper

quadrant

-LRB-

b

-RRB-

left

lower

quadrant

-LRB-

c

-RRB-

right

upper

quadrant

and

-LRB-

d

-RRB-

right

lower

quadrant

of

the

mesh

Reduced

Contention

for

Multiple

Multicast

Let

us

take

the

example

from

Fig.

Multicasts

A

and

B

have

identical

destination

sets

with

sources

at

-LRB-

-RRB-

and

-LRB-

-RRB-

respectively

With

the

SQHL

scheme

the

message

pattern

for

multicast

A

remains

the

same

as

that

of

Fig.

-LRB-

a

-RRB-

However

for

multicast

B

the

new

pattern

is

shown

in

Fig.

Since

A

and

B

have

their

sources

in

different

quadrants

of

the

mesh

their

and

sets

are

different

This

leads

to

zero

node

contention

Thus

in

general

the

SQHL

scheme

is

found

to

be

better

than

the

HL

scheme

for

multiple

multicast

Analytical

Estimate

of

Node

Contention

Let

us

analyze

an

estimate

of

the

node

contention

when

using

the

SQHL

scheme

for

multiple

multicast

on

a

k-ary

n-cube

Again

let

us

assume

that

there

are

S

multicasts

and

all

multicasts

have

identical

source-destination

sets

of

size

D

-LRB-

S

D

-RRB-

which

leads

to

identical

leader

sets

Theorem

The

latency

of

a

multiple

multicast

pattern

with

complete

overlap

of

source-destination

sets

-LRB-

S

sources

and

-LRB-

D

-RRB-

destinations

for

each

source

with

identical

source-destination

sets

S

D

-RRB-

on

a

k-ary

n-cube

using

the

SQHL

Scheme

is

-LRB-

min

-LRB-

S

kn

-RRB-

e

Pni

min

-LRB-

kn

i

S

-RRB-

-RRB-

and

is

-LRB-

max

-LRB-

S

-LRB-

-RRB-

-RRB-

-RRB-

steps

Proof

Based

on

the

grouping

characteristics

of

the

SQHL

scheme

it

can

be

easily

observed

that

the

lower

bound

occurs

when

there

is

uniform

distribution

of

sources

in

the

mesh

On

the

other

hand

the

upper

bound

occurs

when

maximum

number

of

multicasts

have

their

sources

in

the

same

quadrant

Similar

to

the

reasoning

in

the

proof

for

Theorem

let

us

consider

a

node

x

from

the

one

of

the

highest

leader

sets

Lg

The

number

of

sources

in

the

same

quadrant

is

for

the

uniform

distribution

case

For

high

values

of

D

jLgj

ss

and

g

n.

Therefore

the

number

of

start-ups

that

x

undergoes

is

given

by

S

This

gives

us

the

lower

bound

of

-LRB-

max

-LRB-

S

-LRB-

-RRB-

-RRB-

-RRB-

steps

Consider

maximum

number

of

multicasts

having

their

sources

in

the

same

quadrant

This

number

will

be

atmost

min

-LRB-

S

kn

-RRB-

All

these

multicasts

have

identical

jLgj

sets

Therefore

x

undergoes

atmost

e

start-ups

for

the

Umesh

phase

of

each

of

the

multicasts

Subsequently

x

undergoes

min

-LRB-

S

kn

-RRB-

start-ups

as

an

Ln

leader

min

-LRB-

S

kn

-RRB-

start-ups

as

an

Ln

leader

and

so

on

Like

in

the

proof

for

Theorem

there

is

an

extra

start-up

added

to

the

upper

bound

independent

of

whether

x

is

one

of

the

sources

This

gives

us

the

upper

bound

of

-LRB-

min

-LRB-

S

kn

-RRB-

e

Pni

min

-LRB-

kn

i

S

-RRB-

-RRB-

steps

A

small

experiment

was

conducted

to

estimate

the

number

of

sequentialized

steps

that

multiple

multicast

incurs

for

the

SQHL

Scheme

with

complete

overlap

of

source-destination

sets

on

a

mesh

Like

previous

experiments

all

system

parameters

were

ignored

and

only

a

high

startup

time

was

considered

The

experimental

values

were

averaged

over

runs

and

rounded

off

to

the

nearest

integer

The

fourth

row

of

Table

shows

the

results

of

this

experiment

The

analytical

values

in

the

third

row

of

Table

represent

the

worst

case

and

therefore

act

as

loose

upper

bounds

Nevertheless

the

experimental

values

fall

between

the

two

bounds

Based

on

the

lower

and

upper

bounds

derived

in

Theorems

and

it

can

be

easily

observed

Observation

The

upper

bound

on

the

latency

of

a

multiple

multicast

pattern

while

using

the

SQHL

scheme

is

less

than

the

upper

bound

on

the

latency

of

the

same

pattern

while

using

the

HL

scheme

Also

the

lower

bound

for

the

SQHL

scheme

is

consistently

below

the

lower

bound

for

the

HL

scheme

Therefore

the

SQHL

scheme

promises

much

better

performance

than

the

HL

scheme

for

multiple

multicast

Potential

for

Improvement

Though

the

SQHL

scheme

promises

better

performance

there

are

some

cases

where

the

SQHL

scheme

performs

poorly

For

example

if

two

-LRB-

or

more

-RRB-

multicasts

have

their

sources

in

the

same

Table

Experimental

and

analytical

latency

of

multiple

multicast

-LRB-

in

number

of

steps

-RRB-

using

the

SQHL

scheme

on

an

mesh

for

different

values

of

S

and

D.

Complete

overlap

of

source-destination

sets

is

considered

-LRB-

S

D

-RRB-

S

D

Max

Exp

Min

quadrant

of

the

mesh

the

groupings

and

the

choice

of

leader

sets

for

both

-LRB-

or

all

-RRB-

the

multicasts

are

identical

This

leads

to

high

node

contention

Let

us

consider

two

multicasts

C

and

D

with

identical

destination

sets

and

-LRB-

-RRB-

and

-LRB-

-RRB-

as

their

respective

sources

on

a

mesh

The

destination

distribution

and

resultant

multicast

patterns

are

shown

in

Fig.

Since

both

sources

lie

in

the

same

quadrant

the

and

sets

are

identical

This

leads

to

node

contention

and

results

in

higher

latency

for

both

multicasts

An

ideal

algorithm

should

use

source-dependent

information

to

minimize

node

contention

further

We

achieve

this

in

our

next

algorithm

destinationsource

BRCP

worm

Unicast

worm

-LRB-

a

-RRB-

level

leader

level

leader

destinationsource

BRCP

worm

Unicast

worm

-LRB-

b

-RRB-

level

leader

level

leader

Figure

Message

pattern

indicating

node

contention

between

two

multicasts

with

identical

destination

sets

using

the

SQHL

scheme

-LRB-

a

-RRB-

multicast

C

with

source

at

-LRB-

-RRB-

and

-LRB-

b

-RRB-

multicast

D

with

source

at

-LRB-

-RRB-

The

Source

Centered

Hierarchical

Leader

-LRB-

SCHL

-RRB-

Scheme

The

Algorithm

Let

us

consider

a

multicast

with

source

s

and

destination

set

D.

Let

D

-LSB-

fsg

Like

the

HL

and

SQHL

schemes

the

groupings

are

done

along

the

dimensions

for

systems

supporting

dimension

order

routing

Each

grouping

is

partitioned

into

atmost

two

sets

and

each

of

them

is

given

a

leader

node

according

to

the

algorithm

in

Fig

Once

the

groupings

have

been

done

the

remaining

steps

are

the

same

as

the

HL

scheme

The

SCHL

Scheme

Input

Source

s

-LRB-

sn

-RRB-

si

is

the

coordinate

of

s

in

the

ith

dimension

si

ki

ki

is

the

radix

in

the

ith

dimension

A

given

grouping

of

w

destinations

in

the

rth

dimension

g

dwg

where

ti

j

is

the

coordinate

of

di

in

the

jth

dimension

-LRB-

ti

j

kj

kj

is

the

radix

in

the

jth

dimension

-RRB-

Output

Leader

nodes

for

g

such

that

they

can

cover

all

the

members

of

g

in

a

single

startup

Procedure

Partition

g

into

and

such

that

di

-RRB-

ti

r

sr

and

di

-RRB-

ti

r

sr

if

is

non-empty

then

Choose

dl

-LRB-

dl

-RRB-

as

the

leader

for

that

di

-RRB-

ti

r

tl

r

endif

if

is

non-empty

then

Choose

dl

-LRB-

dl

-RRB-

as

the

leader

for

that

di

-RRB-

ti

r

tl

r

endif

Figure

Leader

node

choice

in

the

SCHL

Scheme

In

effect

while

dealing

with

the

rth

dimension

each

group

g

from

the

HL

scheme

is

partitioned

into

atmost

two

groups

and

as

follows

An

-LRB-

n

-RRB-

dimension

plane

Pr

is

drawn

through

the

source

perpendicular

to

the

rth

dimension

All

the

nodes

in

g

lying

to

one

side

of

the

plane

-LRB-

including

nodes

on

the

plane

-RRB-

are

added

to

and

the

remaining

to

Each

of

these

groups

has

a

leader

node

as

close

to

Pr

as

possible

Let

us

consider

a

mesh

with

dimension-order

routing

and

an

example

multicast

with

source

at

-LRB-

-RRB-

as

shown

in

Fig

It

is

assumed

that

grouping

is

done

first

along

the

x

-LRB-

-RRB-

dimension

The

nodes

are

on

both

sides

of

-LRB-

plane

corresponding

to

the

th

dimension

-RRB-

which

is

the

vertical

line

V

drawn

through

the

source

The

leader

nodes

are

now

grouped

column-wise

The

nodes

lie

on

both

sides

of

the

horizontal

line

H

drawn

through

the

source

Thus

the

set

will

be

located

close

to

and

around

the

source

node

In

the

SCHL

scheme

it

can

be

observed

that

jLij

can

potentially

be

atmost

times

the

jLij

of

the

HL

scheme

Assuming

m

levels

of

groupings

and

Lm

as

the

leader

set

in

the

original

HL

scheme

-LRB-

-RRB-

e

startups

are

required

for

the

Umesh

stage

of

the

SCHL

scheme

The

remaining

m

multidestination

worm

phases

take

m

startups

Therefore

the

maximum

number

of

destinationsource

BRCP

worm

Unicast

worm

V

H

level

leader

level

leader

Figure

Message

pattern

for

an

example

multicast

using

the

SCHL

scheme

startups

required

for

single

multicast

is

-LRB-

-LRB-

-RRB-

e

m

-RRB-

-LRB-

-LRB-

jLmj

-RRB-

e

-RRB-

Reduced

Contention

for

Multiple

Multicast

Let

us

take

multicasts

C

and

D

from

Fig.

with

sources

at

-LRB-

-RRB-

and

-LRB-

-RRB-

respectively

The

multicast

message

patterns

for

C

and

D

using

the

SCHL

scheme

are

shown

in

Fig.

It

can

be

observed

that

the

intersection

of

the

leader

sets

of

C

and

D

is

almost

negligible

This

results

in

reduced

note

contention

as

compared

to

the

SQHL

scheme

Therefore

the

SCHL

scheme

shows

significant

potential

for

performance

improvement

for

multiple

multicast

destinationsource

BRCP

worm

Unicast

worm

-LRB-

a

-RRB-

level

leader

level

leader

destinationsource

BRCP

worm

Unicast

worm

-LRB-

b

-RRB-

level

leader

level

leader

Figure

Message

patterns

for

two

multicasts

with

identical

destination

sets

using

the

SCHL

scheme

-LRB-

a

-RRB-

multicast

C

and

-LRB-

b

-RRB-

multicast

D.

Multicasts

C

and

D

are

from

Fig

Building

an

analytical

model

for

the

SCHL

scheme

is

a

very

complex

task

To

estimate

the

node

contention

a

small

experiment

was

conducted

to

estimate

the

number

of

sequentialized

start-ups

that

multiple

multicast

incurs

for

the

SCHL

Scheme

with

complete

overlap

of

source-destination

sets

on

a

mesh

Like

previous

experiments

all

system

parameters

were

ignored

and

only

a

high

startup

time

was

considered

The

third

row

of

Table

shows

the

results

of

this

experiment

These

results

were

compared

with

the

number

of

steps

using

the

SQHL

scheme

It

can

be

easily

observed

that

SQHL

produces

lower

latency

for

smaller

values

of

D

but

as

the

value

of

D

increases

the

SCHL

scheme

promises

much

better

performance

than

the

SQHL

scheme

for

multiple

multicast

Table

Experimental

latency

of

multiple

multicast

-LRB-

in

number

of

steps

-RRB-

using

the

SCHL

scheme

and

the

SQHL

scheme

on

an

mesh

for

different

values

of

S

and

odd

and

even

D.

Complete

overlap

of

source-destination

sets

is

considered

-LRB-

S

D

-RRB-

s

D

j?j

SCHL

SQHL

Simulation

Experiments

and

Results

We

simulated

and

compared

the

performances

of

the

various

algorithms

for

multiple

multicast

A

flit-level

wormhole-routed

simulator

WORMULSim

-LRB-

built

using

CSIM

-LSB-

-RSB-

-RRB-

was

used

to

model

the

communication

steps

and

evaluate

the

algorithms

For

all

the

experiments

we

assumed

system

parameters

representing

the

current

trend

in

technology

The

parameters

used

were

ts

-LRB-

communication

start-up

time

-RRB-

microsecs

tphy

-LRB-

link

propagation

time

-RRB-

nanosecs

tnode

-LRB-

router

delay

at

node

-RRB-

nanosecs

tsw

-LRB-

switching

time

across

the

router

crossbar

-RRB-

nanosecs

tinj

-LRB-

time

to

inject

message

into

network

-RRB-

nanosecs

and

tcons

-LRB-

time

to

consume

message

from

network

-RRB-

nanosecs

The

message

length

was

assumed

to

be

a

constant

flits

A

mesh

was

used

for

the

simulations

and

mesh

for

the

simulations

The

tnode

parameter

for

the

multidestination

simulations

was

fixed

at

nanoseconds

to

take

care

of

the

additional

routing

overhead

compared

to

unicast

message

passing

-LSB-

-RSB-

To

keep

the

comparison

fair

the

number

of

consumption

channels

was

fixed

at

for

and

for

in

both

unicast

and

multidestination

schemes

-LSB-

-RSB-

Each

experiment

was

carried

out

times

to

obtain

average

results

For

each

simulation

we

considered

the

two

cases

a

-RRB-

worst

case

scenario

of

total

overlap

of

destination

sets

and

b

-RRB-

random

overlap

of

destination

sets

In

reality

concurrent

multiple

multicasts

of

an

application

could

lie

anywhere

in

the

range

of

random

to

complete

overlap

of

destination

sets

For

each

of

the

above

cases

we

plotted

multiple

multicast

latency

against

a

-RRB-

number

of

destinations

-LRB-

d

-RRB-

while

keeping

number

of

sources

-LRB-

S

-RRB-

fixed

and

b

-RRB-

number

of

sources

-LRB-

S

-RRB-

while

keeping

d

fixed

The

former

shows

the

effect

of

the

number

of

multicasts

on

latency

while

the

latter

shows

the

effect

of

the

size

of

the

multicasts

on

latency

In

the

complete

overlap

case

the

source-destination

sets

were

chosen

identical

for

S

D.

For

S

D

it

can

be

noted

that

it

is

not

possible

to

generate

identical

source-destination

sets

Therefore

in

our

complete

overlap

experiments

we

generated

source-destination

sets

as

close

to

identical

as

possible

for

S

D.

Multicast

Algorithms

using

Unicast

Messages

We

simulated

multiple

multicast

for

different

values

of

S

and

d

on

and

systems

using

both

Umesh

and

SPUmesh

algorithms

and

compared

them

Meshes

Figures

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

show

the

comparison

for

complete

overlap

of

source-destination

sets

It

can

be

observed

from

Fig.

-LRB-

a

-RRB-

that

the

SPUmesh

algorithm

outperforms

the

Umesh

algorithm

for

both

sizes

of

multicasts

and

for

both

ranges

S

D

and

S

D.

For

a

large

number

of

multicasts

-LRB-

S

-RRB-

the

SPUmesh

algorithm

is

about

times

better

for

a

large

d

-LRB-

-RRB-

and

about

times

better

for

a

small

d

-LRB-

-RRB-

Even

for

a

small

number

of

multicasts

-LRB-

S

-RRB-

the

SPUmesh

algorithm

is

about

times

better

Also

the

SPUmesh

algorithm

performs

equally

well

for

both

large

and

small

d

This

is

because

increasing

d

does

not

increase

the

node

contention

as

the

multicast

trees

are

perfectly

staggered

for

identical

destination

sets

On

the

other

hand

there

is

a

marked

difference

between

the

d

-LRB-

-RRB-

and

d

-LRB-

-RRB-

using

the

Umesh

algorithm

This

is

because

increasing

d

increases

the

number

of

startups

the

common

center

node

will

have

to

sequentialize

This

trend

can

be

seen

more

clearly

in

Fig.

-LRB-

b

-RRB-

As

predicted

in

Theorem

the

Umesh

algorithm

performs

worse

for

even

d

-LRB-

odd

D

-RRB-

than

for

odd

d

-LRB-

even

D

-RRB-

in

the

S

D

range

Figures

-LRB-

c

-RRB-

and

-LRB-

d

-RRB-

show

the

comparison

for

random

overlap

of

source-destination

sets

It

can

be

observed

in

Fig.

-LRB-

c

-RRB-

that

the

SPUmesh

algorithm

still

outperforms

the

Umesh

algorithm

for

large

d

by

a

factor

of

about

This

is

clear

in

Fig.

-LRB-

d

-RRB-

where

the

Umesh

latency

shoots

up

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

a

-RRB-

identical

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

b

-RRB-

identical

destination

sets

fixed

S

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

c

-RRB-

random

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

d

-RRB-

random

destination

sets

fixed

S

Figure

Multiple

multicast

latency

on

a

mesh

using

Umesh

-LRB-

um

-RRB-

and

SPUmesh

-LRB-

sp

-RRB-

algorithms

Complete

overlap

of

source-destination

sets

with

-LRB-

a

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

b

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

Random

overlap

of

source-destination

sets

with

-LRB-

c

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

d

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

for

larger

d

whereas

the

SPUmesh

latency

remains

more

or

less

constant

The

latency

values

are

not

as

high

as

the

identical

destination

case

because

the

node

contention

is

less

-LRB-

since

destinations

are

chosen

randomly

-RRB-

It

can

also

be

observed

that

unlike

the

complete

overlap

case

the

Umesh

algorithm

performance

does

not

depend

on

whether

d

is

odd

or

even

Meshes

Figures

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

show

the

comparison

for

complete

overlap

of

source-destination

sets

The

results

are

similar

to

the

case

Figure

-LRB-

a

-RRB-

shows

that

like

in

the

case

the

SPUmesh

algorithm

outperforms

the

Umesh

algorithm

by

a

factor

of

for

a

large

S

-LRB-

-RRB-

and

by

a

factor

of

for

a

small

S

-LRB-

-RRB-

for

both

ranges

of

S

D

and

S

D.

Also

similar

to

the

case

there

is

minimal

change

in

latency

as

d

increases

for

the

SPUmesh

algorithm

as

compared

to

the

Umesh

algorithm

This

is

shown

in

Fig.

-LRB-

b

-RRB-

Also

as

predicted

the

Umesh

performance

is

worse

for

even

d

-LRB-

odd

D

-RRB-

than

for

odd

d

-LRB-

even

D

-RRB-

in

the

S

D

range

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

a

-RRB-

identical

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

b

-RRB-

identical

destination

sets

fixed

S

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

c

-RRB-

random

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

d

-RRB-

random

destination

sets

fixed

S

Figure

Multiple

multicast

latency

on

a

mesh

using

Umesh

-LRB-

um

-RRB-

and

SPUmesh

-LRB-

sp

-RRB-

algorithms

Complete

overlap

of

source-destination

sets

with

-LRB-

a

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

b

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

Random

overlap

of

source-destination

sets

with

-LRB-

c

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

d

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

Figures

-LRB-

c

-RRB-

and

-LRB-

d

-RRB-

show

the

comparison

for

random

overlap

of

source-destination

sets

Again

the

results

are

similar

to

the

case

The

figures

show

that

the

SPUmesh

algorithm

outperforms

the

Umesh

algorithm

by

a

factor

of

about

and

the

Umesh

latency

shoots

up

for

larger

degrees

of

multicast

sets

whereas

the

SPUmesh

latency

remains

more

or

less

constant

Again

unlike

the

complete

overlap

case

the

Umesh

algorithm

performance

does

not

depend

on

whether

d

is

odd

or

even

The

above

results

lead

to

the

conclusion

that

node

contention

in

the

Umesh

and

the

SPUmesh

algorithms

does

not

vary

with

change

in

number

of

dimensions

of

the

mesh

This

is

because

the

multicast

tree

is

generated

according

to

the

dimension-ordered

chain

and

is

not

affected

by

the

dimensionality

of

the

mesh

Multicast

Algorithms

using

Multidestination

Messages

We

simulated

multiple

multicast

for

different

values

of

S

and

d

on

and

systems

using

HL

SQHL

and

SCHL

schemes

and

compared

them

Meshes

Figures

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

show

the

comparison

for

complete

overlap

of

source-destination

sets

It

can

be

observed

in

Fig.

-LRB-

a

-RRB-

that

for

large

d

the

SCHL

scheme

is

about

times

better

than

the

the

SQHL

scheme

and

about

times

better

than

the

HL

scheme

A

property

of

the

HL

scheme

is

that

latency

reduces

with

increase

in

d

-LSB-

-RSB-

This

can

be

noted

from

Fig.

-LRB-

a

-RRB-

as

the

graphs

for

d

are

lower

than

the

graphs

for

d

This

basic

property

of

the

HL

scheme

is

best

exploited

by

the

SCHL

scheme

This

can

be

noticed

in

Fig.

-LRB-

b

-RRB-

where

on

increasing

d

the

latency

of

SCHL

drops

substantially

It

goes

from

microsec

-LRB-

d

-RRB-

to

about

microsec

-LRB-

d

-RRB-

a

reduction

by

a

factor

of

about

For

smaller

d

the

SQHL

scheme

outperforms

the

SCHL

scheme

since

the

number

of

startups

at

the

Umesh

stage

of

the

multicast

is

more

for

the

SCHL

scheme

A

larger

d

offsets

this

overhead

and

the

SCHL

scheme

starts

outperforming

the

SQHL

scheme

at

d

greater

than

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

a

-RRB-

identical

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

b

-RRB-

identical

destination

sets

fixed

S

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

c

-RRB-

random

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

d

-RRB-

random

destination

sets

fixed

S

Figure

Multiple

multicast

latency

on

a

mesh

using

HL

-LRB-

hl

-RRB-

SQHL

-LRB-

sqhl

-RRB-

and

SCHL

-LRB-

schl

-RRB-

schemes

Complete

overlap

of

source-destination

sets

with

-LRB-

a

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

b

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

Random

overlap

of

source-destination

sets

with

-LRB-

c

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

d

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

Figures

-LRB-

c

-RRB-

and

-LRB-

d

-RRB-

show

the

comparison

for

random

overlap

of

source-destination

sets

The

HL

scheme

undergoes

a

large

degree

of

node

contention

even

for

randomly

chosen

destinations

It

can

be

observed

in

Fig.

-LRB-

c

-RRB-

that

the

SCHL

scheme

still

does

about

times

better

than

the

SQHL

scheme

and

about

times

better

than

the

HL

scheme

Also

like

the

identical

destination

sets

case

the

SQHL

scheme

outperforms

the

SCHL

scheme

for

small

d.

Again

there

is

a

crossover

point

after

which

the

SCHL

scheme

performs

better

This

can

be

seen

Fig.

-LRB-

d

-RRB-

Meshes

Figures

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

show

the

comparison

for

complete

overlap

of

source-destination

sets

The

SCHL

and

SQHL

schemes

outperform

the

HL

scheme

Like

in

the

case

the

SCHL

scheme

best

exploits

the

basic

HL

scheme

property

This

can

be

observed

Fig.

-LRB-

a

-RRB-

and

more

clearly

in

Fig.

-LRB-

b

-RRB-

It

can

be

seen

in

Fig.

-LRB-

b

-RRB-

that

on

increasing

d

the

latency

of

the

SCHL

decreases

substantially

from

microsec

-LRB-

d

-RRB-

to

about

microsec

-LRB-

d

-RRB-

Also

as

in

the

case

the

SCHL

graphs

crossover

to

outperform

the

SQHL

graphs

but

at

a

much

larger

d

This

is

because

the

radix

of

each

dimension

in

the

mesh

is

only

which

does

not

allow

the

grouping

potential

of

the

SCHL

scheme

to

be

realized

completely

as

compared

to

a

mesh

Figures

-LRB-

c

-RRB-

and

-LRB-

d

-RRB-

show

the

comparison

for

random

overlap

of

source-destination

sets

As

in

the

case

the

HL

scheme

undergoes

a

large

amount

of

node

contention

and

performs

poorly

It

can

be

observed

in

Fig.

-LRB-

c

-RRB-

that

the

SCHL

scheme

still

does

about

times

better

than

the

SQHL

and

about

times

better

than

the

HL

scheme

Here

too

the

SQHL

scheme

outperforms

the

SCHL

scheme

for

small

d

but

there

is

a

crossover

point

after

which

the

SCHL

scheme

performs

better

as

can

be

observed

in

Fig.

-LRB-

d

-RRB-

The

above

results

in

Section

and

Section

lead

to

the

conclusion

that

node

contention

is

one

of

the

principal

reasons

for

poor

performance

of

existing

multicast

algorithms

for

multiple

multicast

Reducing

node

contention

should

be

the

primary

focus

while

designing

multicast

algorithms

Comparing

Unicast-based

and

Multidestination-based

Schemes

We

compared

multicast

algorithms

using

unicast

messages

with

those

using

multidestination

messages

This

measures

the

potential

benefits

that

a

system

provides

if

it

supports

multidestination

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

a

-RRB-

identical

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

b

-RRB-

identical

destination

sets

fixed

S

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

c

-RRB-

random

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

d

-RRB-

random

destination

sets

fixed

S

Figure

Multiple

multicast

latency

on

a

mesh

using

HL

-LRB-

hl

-RRB-

SQHL

-LRB-

sqhl

-RRB-

and

SCHL

-LRB-

schl

-RRB-

schemes

Complete

overlap

of

source-destination

sets

with

-LRB-

a

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

b

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

Random

overlap

of

source-destination

sets

with

-LRB-

c

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

d

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

wormhole

routing

We

compared

the

SPUmesh

-LRB-

the

best

unicast-based

algorithm

-RRB-

with

the

SCHL

-LRB-

the

best

multidestination-based

scheme

-RRB-

scheme

Meshes

Figures

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

show

the

comparison

for

random

overlap

of

source-destination

sets

For

large

d

the

SCHL

scheme

outperforms

the

SPUmesh

algorithm

as

S

increases

by

a

factor

of

almost

This

can

be

observed

in

Fig.

-LRB-

a

-RRB-

and

Fig.

-LRB-

b

-RRB-

The

figures

also

shows

that

the

SPUmesh

algorithm

shows

better

results

for

small

d

There

is

a

crossover

point

at

around

d

after

which

the

SCHL

scheme

keeps

improving

The

reason

is

hierarchical

grouping

of

destinations

gives

relatively

large

leader

sets

for

small

d

This

is

because

the

destinations

are

randomly

scattered

in

the

mesh

On

the

other

hand

SPUmesh

generates

perfectly

staggered

trees

for

identical

destination

sets

But

as

d

increases

the

advantage

of

efficient

grouping

overcomes

the

node

contention

This

results

in

better

performance

by

the

SCHL

scheme

As

noted

earlier

the

latency

of

SPUmesh

remains

steady

with

increase

in

number

of

destinations

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

a

-RRB-

identical

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

b

-RRB-

identical

destination

sets

fixed

S

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

c

-RRB-

random

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

d

-RRB-

random

destination

sets

fixed

S

Figure

Multiple

multicast

latency

on

a

mesh

using

SPUmesh

-LRB-

sp

-RRB-

and

SCHL

-LRB-

schl

-RRB-

Complete

overlap

of

source-destination

sets

with

-LRB-

a

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

b

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

Random

overlap

of

source-destination

sets

with

-LRB-

c

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

d

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

Figures

-LRB-

c

-RRB-

and

-LRB-

d

-RRB-

show

the

comparison

for

random

overlap

of

source-destination

sets

Unlike

the

identical

destination

set

case

the

SCHL

scheme

outperforms

the

SPUmesh

algorithm

for

both

large

and

small

d

This

can

be

seen

in

Figure

-LRB-

c

-RRB-

and

Fig.

-LRB-

d

-RRB-

-LRB-

to

a

larger

extent

-RRB-

Since

the

destination

sets

are

completely

random

the

phenomenon

of

identical

sets

does

not

occur

for

sources

on

the

same

column

Hence

there

is

no

node

contention

to

degrade

the

performance

of

the

SCHL

scheme

Also

the

grouping

effect

takes

over

for

larger

d

to

give

a

factor

of

about

improvement

over

the

SPUmesh

algorithm

Meshes

Figures

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

show

the

comparison

for

complete

overlap

of

source-destination

sets

Figure

-LRB-

a

-RRB-

shows

that

for

large

d

-LRB-

-RRB-

the

SCHL

scheme

outperforms

the

SPUmesh

algorithm

but

for

smaller

d

-LRB-

-RRB-

this

is

not

true

This

is

because

as

explained

before

the

radix

of

each

dimension

in

the

mesh

is

only

which

does

not

allow

the

grouping

potential

of

the

SCHL

scheme

to

be

realized

completely

Figure

-LRB-

b

-RRB-

shows

the

crossover

point

-LRB-

d

-RRB-

where

the

SCHL

scheme

starts

performing

better

than

the

SPUmesh

algorithm

The

reason

for

the

large

crossover

value

is

again

the

small

radix

of

the

mesh

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

a

-RRB-

identical

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

b

-RRB-

identical

destination

sets

fixed

S

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

sources

-LRB-

S

-RRB-

-LRB-

c

-RRB-

random

destination

sets

fixed

d

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

Number

of

destinations

-LRB-

d

-RRB-

-LRB-

d

-RRB-

random

destination

sets

fixed

S

Figure

Multiple

multicast

latency

on

a

mesh

using

SPUmesh

-LRB-

sp

-RRB-

and

SCHL

-LRB-

schl

-RRB-

Complete

overlap

of

source-destination

sets

with

-LRB-

a

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

b

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

Random

overlap

of

source-destination

sets

with

-LRB-

c

-RRB-

fixed

d

-LRB-

-RRB-

with

varying

S

and

-LRB-

d

-RRB-

fixed

S

-LRB-

-RRB-

with

varying

d.

Figures

-LRB-

c

-RRB-

and

-LRB-

d

-RRB-

show

the

comparison

for

random

overlap

of

source-destination

sets

Figure

-LRB-

c

-RRB-

shows

that

the

SCHL

scheme

outperforms

the

SPUmesh

algorithm

for

both

large

d

-LRB-

-RRB-

and

small

d

-LRB-

-RRB-

Figure

-LRB-

d

-RRB-

clearly

shows

that

the

SPUmesh

algorithm

performs

almost

as

well

as

the

SCHL

scheme

till

about

d

After

that

point

the

SCHL

clearly

outperforms

the

SPUmesh

Like

the

case

this

is

due

to

phenomenon

of

identical

sets

not

occuring

for

sources

on

the

same

column

for

the

SCHL

case

These

results

lead

to

the

conclusion

that

systems

providing

multidestination

message

passing

can

support

more

efficient

multicast

than

systems

providing

unicast

message

passing

only

Scalability

with

Increasing

System

Size

Finally

we

compared

the

five

multicast

algorithms

for

varying

system

sizes

We

simulated

multiple

multicast

patterns

using

system

sizes

of

and

for

a

-RRB-

S

and

d

N

and

b

-RRB-

S

N

and

d

where

N

for

an

n

n

system

size

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

System

size

-LRB-

a

-RRB-

identical

destination

sets

S

d

N

um

sp

hl

sqhl

schl

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

System

size

-LRB-

b

-RRB-

identical

destination

sets

S

N

d

um

sp

hl

sqhl

schl

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

System

size

-LRB-

c

-RRB-

random

destination

sets

d

N

S

um

sp

hl

sqhl

schl

50 100 150 200

Multiple

Multicast

Latency

-LRB-

in

microseconds

-RRB-

System

size

-LRB-

d

-RRB-

random

destination

sets

S

N

d

um

sp

hl

sqhl

schl

Figure

Multiple

multicast

latency

on

varying

system

sizes

using

Umesh

-LRB-

um

-RRB-

SPUmesh

-LRB-

sp

-RRB-

HL

-LRB-

hl

-RRB-

SQHL

-LRB-

sqhl

-RRB-

and

SCHL

-LRB-

schl

-RRB-

schemes

Complete

overlap

of

source-destination

sets

with

-LRB-

a

-RRB-

d

N

and

S

and

-LRB-

b

-RRB-

d

and

S

N

Random

overlap

of

source-destination

sets

with

-LRB-

c

-RRB-

d

N

and

S

and

-LRB-

d

-RRB-

d

and

S

N

Figures

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

show

the

comparison

for

complete

overlap

of

source-destination

sets

Figure

-LRB-

a

-RRB-

shows

that

the

new

algorithms

scale

better

with

increasing

system

size

than

the

existing

algorithms

for

multiple

multicast

It

can

be

observed

in

this

graph

that

the

SCHL

scheme

is

the

only

one

which

shows

constant

performance

with

increasing

system

size

for

large

values

of

d.

Figure

-LRB-

b

-RRB-

also

shows

that

the

new

algorithms

are

more

scalable

with

increasing

system

size

for

large

values

of

S.

Figures

-LRB-

c

-RRB-

and

-LRB-

d

-RRB-

show

the

comparison

for

random

overlap

of

source-destination

sets

Like

in

the

complete

overlap

case

Fig.

-LRB-

c

-RRB-

shows

that

the

new

algorithms

scale

better

with

increasing

system

size

than

the

existing

algorithms

for

multiple

multicast

Again

it

can

be

observed

in

this

graph

that

the

SCHL

scheme

is

the

only

one

which

shows

constant

performance

with

increasing

system

size

for

large

values

of

d.

Figure

-LRB-

d

-RRB-

also

shows

that

the

new

algorithms

are

more

scalable

with

increasing

system

size

for

large

values

of

S.

Conclusions

In

this

paper

we

have

introduced

the

concept

of

designing

multicast

algorithms

with

minimized

node

contention

so

that

multiple

multicast

patterns

can

be

implemented

with

reduced

latency

We

have

shown

that

the

existing

multicast

algorithms

-LRB-

Umesh

and

HL

schemes

-RRB-

for

wormhole

k-ary

n-cube

systems

lead

to

poor

performance

when

multiple

multicasts

are

involved

The

poor

performance

has

been

explained

by

node

contention

Node

contention

arises

from

the

fact

that

the

existing

multicast

algorithms

were

designed

with

single

multicast

in

mind

We

have

presented

a

detailed

analysis

of

the

node

contention

that

results

when

existing

multicast

algorithms

are

used

for

multiple

multicast

A

new

approach

has

been

proposed

which

uses

source-specific

information

to

create

multicast

trees

Using

the

new

approach

three

new

multicast

algorithms

-LRB-

SPUmesh

SQHL

and

SCHL

-RRB-

have

been

developed

The

node

contention

in

these

algorithms

have

been

analyzed

for

multiple

multicast

patterns

and

shown

to

be

less

with

respect

to

the

existing

algorithms

The

new

algorithms

have

been

demonstrated

to

perform

times

better

than

the

Umesh

and

the

HL

schemes

under

multiple

multicast

patterns

Thus

these

algorithms

demonstrate

a

lot

of

potential

for

designing

scalable

collective

communication

libraries

on

current

and

future

wormhole

systems

By

comparing

the

SPUmesh

with

the

SCHL

it

has

been

demonstrated

that

multidestination

message-passing

is

a

desired

feature

for

future

wormhole

systems

to

implement

efficient

collective

communication

operations

In

this

paper

we

have

only

emphasized

on

multiple

multicast

patterns

We

are

currently

evaluating

other

patterns

like

multiple

barriers

and

reduction

operations

It

will

also

be

interesting

to

see

how

much

benefits

these

new

algorithms

can

provide

when

integrated

with

application

characteristics

on

distributed

shared

memory

systems

with

cache-coherency

Additional

Information

A

number

of

related

papers

and

technical

reports

are

available

electron

ically

through

the

home

page

of

Parallel

Architecture

and

Communication

-LRB-

PAC

-RRB-

research

group

The

URL

is

Alternatively

the

papers

can

be

obtained

by

anonymous

ftp

from

ftp.cis.ohio-state

edu

cd

pub/communication

References

-LSB-

-RSB-

A.

Bar-Noy

and

S.

Kipnis

Multiple

Message

Broadcasting

in

the

Postal

Model

In

International

Parallel

Processing

Symposium

pages

-LCB-

-LSB-

-RSB-

M.

Barnett

S.

Gupta

D.

G.

Payne

L.

Shuler

R.

van

de

Geijn

and

J.

Watts

Interprocessor

Collective

Communication

Library

-LRB-

Intercom

-RRB-

In

Scalable

High

Performance

Computing

Conference

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broadcast
collective communication
meshes
multicast
wormhole routing

unicast and multidestination message passing