Cost-Optimal Multicast Trees for Multi-Source Data Flows ... - IRL@tamu

Cost-optimal Multi-source K.

Multicast

Data

Ravindran,

A.

Flows

City Convent

D.

at

Contact

of New

138th

E-mail

Street,

address:

York

(City

New

a graph-theoreticti

constructing ‘steiner

multicast

trees’,

dktribution

is NP-complete.

the

paths,

So, many

of

rate

modeled

as

for a link

heuristics-based

are available to generate near-optimal trees. Typically, an algorithm first assigns the edge costs for til

links,

and then

terconnecting

examines a given

various

set of nodes.

not work well for the evolving figurations

link

for in-

strategy

does

application

con-

and image

distributions)

necessary to construct multiple dktribuone per media data stream. This is be-

cause the overlapping mon

paths

This

multimedia

(such aa audio-video

where it is often tion paths, viz.,

candidate

of multiple

forces the link

lay and bandwidth

tree segments

cost to change,

characteristics

over a com-

based on the de-

of data

streams

flowing

over these segments. Accordkgly, algorithms that hitherto have assumed non-varying link costs during various phases of a run now need to take into account costs as candidate lapping

trees with

are examined

algorithm

runs

different

in a given run.

by examining

the variability

of link

levels of path

over-

In other

words,

as an

paths,

the link

costs

various

change. The paper embarks on a study of heuristics-baaed algorithms to tackle thk ‘modified steiner tree’ problem. The

algorithms

than

feasible

‘steiner

allow

more

otherwise

tree’

with

cost-efficient

routing

a classical

treatment

of data of the

problem.

Keywords:

Multimedia

path’ routing, computation,

distribution

trees,

‘shortest

flow & QOS baaed link costs, ‘steiner tree’ overlapping trees, heuristic tree algorithms.

NY

connecting

should

problem

algorithms

S. Bloom

College)

York,

The links

perspective,

G.

10031

(USA)

edu

engr. ccny. cuny.

tree, From

&& Science

raui@cs.

Abstract

distribution

Loguinov,

of Computer

University

Avenue

for

in Multimedia

Sabbir,

Department

Trees

the nodes in a tree, i.e., edges of the

have enough

of the source

bandwidth

to carry

compressed

video

by dktributed cated

applications

mult imedla

another

in a conference

From routing

session)

may

of data

is often

rezdized

by

tm.e-

structured paths over switching nodes and inter-node links. A tree-structured path consist of a root node where the data

of a source

nodes that nations

is made

perform

at leaf nodes

data

available, routing,

a set of intermediate and one or more

of the tree that

0-7803-7018-8/01/$10.00 (C) 2001 IEEE

consume

desti-

the data.

accessing

[1]. See Figure

perspective,

be defined

network

the

as follows.

consisting

a repliwith

one

1. multipoint Let

P(V,

f)

of nodes V intercon-

nect ed by communicant ion links S, with a link cost assignment function Ce Iv= ~ g (say, baaed on bandwidth dlOCElt ion).

Given

a set of nodes U c V that

contain

user enti-

ties, i.e., nodes representing data sources and de~tinations, it is necessary to find a tree spanning the nodes U and links ~such

that

U g ~ G V and

tree is called ‘P [2].

to construct

for multicaating stream

for

cet is a minimum.

a configuration, multiple

destinations

~

it may

dk.tribution

from

Such a

~e/ &steiner tree for U in the topology

a minimum

Given

paths:

multiple

(e.g., multimedia multicasting

often be necessary one-per-source,

sources to a common conferencing),

multiple

source to a set of destinations TV broadcast).

streams

set of

and one-perof data

(e.g., audlo+video

from

a

in digital

Combinatorics researchers have shown the steiner tree (ST) problem to be NP-complete [3, 4]. Since then, many heuristic-based

algorithms trees

ous candidate

routing

needed

The tree that

users interacting

a graph-theoretical problem

be an undkected

entities. Multipoint

data).

(e.g., clients

web server,

have been proposed

[5, 6, ?].

TypicaJly,

assigns the edge costs for all links,

Introduction

the data

of 2.5 mbps

carries data, i.e., mult icast tree, provides the basic network capability for multi-destination data delivery, as required

near-optimal

1

to support

(e.g., a bandwidth

a path

paths

From with

these paths,

the minimum

cost.

to generate

algorithm

and then examines

for interconnecting

among

a

fist vari-

a given set of user

the algorithm

It is not guaranteed

chooses how-

ever that the algorithm exhaustively searches the space of all possible paths. So there may exist paths that incur less cost thao the path declared as ‘cost-minimal’ by the algorithm. In this paper,

we malyze

the scope of the ST problem

IEEE INFOCOM 2001

+

u

A

d-x

source

source 51

source s?, >..

‘.

Destination .:

u

,..

P

● -.”

... ”

entities for dati

d-v

t

Source of data

u

P.,$ible UIaP in to network ffeve

A

52

(U5W level view)

d-x, d-y, d-z, dv: d-z

d-y

source

sl

“Jg.f&;&%’&Rw

of u?er !evel cOmmunicatiQns realmtton of data paths

(path

P-I)

@ath P-11)

w destination

d

destination

d

,

D

Network

node

—

Nehvork links

—

Network

~

Network node

O link

x : Node where streams of S1and $2are mulhplexed —

d

of tree T ~ Vertex

— Direction

e te

Figure

2: ‘path

sharing’

to reduce

paths

for

multicasting

tions

that

in light

require

and the network

of the evolving

multipoint

multimedia

communications

strategies

that

to optimize

and that

by multiplexing

applica-

among users

attempt

transport

cost

set up for connectover the physical

topolo~, as shown in Figure 2. Suppose s 1 and S2 generate bursty data flows with a per-hop bandwidth of 1 unit (normalized),

and solutions,

Dstspathfor$l

of data flow

Consider, as illustration, a channel ing sources SI and sz to a destination 1: Tree-structured

Dstspatkfnrs2

---->

Edge

@

Figure

Pathfo~#&JtJsxxddats

. . . ...>

gain is achievable

over a shared

data of S1 rmd 92 are considered consisting

the

a 35% bandwidth

these streams

linkl.

individually,

If the

the path

of 4 hops and 3 hops respectively

will

P-I

be set up

consumption of underlying communication resources. As we shall see, the emerging characteristics of multimedia applications and multi-service networks casts the ST problem

wit h a total bandwidth of 7 units (normahzed). With link shaxing between these data taken into account, the path P-II consisting of 5 hops and 4 hops respectively with an

with

overlap

a more complex

what

model

has been assumed

of link

cost assignments

in a hitherto

classical

than

is

streams other

possible

that

the

have some of their

on a common

link.

trees

generated

paths

overlapping

An overlapping

sharing

of the underlying

link

ing the

overall

assignment,

generation link (e.g.,

link

cost

for with

path

resources,

finds

these paths.

with,

A treats

one an-

eliminates

depicts

thereby

the

across the streams

that

and u when

tree

share this link.

Fur-

thermore., a statistical flowing over the link

multiplexing of vwious data streams is possible, which allows reducing the

per-stream

allocations,

data

resource

streams.

For instance,

sources can be supported tree thzm that

possible

video

with

more efficient the variability

with

ined. The tree setup. ceiver

different

levels

from

of path

overlapping

changes

its bandwidth

demand,

timality of the tree may be afected, tree reconfiguration.

0-7803-7018-8/01/$10.00 (C) 2001 IEEE

the global possibly

cost op-

triggering

A next

a

considers of the

to start and

through

P, t

A

needs

For doing

92.

allocation

as apportioned

sz also, it needs path

so,

cost of the links p-to-t, to .s1 from

1 unit

to 0.7

now.

degree of cost optimality by taking

into

account

achievable the

on link costs allows resource-efficient dktributed

sense, particularly multimedia

impact

in tree of path

routing

— in

in the case of geograph-

applications

(such as remote

class rooms over Internet ). The change of link costs with respect to the degree of link sharing however presents a new dimension

computing

changing link costs may influence the overall Also, when a new source joins or when a re-

$1 alone

the nodes p, t and u since it

to the complexity

of ‘tree generation’

because of the need to compute link with respect to the number

are exam-

account

A that

of P-I as cost-minimal

by S1 ad

more cost-minimal

ically

cost variabilas carddate

a total

an ST algorithm

unit each (and so for sz also). With this changed link cost assignment, t hls shared path does in fact turns out to be

a network-wide

trees.

routing will result if one takes of link costs arising from the trees. So link ST algorithms,

and u-to-d

sharing

on a shared

are sent on separate

When

to reduce the bandwidth t-to-u,

into path

through

shared

constructions

2 MPEG

Taking

the cost efficacy

The Klgher

bursty

have assumed constant link of a given run. We however

sharing of links by multiple ity needs to be factored into trees

streams

with

less bandwidth

if they

ST algorithms previously costs during various phases believe that into account

particularly

5 hops.

the

itself. For instance, the fixed cost of using a network tariff per unit of ‘connect time’) can

get amortized

contains

to m-evaluate

be set up, requiring

Consider

the 4-hop

the path

influenc-

and hence

edges will

of 6.9 units.

various

of the problem. It

of 3 of their

bandwidth

treatment

a minimal

path

problems,

the cost variability of each of flows sharing it (besides

length)2.

This complexity

calls

1For example, a link carrying 2 compressed video data streams with a peak rate of 3 mbps each will need to allocate a sustained bandwidth of 4.2 mbps. ‘ZEven with using a weighted sum of tree edges, an ST algorithm

may take into account

only the cost differences

from

IEEE INFOCOM 2001

for a re-exarnination and a possible ification

of currently

introduction

of existing

available

heuristics.

Our

1, as given

ST algorithms

of new heuristics paper

and/or walks

by a relation:

modc:B+Z

through

(1)

for Z~K?+,

these problems and offers solutions. The paper first develops a model of determining the cost of multicast trees. The model factors in the topological

where the cost of an incremental satisfies the condition:

configuration

of trees and the sharing

data streams.

Using

[c(bz + c$b)t – c(bz)t] > [c(bl + c$b)i – c(bl)i] for bz > tn. To satisfy this condkion, c may be drawn from a space

of heuristic-based tree’

problem.

2

‘cost’

the model,

algorithms

of paths

by various

we then embark

on a study

to tackle

the ‘modified

steiner

of ‘monotonic policy

assume only

not ions

in

mult

icast

2.2

net works

communication

strategies.

algorithms

be seen through

That

the development

has assumed

twist

the limiting

steiner

can

allocation

C, to represent

network.

For

trees’

for

a specific

simplicity,

multicast

tree for a configuration

we

routing

consisting

topology

connect ing a set of nodes ~ through

links ~such

that

c(b(e, .)).

of user

‘P(V, &) is an acyclic

graph G(~)

~

6b

case in this paper.

nodes U placed in a physical

of %hort-

an addkional

functions by the

‘st einer

The We first present a canonical view of multicast functions in the network that epitomizes the evolving multimedia est tree’

convex’

implemented

bandwidth

is a minimum,

a set of

where ~ ~ C

VeG~

this view. and UG~GV.

Macro-level

2.1

parameters

for tree

Let L(z,

setup

links The

physical

topology

7(V, E) consists another through

of

the

of a set of nodes a set of links &.

underlying

network

V, connected with one The user entities, viz.,

sources and destinations of multimedia data, reside in dktinct nodes U ~ V, and form the communication endpoints. and

If U, and ud axe the nodes containing

data

destinations

respectively,

source can generate timedia

then

generating

video

~ U.

a configuration,

and destinations

relative

and audio

i.e., the placement to one another

streams),

of sources

in physical

topol-

ogy of the network. When

a link

say that part

1 ~ & is included

1 supports

of multiple

of 1 — denoted b(l,’Z),

b(LT2),

a path

trees

segment

as CAP(1) as

“ ““

over these trees and allocated

as part

b(l, T“’) < [CAP(l)

Since 1 may be capacity into

chunks

by the data streams

flowing

therein

(e.g., a 45 mbps DS3

video streams of 2 mbps for determining whether

of a new

–~b(l,

(1, T).

tree ‘T, we

the bandwidth

— is partitioned

required

line carrying 3 MPEG-2 An algorithmic constraint be included

in a multicast

‘TI, Z,... ,

mult icast

tree

T’

each). 1 can is then:

~) — i.e., 1 should have enough

vi left-over bandwidth capacity to transport the data flowing over ‘T’. Thus, that 1 lies in the shortest path (in terms of the number

of hops)

does not imply

that

1 will

between

a source

be included

and a receiver

as part

of the tree

connect ing them. The

cost of (1, T)

bandwidth

allocated

is dkectly

related

by 1 to transport

one link to another, but not the non-linear sharing across flows in each link.

0-7803-7018-8/01/$10.00 (C) 2001 IEEE

a pair

consisting

of nodes

of a set of adjacent

z, y E ~.

Since

G is

acyclic, there is exwtly one L(x, y) for aay given z and y. A multicast path for carrying a data flow q from a node p to a set of receivers with

root

Ud is then

a subtree

projected

at p and leaves at nodes LLi, given

from

G

as:

A

(e.g., a mul-

and a receiver may consume the data streams generated by various sources. The tuple (Us, Ud, V, S) may be viewed w prescribing

y) C ; be a path connect

data sources

U,, ud

one or more data streams

workstation

that

to the amount the data

of

of ‘T over

effects of bandwidth

To support

the muk icast %OW of q from

tree is first

created

to connect

a set of nodes U and links a multicaet cent ained that

dktribution in

where

!?./d,

p need not

S, which tree

Ud

p to ud, a Steiner

the nodes w~th

C

{p, ud } through

is then root

projected

at ~

U C V and

be the same as the node

and

& C &.

into leaves Note

u E U, where

the source that generates q resides (such as nodes 1 and In3 Figure 3, the multicast trees are 9 in Figure 1). ~; ture

= ; fib = of a multicast

rameter

(~ – .L(x~, w). The topological strucpath depends on the configuration pa-

(US, Ud, V, S) and the bandwidth

cost parameter

{c(b(e))}vecs. For the cases of {P,ud} = V and Iudl = 1, the tree construction can be done in polynomial time — such as Kruskal’s ‘minimum spanning tree’ (MST) algorithm in the former case and Dijkstra’s ‘shortest path’ algorithm in the latter case [7]. !J1-ee construction in other cases is an NP-complete problem [3, 4]. So heuristic-based algorithms

are employed

trees.

A variety

where

[8, 9, 6, 10, 11].

3A

word

~b~ut

that

construct

of such algorithms

the

tfjrm

%ree’,

close-to-optimaf

have been studied

as

else-

used in this paper, is in

place here. ‘steiner tree’ is a graph-theoretic term referring to an acyclic graph constructed over a network of nodes and links, whereas ‘multimat tree’ is a (network) protocol-oriented term referring to the data flow path from a soume to a set of receivers. This paper trsate ‘multicaet treee’ as graph-theoretic projections derived from ‘steiner trees’.

IEEE INFOCOM 2001

2.4

Current

‘st einer

To construct signing

tree’

algorithms

a tree T?, an algorithm may start by asc(~(q))l V1 E S. This requires

edge costs as:

ILL I x ISI steps. The statically assigned link costs and the placement of U in physical topolo~ are used as input pa rameters to the algorithm. Some of the algorithms

are ‘shortest

distance

routing’

and (truncated MSTS’. In the ‘shortest dkt ante routing’, the least cost path from p to each u’ c ud is determined, ‘-’’it=’pa’hf”r””w

........-

Tq’;”

{multicast

path

for

flow’

‘st.rn.r’~.’

‘-”

) II lletwork-orieIlkdview

and then all these paths

q-b

)

a MST

G

II

(jjmph-theoret

is first

In ‘truncated

for (V, E), and then,

MSTS’, edges that

p to any of the nodes Ud are removed from

do not connect

icvim)

are merged.

constructed

the MST. ~-o,x-b:

Node where ilowq-al the

algorithm

considers to Stali

q-b

tl,

f2, t3 :Nodes

containing

receivers

Another which

Figure3:

Multicast

paths

projected

from

steiner

cast tree. Cost A

incurred

cost

specific

assignment

resource

be used

as input

A reiation

to

~ = F

maps

over the link

1:

+

B

(1, ‘T) and t(r) ‘5 mbps

the

rate

flow-

trees.

to a form

computing

to the bandwidth

through

is the required with

positive

So, if the routing

that then

~(r)

self from

~(dr)

for dr >0.

flow

q from

cost minimal

~[~

~

condition:

When

(3)

with

costs

(or

the same

by a chosen algorithm4.

(ST)

problem

complexity

manifests

standpoint

applications

in light

is described

itof

next.

of multicast

trees

a ‘steiner

tree’

is created

multiple

with

sources,

the intent the traffic

of carinterac-

provide

a single

a graph-theoretic

3.1

Sharing

of

data

%OW5. In this

treatment multicast

section,

we

of this problem. paths

bandwidth

the

data

allocation

are the trees that

lapping

attributed

can carry a tree T/[i]

the is

point

problem

tOt-COSt(~~])lj=

in multicast

0-7803-7018-8/01/$10.00 (C) 2001 IEEE

segments

say, q. and

segments node

of T~

and Tz

q~ respectively

l,...,K.

routing.

of the multicast

z and carrying

ceiver t. Referring 4 cOnSt


f,c),

various

feasible

Consider (by

over each link

ity that

link

of qs (generated

an aggregated

non-overlapping former

with

by 93) overlap

allocation

tree for qs with

case is possible

placemeti

the additional

the existing

or to create

individual

if p continues

session. either to a separate

allocation.

The

to be a cost-feasible

bandwidth

allocation

for qs,

and the latter case is necessmy if no cost-feaablle place ment can be found. In other words, if the incremental cost of aggregating rent

tree

sending

out

q3 over a separate

the algorithm

ql @ q2 to send over the cur-

qs with

~1 ~~z turns

non-overlapping

tree should

to be higher

than

In other

alternate

caddate

earlier

tree, the algorithm

for qs. establish

tended

steiner

of final

tree

ity, A’

should

examining link

Before

either

overlap

of

an algorithmic

treatment

of the

ex-

tree problem.

s Note that a cost-feaaible placement some toplogies, such as ‘ring’.

0-7803-7018-8/01/$10.00 (C) 2001 IEEE

of p may not exist in

cost

of paths

with

cost-feasible

considered of link

already

influencing

to achieve

the

better

consid-

generation

cost optimal-

the functionality

paths

are

explores

changeability

of the paths

thereby Thus

paths

A’

a given run of The

revalidation

a possibil-

candidate

in the

of re

presence

of

cost changes. An

OVT

enumerate

construction the

candidate

paths

about

each link.

The flow

able to A’

what

costs

during

a run.

data

are sources

and which

and ud in the physical

is found

that

connects

A general We treat

to which

availof U

of them

Accordingly, with

2.4.

of this

ad-

new candidate paths of various subsets of

ud, in order

This

inover

plezement

Equipped

subtree.

of the new subtree.

is not

are receivers.

of previously

through

requires

placement

the relative

information, A can examine the ag~egated data flows a cost-minimal

This

the

regard

topology.

sources E U, to the receivers

be able to

it sifts

information

without

account

u

can be aggregated

considers

of them

A needs to take into

context

flows

aggregation

topology,

A should

in links

since it merely

in the physical

ditional carrying

algorithm

changes

formation

4.2

is faced with

the

be augmented

the cost minimality

or oth-

when

paths,

various

creates this

the cost feasibility

algorithms

A’

can change.

itself.

estimate

case however,

tree

an OVT, words,

in the run

the cost of

erwise of p for the flow ql @ q2 Q?qs. We are interested in determining the algorithmic complexity involved in this extra computation. We now provide

aa determined

due to topological

costs change as different

explored.

various

Suppose a source 93 needs to join the multicast Two possibilities of bandwidth allocation arise: tree with

creations,

this limcompose

a classical ST algorithm A’ that generates considering each source E Z& sep~ately). when

used for generating

levels of flow aggregation

of ql ~d qx (viz., @{ql }, @{q2}, @{ql, q2}) in the physical topology.

have the path

of NVT

steiner

costs may require

where pset ({s1, s2}, ud, f, c) is the set of cost-feasible placements. To determine pset, an algorithm needs information

by overlap

paths.

ered as cost minimal, {Jqsl,P)

does not take into

are feasible

(4). To overcome algorithm should

criteria

Modified

NVTS

the cost-feasible placements of p under the policy c E C. The candidate paths to be examined by a routing algorithm

the multicast

to a

[c(f (q, Iaq2))t – c(f (q,))t] when m stats sh~iw z with Accordingly, the cost comparison given by (5) requires enumeration

need not be cost-optimal

VW