Finding Disjoint Paths in Networks - CiteSeerX

Finding

Disjoint

Deepinder

Sidhuand

Paths in Networks* Raj

Department University

Nairand

of

Computer

of

Maryland

Baltimore,

Shukri

Abdallah

Science —

MD

BC

21228

and Institute

for

Advanced of

University

Computer Maryland

Park,

college

Abstract Routing

is an

been

extensive

rithms

for

it

is

important

single

tween

pairs

paths

can

presents

rithm

that

of the

rithm,

the

and

compare

ing

algorithm.

requires paths

congestion

3 to

of

disjoint

it

paths.

We

the

We

of comparable

and

algo-

shortest

describe

the

conclude

that

fewer

messages

to

have

initial-link-disjoint

The

method

find

nate

discover

research

was supported

in part

of Defense

at the University

of Maryland

and conclusions

contained

those

of the authors

resenting

the official

the Department Permission granted direct title that

commercial

and should policies,

of Defense

to copy

provided

that

without

not

either

by the Department Baltimore

in this

be interpreted expressed

are

of

the

that

first

link).

[3, 4] was

disjoint

paths

node

to

of minimum

work

in

total

cost

Alternate

algorithms

of

a desti-

the

a destination.

non-disjoint

or implied,

are not made

the ACM

copyright

or distributed notice

otharwise,

and/or specific permission. @ 1991 ACM O-8979 J.444.919J

or to republish,

requires

cost

algorithm

of

tion

is

we

path

[7, 8] find

alter-

paths.

Section

and the

In this

a fee

1.50

43

node.

algorithm

the

paper the

the

present

from

other

any

design

the

node this

transforma-

in

the

set

SZ.

as follows. of

of

every

the

simulation

algorithm

a disS’r,

methods,

graph

path

the

of

a set,

is organized

compare

and 4, we

paths Unlike

shortest

present

3, we

design

constructs

require

the

of the

2, we

Section

Section

not

retains

rest

present which

node-disjoint

does

and

The

for

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paper, algorithm

z to a destination

as rep-

fee sII or part of this material

this

minimum

County.

document

or the U.S. Government.

the copies

advantage,

To copy

and

topol-

paths

extended

paths

to

in

every [6]

routing

disjoint

tributed

of the publication and its date appear, and notice is given copying is by permission of the Association for Computing

Machinary.

node

network

node

quality.

The

views

from

1< disjoint

every

the

augmentation

paper

disnode

every

multiple

a pair

cost

later

distance-vector

algorithm

of

[2] finds

construct

design

multiple

requires

(disjoint

total A

In *This

to

the

a destination

[1]

of link-disjoint

to

finding to

knowledge

are

to

for

scheme

[5]

devoted

node

solution

complete

in

been

every

One

Another

from

rout-

to

a network.

[5]

simulation

our

in

nation.

algo-

disjoint-path

from

minimum

path

has

algorithms

paths

used

a desti-

effort

distributed

ogy.

This

to

using

another

4 times

between

paths

performance

with

disjoint

distance-vector

includes

of joint

be-

packets.

disjoint

algorithm

paths

a network

dropped

multiple

its

in

Considerable

instances,

Multiple

Introduction

has algo-

bandwidth

a distributed

evaluate

many

disjoint

effective

reduce

finds

The

as one

multiple

probability

paper

In

in a network.

increase

the

nation.

20742

in

There

in distributed

routing.

have

of nodes

of nodes,

reduce

interest

path to

implemented

net works.

research

desirable

pairs

function

communication

CP

—

MD

1

computer

Studies

described

conclusions.

In

algorithm. results in

of [5].

2

Constructing

In

this

section,

strutting every

of minimum

to

a destination

Notations

A

network

is denoted

this

link

(1)

both

distinguished

node;

free

FIFO

but

changes

occur

destination

that

which

is the

arbitrary

the

destina-

provides

error-

its propagation node

de-

sends,

within

if Figure

finite

and

and (5) no topolog-

the

construction

1:

Example

branch

of an

identification

f-segment,

branch-link

list

of the

links,

where a node

a tree

of node node

link path

WC.

of its

shortest

node

z to

destination

z) and

any path A jerk

Q is denoted

bors.

An

ment

that

and

a leaf

node.

shortest its

cost

branch

a pair link,

together

tree

~2)(u1, uniquely

tree, the

from

(u,

f),

the

identifies

maximum In

The

branch

prefix node

h

neigh-

node

seg-

between

44

branch lists

c(l”(h,

z))

h is up-tree

are different

or a fork

or neither

and

the other,

node

from

z .

node

with

links

(U2, ~2) ~2 and

h,

of node

with

is greater (down-tree) neither

respect

h is neither

1.

to node

z. If

h is a proper

list

of node

x,

r.

If

the

branch

and

h

are

identi-

(less)

than

of node list

tree

x can be up-

of node

z

the

the leaf node

node

nodes

(~2, U2, ~1) link

and

list

=

the only list of z

identifies

j2)

z)

the forks

f2

from of

T(2,

the

(U2, ~2) and

branch

identification

is down-tree

identification

(u3,

the fork

identification

of the

overlap

and ~Z are identification

The

neighbor,

down-tree branch

of branch

1,

between

link

identifi-

f-segments

the f-segments

segment

identifies

the

Figure

branch

identify

between

u,

A branch sequence

z) z).

~l)).

a tree

cost of

path

T(z,

A horizontal

are denoted

of forks

path

(us, j2)

a

y. Ev-

up-tree

x(+ provide

is ((u2,

cal and

two or more

is a fork-free

which

segment

path The

a node

~1, and

by a

is the iden-

respectively.

either A

is

neighbors.

of node

neighbor, (u, j).

a minimum-length

list,

of

identifies

by c(Q).

with

f-segment

connects

connected

which

The

z and z))

link

up-tree

and (~2, U3, 1) respectively.

path

y, z) is assigned

identifier

c(T’(z,

is a node

nodes

(z,...,

path,

link

prejerred

A non-tree

its

containing

jI ($, U1, jl, u2, ~2, Z) where forks in 7’(z, z). The branch

path

shortest

and

links, shortest

neigh-

the

the

is the identifier

tifier

shortest

y is called

horizontal

z) =

x has a path

a link up-tree.

(down-tree)

z.

cation

and non-tree

links

on the

Two

are called Q(z,

whereas

j,

the f-segment

z, to

z. For a down-tree

Node

which

identifier

node

lies

destination

a horizontal

Every

tree

a fork,

z

towards

is pointing

z as it provides

z to

horizontal

E into

link

y (cc).

of node

neighbor

pointing

is an up-tree

x (y)

each node,

down-tree

to destination

(z, y), node

from

the leaf nodes of

at destination

G. A link

is pointing

the links

from

by T(x,

2’, rooted

cost paths

towards

T divides

ery

V

with

assume

an

called

(4) each

during

tree,

the destination

pdh

is

link and

z in graph

pointing

called

y, in

We

E.

paths.

cent ains minimum

from

V

a message;

A shortest-path

bor

z and associated

there

z, in

graph, set,

to its neighbor(s)

receiving

from

a link

number

(2)

finite;

a message

alternate

cost

C(Z, y).

communication

after

ical

by

and

of nodes,

an operational

is arbitrary,

paths a network.

undirected

V,

The

node,

(3)

required, time

y).

directions;

and

in

an

set,

C(Z, y) is a positive

cost in

tion

lay

(z,

is represented

the

same

as

a pair

by

disjoint

con-

Concepts

a vertex

connects

and

for

approach

node

modeled

with

an cost

and

is

(V, E), link in E

G =

Paths

propose

a set node

2.1

A

we

Multiple

is a proper up-tree

c(T(z, z)),

z. If the lists nor

prefix down-tree

of

2.2

Problem

Statement

bil : branch

identification

sending The goal of this

paper

is to find

for every

node

node

list

with

of

the

the form

z # ((nl,n~)(n2jn~)...(nm,~~))

z a set S$ of disjoint node

z in a graph

The

PI

P2

paths G with

node

Any

two

the

x to the paths

x to a destination

the following

set S$ contains

from

from

shortest

path

destination

where

properties:

T’(z,

z)

A path more

For

P4

k horizontal

each

{2’(x,

path

z)}

and

node-disjoint c(P)

~

any path from

c((z,

(z, Y)Q’(y,

z)}

z)

contains

y)Q(y,

the

path

z))},

Z) contains

in

(~, y) Q’(y,

every

only

branch-links

path

of the node

ALT messages no

links.

(z, y)Q(y,

than

ALT(nid,

constructs P4

from

the set SZ of paths every

node

c(Q)

destination

Distributed

to the multiple

assumes

each node

tree neighbors. P4

graph

disjoint

knows

Disjoint

are constructed

from

or

of the

the

: path

identifier

its preferred

paths

by the

paths

PID messages

propagate

est path

path

with

and up-

properties

exchange

of two

An

Pl-

PID(nid,

nid : identifier

possible

al-

possible

and

alternate

links

in

the

path.

important

information

about

and branch

The

general

form

types of the

are used

short-

node

sending

Cst : cost of the shortest node

identifier

path

from

This about

of this

scheme

distributed

algorithm

the

node

z starts

path

a

identi-

a distributed

neighbors

after

PID messages from

receiving

sages move

up-tree

node

sends

a

and

horizontal

towards

links

after

Each

in the up-tree camies

of the

two

destination

the

PID message

down-tree.

link

to inform

the

com-

sending

by the up-tree

tal

45

and

PID(z, O,–, ()) messages to z. The PID messages are propagated

by

message

and

PID messages

identifier

of paths

in a set of disjoint

neighbors.

destination

from

node

disjointedness

a path

uses

its path

each neighbor

the

the

identify

of its horizontal

putation

to the destination, of the sending

of our

to determine

to uniquely

The

identifi-

cst, pid, bil) where: of the

aspect

Scheme

labeling scheme that labels each node with an Path identifiers identifier called the path identifier,

message,

pid : path

des-

is a

node

identifiers

to all nodes.

sending

path

to the

in the

Labeling

fiers

is

links

of horizontal

problem

Messages

message

node

of the

path

nhl : number

and

lists

the

path,

alternate

P 1 –

paths.

cation

sending

alternate

sending

bis : set of branch

G.

costs,

node

tination,

z.

PID and ALT, over the links

of messages,

or hor-

of such a mes-

Algorithm

The solution that

form

the

cst, pid, his, nhl) where:

possible

2.3

about

down-tree

general

Cst : cost of the possible

[

that

properties

information

message,

k horizontal

an algorithm

with

z to the

the mes-

up-tree,

The

ternate we propose

shortest

sending

propagate

: identifier

nid

is

SC – {P z

sage is

links.

section,

on the

through

neighbors.

S$ –

z) which

in

C(Q))

more

set

paths

izontal

pid

In the next

1 < i < m are the

sage.

Q, Q’ in SC are node-disjoint.

Q in the set S. – {?’(z, than

n~),

z.

alternate P3

(ni,

node

leaf

nodes.

Every

of its

up-tree

link

direction,

PID messages

a

z

PID mes-

on each

receiving tree

z. The

PID message one PID

carries

and each horizonsent

by

the

end

nodes.

PID message

No

is sent

over

a down-tree

Node

z,

from

link. An

up-tree

receiving sends

neighbor,

z,

PID message

a

a PID(z,

of the

over

destination,

the

C(Z, z), z, bilZ)

tree

link

message

disjoint

on

neighbor,

u.

((u, x));

otherwise

message

it receives

On receiving

a

sage from

its

ate

z learns

node

over

to each up-

The branch

about

in the

leaf

(c(T(y,

link,

2.o

identification

of the

neighbor

b“ t

uses

PID messages disjoint

link.

The

3.o

neighbor

iden-

for

each up-tree b. +

3.2

the

if v is preferred

leaf 3.2.1

are

3.3

its neighbors

used

to inform

more

than

from

neighbor.

over

hori-

destination,

its

identifier

z)}

z,

h,

learns

checks

path, C(X, z)

on

about

if

Figure

x does

received

over

not

and

its

horizontal

Q

z

saves

the

its down-tree

messages

and horizontal

ALT message node

PID

it receives

by

=

cludes

from

in

has

a cost

y of node through

(a, h)

x receives

a

it

the

re-

the branch

identi-

path

a neighboring

each

identifiers

46

node

cost

and

node

y.

greater

from

path

neighbor

two

alternate are

which

y.

any

other

including

in

This

is achieved

received

in the

ex-

Q and neighbor

a path S.

in

path

Sm, it

every

have

Q in

path

intersects

For

an

path

other

Q in

x does not

information from

than

c(Q).

the

any

x includes

than

y describing

z includes

y, x reconsiders by

node

cheaper If

S. every x, if node

saved

Any

PID or an ALT

Q,

if Q is disjoint

ALT messages

neighbors through

for constructing

path

set S.

offered

PID

the

from

Sz of lower

a

a hor-

link

uofzdo

U,Z)}

Ub’

set for an alternate

alternate

x

be included

propagate

LJ {(

b’

2: Rules

message

(z, z),

receiving from

neighbor

h

link

a path,

Q can

each up-tree

on

horizontal

are

node,

Node

When

b’ U bisZ U {(v,

the

one

offered

and the last

neighbor

then

else bv +

any further.

ceives

u of z do

~)}

if bisz # 0 then b. t

one horizontal

it has a disjoint

path

and

node

or horizontal

bv+--bv

with

z, of the

that

neighbor

Every

neighbor

and z is a fork

the

it

then

z), (v, z)}

L U {(u,

b. +

to inform

with

neighbor,

intermediate

message

b’ U bisz U {(y,

PID

Pat hs

PID message and

set SC.

of x then

of x and z is a fork

b. t

at the

after

PID(h, c(l’(h, z)), pid~, bil~) message

the

then

else

fication

z)

x then

if bisz # 0 then

it

respectively.

(x, h)T(h,

of node

bu+-bvUb’

tree

paths

paths

(z, z), determines

izontal

z then

Ubism

z)}

if y is up-tree 3.1.1

zont al link.

An

of node

neighbor

if v is an up-tree 3.1

PID messages on their

ALT messages about

cost

z.

as follows:

else

alternate

whose

of node

b’ U bisz

b“ *

and

outgoing

terminates

shortest-path

about

a

bisz is the

path

else

this

x is a fork,

neighbor

b’ U {(y,

u.

propagation

Disjoint

A horizontal

o. In this figure,

of z and x is a fork

for

receiving

alternate

b’

links.

neighbors

cost

y), p’, b., no) mes-

set bv is constructed

if y is up-tree

to each of its hor-

of the

nodes have sent and received

A node

c’+c(z,

set of the shortest

if u is a horizontal 2.1

@ZV,

.z)) + c(y, x)),

If node

bd field

in the

identifier

(u, z), to its branch

to each up-tree

horizontal

c’, p’, b’, n’ ) message

the minimum

neighbor

if v is the preferred ~v +

(z, z).

bdv. It propagates

list

PID message

nodes

path

neighbors.

the branch

The

link

1.0

y, an int ermedi-

PID messages

and up-tree

message

its

path

information

list

a horizontal

ree neighbor

identification

tification

ALT(g,

sends an ALT(z,

identification

PID(y, c(T(g, .z)), pidv, bilv) mes-

down-t

cost of its shortest

appends

path,

branch

an

y describing

If

its branch

izontal

receiving

sage to every eligible

(x, z),

bil$ is equal to x is a fork, bilr is (). Node r also sends a PID(x, C(X, z), z, ()) message to each horizontal neighbor. The destination node z ignores any PID tree

on

neighbor

in S’z

the

path

by

using

PID or last

y. paths

different

are disjoint

and

there

if their is no

path

common

branch

link

absence

that

alternate

and

the

shortest

the branch

and

alternate

of node

node

links

z. Figure

identification this

it sent

such

path

a path.

Neighbor

rules

node

z gets

an

ALT(y,

node

y describing

an alternate

cost

c’ + C(Z, y)

which

disjoint

neighbors

is less

path,

about

node

the

neighbor

path.

In

the

disjoint

neighbor

is equal

than

new In

first

set,

of – 1 to

the previously

second

node

v.

If v is a horizontal

identification

Figure

2.

The

message

describing

set

neighbor

may

not

Figure

4 shows

the

the branch where

the

receive

the

the

application

R5

function

bor.

identification

k’igure bors

projects

In

in a PID from ber of

a leaf

neighbors message

before

all its neighbors.

ALT messages

sent by a leaf up-tree

neighbor

agated

any

node.

node

about

it receives

This

helps

containing Any

z does

any path

any

nV

reduce

ALT message

Destination

3: Rules

not

node v

received

that is either

If node

from

up-tree

v is up-tree

from

of

the shortest

of z, v will

node

learn

cc propagates

about

I

information

the tree path

P is not node

from

x to v.

disjoint

v will

from

the shortest

disregard

this

message.

for determining

eligibility

of neigh-

ALT messages

ALT messages. it receives.

sage propagation The

“else”

3.2.1

will

not

topology Figure

the

and 2 may

ternate

disjoint

details

and

node

may This

will

part of

in some special

con-

Any

set is empty

message

not

“if” It

cases which cost

cause path

from

solutions,

ALZ’ mes3.1.1

be noted the

z to exclude its

2. and

that,

on the network

assignments,

a node

ALT

in Figure in

should depend

iden-

any

to

the rules statements

be needed. link

branch

modification

simplify

the

whose

propagate

set Sz.

rules

in

some

al-

For

further

see [9].

Termination The

the num-

z is not z does

node z.

P along

path

inform

sent

neighbor

P is not disjoint

of v, then

to receive

tification

P it received PID messages

sub-optimal

z of the destination

further.

not

z)) which

message

P

node v will’

an

set of an

addition,

its eligible

that

eli-

sets of alternate

“fbi~(m)”

path

node x.

the path

Node v is not an up-tree neighbor of x, and path P is, non- disjoint from the shortest path of node v. If the

of-

ALT message m. A node, x, does not send ALT messages before neighit receives a PID message from its preferred branch

a PID

than

v about

alternate

P because

alternate

ALT message for

ji-om

the path

path

cancel-

of rules

identification

identifier

node

of v. If v is down-tree

about

as shown

receiving

path

inform

from

the path

the path

P.

the new path

structing

b. is determined

in

I

path

n’. The

to

of

the constraint

node z is ((v, z)Z’(Z,

about

or down-tree

ev-

neighbor,

nV is equal

x will

the number

z

z sends to every

node

because

P exceeds

is a horizontal

z, then

two

node

to v. in the path P exceeds

statement.

the cheapest

Node v

eligi-

w to cancel set,

R4

cheapest using

in path

z does not

know

forma-

z.

from

its

P

a cost

will

P with

its

path

the

with

to n’ + 1; otherwise,

branch

links

because

3. When

z informs

v

disjoint

prevents

neighbor,

message

Node v has a diflerent Node

them

path

.4LZ’(Z, c’ + C(C,y), p’, b., n.) message

paths,

R3

path links

v is a horizontal

the problem

is deter-

in Figure

disjoint

of horizontal

If node

horizontal

neigh-

c’, p’, b’, n’) message

ALT messages. sends an ALT message

ery eligible

eligible

eligibility

outlined

of

lation

k.

set of an

ALT messages informing

the

alternate

its

to a neighbor alternate

to z by v. This

not send it an ALT

2 cent ains the rules

cost

In this case, node x sends the second

alternate

The number

path. iden-

and

using

in

cheapest RZ

set with

in the branch

message

minimum

F’ath P was oflered tion of a loop.

of the cost of its cheap-

mined

gible

RI

set

path

send an ALT

of a new

P if:

is

shortest

it

have through

bors to whom

fered

for a node

x does not

path

of an

identification

z keeps track disjoint

sets

path

set of the

the branch

est alternate

ble

informing

path.

Each

about

sets

set of the alternate

Z’(Z, Z) is the set of branch for constructing

Node

two

do not inter-

comparing

identification

list

The

disjointness

a branch

path

sets.

in the

paths

The

identification

branch

tification

link

shortest

by constructing

the

The

branch

f-segment,

path

checked

identification

the two alternate

a common

for

branch

of a common

indicates sect

in their

distributed

minates

when

computation no node

of this

in the network

paths

ALT message to any of its neighbors.

to an

each

prop-

node

complete given

send

47

has found algorithm

in [9].

a set of disjoint and

proof

of its

algorithm

ter-

can send an At this

time,

paths.

The

correctness

is

fbk(ma) = fbk(ml)

u {(y,x),(x,d)]

Figure

5:

Shortest

path

tree

from

a node

to the

for

an example

net-

work

rows fbls(mz) = fbi~(ml)

shortest

u {(x,d),(y,x),(v,x)}

rooted

path

that

node

at z and has leaf

nodes

Figures

from

destination

6 and

7 illustrate

z defines

to

z.

This

e, g, ~ and the

th~

tree

i:

c.

exchange

of PIZ

ALT messages respectively. To trace the prop by PID or othe agation of ALT messages triggered ALT messages, ends of arrows are tagged. A ta~ by concatenating thl of an ALT message is formed tag of the message which triggered this ALT mes and

Figure

4:

cation

sets

Examples

of changes

to branch

identifi-

Example Figure with

5, shows

The

integer

tree

shows

by solid

directed lines.

with

shortest node

of a network a link

paths

A connected

and

the

every

of directed

some

identifier

its neighbors.

by

“lb”

ar-

neighbors

48

leaf and

node “lb”

node

b and

the

h when

node.

in this

z starts F’lD

c sends about

receiving

situations

by sending

to

The

of the

special

destination

computation

The

node links

the

out

The

are indicated

non-tree

sequence

G

on it.

is its cost.

from

Z. The tree links

arrows

graph

T, superimposed

tree,

associated the

the destination dotted

an example

a shortest-path

sage and point

the

messages

distribute to each

ALT

messages,

path

Q

it receives

=

W

example.

(c, z),

c

tagge to it

PID messag~

9 .........m...Kkm7&+.

Y’!’KE4

-’e”)y:~ .y -....!

.-

-z

“---’’-”’ -~..=L&’’=,-,

Figure

6: PID

Node c considers

link.

node h as an eligible

of both

horizontal

branch

f

respectively,

are neither

nor proper

horizontal

F’lD

messages

travel

over

links (~, b) and (h, c) do not initiate

messages because identical

which

path

the nodes

identifiers.

b, j,

the

h, and c have

The same is true for the

node

tagged

“2hc”

message tagged

alternate

path

“2h”

neighbor,

to inform

Q = (~, d, a, z) with

of node

of nodes e and d. The node

b is down-tree because

from

informing

The

it does not

have any

ALT

message

c the

it about

its second short-

Q = (h, f, d, a, z) because node c

path

h with

the minimum

node

disjoint

.P = (h, C, z).

path

f.

message to its

to offer.

h sends node

is providing

3

the ALT

identifier

alternate

link

(a, e). The leaf node ~ sends its down-tree

path

The

est disjoint

ALT

PID messages which travel over the horizontal

neighbor

alternate

pre-

fixes of each other. The

from that

b because

down-tree

b)) and ((h, b))

identical

does not send ALT messages to

leaf node g does not send an ALT

from node c since

lists,((c,

is different

neighbor

the nodes are the same and (3) node identification

f

The node

message exchanges

f cloes not send an ALT message to its horizontal

path of node h, (2) the path iden-

his neither up-tree nor down-tree their

7: ALT

nodes e and d because the path

because (1) the path Q = (c, z) is disjoint

from the shortest tifiers

Figure

z, b and h.

from its neighbors neighbor

message exchanges

Simulation

Results

h,

it of an

In this section, on our

one horizontal

49

we present

algorithm

and

simulation

the

results

Ogier-Shacham

based algo-

Network (N, D,d)

Total Me: Sf 3008 1267 1775 5610 1495 882 1241 1470 1746

(20,3,6) (20,7,3) (20,10,3) (20,2,12) (20,5,3) (20,12,3) (20,9,2) (20,5,3) @&L.

ges Ss

Sf 1344 705 651 2026 747 457 587 850 858

6005 5646 6234 8455 5096 2671 4534 4525 3897

Table

rithm.

The

bitrarily

simulation

chosen

shown

in

was performed

networks

column

whose

1 of Table

(IV, D, d) represents

a network

The

are assigned

arbitrary

link

costs

except

(ours) and Shortest-Sum

gorithms paths

respectively.

found

stretch

factor

as c(P(z, results

for a path

z))/c(T(z,

P(z,

z)).

second

column

messages generated disjoint

of

node.

by each algorithm

These results

algorithm

requires

a half times paths

The

number

column found

for every

shortest

path

had the shortest

paths found The fourth

is the

algorithm.

over 9070 of the time. work topology

number

and the link column

that

disjoint

using our algorithm of the two paths

found

using

paths.

column.

In gen-

Using

by

with joint

we

paths

on the net-

minimized

over all possible

We compare

path

requires

factor

plete discussion tiple

the Ogier-Shacham

50

disjoint

pairs

nec-

as one of its disfinds

a pair

of disjoint using

paths.

simulation.

3 to 4 times fewer messages

of our algorithm

paths,

algorithm

does not

approach

to discover paths of comparable

for each node

pairs of

and reduce

the sum of the costs is

the two algorithms

Our algorithm

fac-

such that

our

that

alternative

of disjoint

can be used

between

compare

the shortest

This

paths

packets.

approach

of

This

as one of the

in a network,

of dropped

include

node-disjoint

path

disjoint

distance-

network.

bandwidth

congestion

simulation,

paths.

multiple

the shortest

an alternative

essarily

a distributed

communication

Multiple

the probability

of

the number

and the average stretch found

of messages per path

for finding

includes

nodes, reduce

cost assignments.

paths

the

of Ogier-

Conclusions

we describe

to increase the effective

is the

shows the average stretch

tor of the two shortest

that

3 to 4 times fewer mes-

and

algorithm

algorithm

path as one of the paths

by each node is dependent

requires

Summary

vector

The Ogier-Shacham

In general,

with

the number

the Ogier-

to compare

is shown in the fifth

paths in a computer

of dis-

percentage

path is disturbed,

using

In this paper,

our a~gorithm.

is not one of the two paths found

the Ogier- Shacham algorithm

to find mul-

node in the network.

in parentheses

of our algorithm

This

than

of

one and a half to four and

shows the total

nodes whose shortest

number

the Ogier-Shacham

more messages than

The third joint

from

aigo-

the

every node to every other

show that

the Ogier-Shacham

sages per path.

4

paths from

for

is defined

1 summarizes

shows the total

that

our algorithm

It is important

eral, our algorithm

we use the

z) which

Table

Shacham found.

al-

the quality

algorithms,

we conclude quality,

finds more paths

algorithm.

performance

of the simulations.

The tiple

two

the results,

Our algorithm

costs.

(Ogier-Shacham)

To compare

by the

link

From

paths of comparable

Shacham

the

and “ss” refers to the Shortest-

7 7.90 7.43 8.20 11.13 6.71 5.96 7.31 5.95 6.30

2.24 1.’78 2.73 2.77 2.00 1.93 2.11 1.73 2.03

rithm.

di-

[10] which

I

P th Ss Sf

uses fewer messages than

The notation

“sf”

have unit

for

Messages/

Results

algorithm.

are

last two networks First

~

D and average node degree d. All networks

ameter

(:.71) (8.95) (10.79) (o) (3.29) (0) (6.13) (o)

finding

notation

of lV nodes with

760 760 760 760 760 448 620 760

1: Simulation

on 9 ar-

parameters

1.

Stret~h Fat or Sf Ss -iEim 2.20 2.27 1.37 1.52 1.32 1.19 1.53 1.68 1.33 1.39 1.62 1.54 1.21 1.30 1.12 1.26

‘Total No. ‘aths

No.

see [9].

quality.

For a com-

to construct

mul-

Acknowledgement The authors for helpful

would

like to thank

Mr.

P. Tsuchiya

discussions.

References [1] A. Itah

and M.

proach

Rodeh.

to reliability

In Proc.

adaptive

1984.

path algorithm

in communications

Transactions

ap-

networks.

on FOCS,

A K shortest

routing

multi-tree

in distributed

25th Symposium

[2] D. M. Topkis.

IEEE

The

for

networks.

on Communications,

36,

1988. [3] J. W. Surballe. Networks, [4] J. W.

Disjoint

Surballe

method paths.

and

of finding

[5] R. Ogier

for finding

[6] C. Cheng,

shortest

A

paths

Urbana,

[7] P. F. Tsuchiya. scription

and analysis.

S. AbdaUah,

Submitted

tributed paths. [10] M. and

algorithm Submitted

Garner, M.

and Com1990.

Report

A dis-

path rout-

1990.

and S. Abdallah.

for finding

multiple

for publication,

Frishkopf.

MTR

and R. Nair.

for alternate

V. Haimo,

De-

June 1987.

for publication, R. Nair,

Confer-

hierarchy:

Technical

for total

Allerton

October

MITRE,

tance vector algorithm

[9] D. P. Sidhu,

algorithm

of minimum

The landmark

W87-87WOO152, [8] D. P. Sidhu,

of disjoint

’89.

Control,

Illinois,

al-

and J. J. Garcia-

distributed

disjoint

ence on Communication,

ing.

pairs

In Proc. 28th Annual

puting,

of disjoint

A distributed

S. P. R. Kumar,

K

length.

pairs

IEEE INFOCOM

Luna-Aceves. finding

A quick

14, 1984.

and N. Shacham.

In Pm.

paths.

in a network.

R. E. Tarjan. shortest

Networks,

gorithm

paths

4, 1974.

I. Loobeek,

Type-of-service

A disdisjoint

1990. D.Davis, routing:

Technical ReModeling and simulation. Corporaport 6364, BBN Communications tion,

January

1987.

51