A Symbolic Algorithm for Maximum Flow in O-1 Networks* {

A Symbolic

Algorithm

for

Maximum

Flow

in O-1 Networks*

Gary D. Hachtel Fabio Somenzi Department of Electrical and Computer Engineering University of Colorado at Boulder, 80309

Abstract We

an

network. tion

matching

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of

than

it

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edge

edges

graphs The

finding

problem

each

layer

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enforced

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network

0

=lm = 10 : = 01 3 = 11

han-

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is

a O-1

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(implicitly)

Disjointness

for

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previously

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defined

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larger

edges).

disjoint

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algorithm

nodes

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1036

algorithm

The

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1 Introduction Sizeis a significant of VLSI

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paths.

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networks

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Icl = IvI = n, and a problem is formulated as

Procedure

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y)

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2

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3

t(Y),

new

i++)

Y))

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F(c,

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g));

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return

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if

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zero;

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v));

t=s

1 4

{U(z,

y),

S(Z,

(E(z, for

y),

(j

=

y))

=makexpotent-ntwk

F(z,

l;j

y),

s(z),

i(y),

r);

< MAX-SWEEPS;

j + +)

{

r’ npush_flow

5

(E(z, if 6

(r

y),

(U(a,

v),

S(C,

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} }

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g),

s(r),

i(y),

2, U(z,y),

T-NOT-REACHABLE)

~

y))

i(Y), Sweep

S(o,

y));

break;

?ls?wO

=update_rpotent_ntwk

~> u(~>Y),

S(*,

newl

new3

newz

Y)); Figure

3:

Edge

Partitioning

in

a Right

Potent

Network.

Phase

} 8.

Eventually, the

Figure

2: Pseudo-Code

for

Procedure

MAXIMLTM_FLOW.

the

augment

the

maximal

amount.

Because erated For

follows. Maximize:

x

algorithms

network. we

set

the

● the

edges

(z,

edge) ● the

or

y

flow

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either . F’(y,

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the

be

tain to

7. The in

least

each

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partitioned

right

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a sequence

of

(BFS).

stored 4),

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from

into

y)

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vertex the

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tion

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what

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for

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to

find

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and

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useful thus

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sets make

U]

6 constitute are

the

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BDD

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backward

Given

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with

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augmenting

(2, 1),

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Section

start-

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graph,

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analysis,

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(3, 6)

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is always

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edges

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right-potent

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found

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layer

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each

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Potent

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+

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the and

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set that

of edges

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make~potent.ntwk

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UB(y,

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a path

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to

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cedure

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of

reached(z)

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backward

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network

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Building

After

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paths.

sweep,

n-L,

in

(2, 3).

most sizes

of a given

example

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shortest.

Network and

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as follows.

3.2

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(O, 2)

u --l(z,

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to

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originally

Because

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than

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sense

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in Figure

Search

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In

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