Binary Decision Diagrams - PowerPoint Presentation

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Binary Decision DiagramsA general-purpose toolkit for combinatorial problems Torsten Mütze[Based on D. Knuth, The Art of Computer Programming, Vol. 4]

Binary Decision Diagrams (BDDs)Data structure to represent a Boolean functionD. Knuth 2008: “One of the only really fundamental data structures that came out in the last 25 years.”Very powerful general-purpose toolkit forall fundamental combinatorial problems:countingrandom samplingcomputing minimum/

maximum weight

independent sets

matchingsspanning treesHamilton paths/cycles

colorings

…

Binary Decision Diagrams (BDDs) - Definition

Data structure to represent a Boolean

function

i

i

R2

R1

i

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3

┴

┬

┴

┬

┴

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┴

┬

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┴

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Directed

acyclic graph

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┴

┬

3

must be

ordered

…

…and

reduced

!

i

i

Virtues of BDDsEvaluate for any in stepsFind lexicographically smallest

with in stepsCount solutions to

Count solutions to with forGenerate uniformly random solutionto

Solve over all with

Efficient algorithms

to construct,

combine and modify BDDs

Encode class of

combinatorial objects as solutions to a

boolean

function

Build BDD for

independent sets

matchings

spanning trees

Hamilton paths/cycles

colorings

…

Count all…

Compute the generating function of all…

Sample uniformly at random…

Compute

maximum/minimum weight…

Dynamic programming

on the BDD

Running time polynomial in and

number of

nodes

of the BDD of

small

→

efficient algorithm,

often number of solutions

Recipe for

combinatorialists

Count solutions to

Bottom-up

dynamic program

Decorate BDD

nodes with

counts ‘=‘ #paths from

node to

Example

Count all independent sets of

┴

┬

1

1

0

→ result

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┬

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4

Count solutions to with

for

Bottom-up

dynamic program

Decorate BDD nodes

with

count vectors

Example

Compute the generating function

of all independent sets of

┴

┬

1

(1,0,…,0)

(0,0,…,0)

→ result vector

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2

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(

0,0,0,0,0,0,0)

(

1,0,0,0,0,0,0)

i

i+1

i+1

i

j

k

weighted sum

with binomial coefficients

(

1,6,9,2,0,0,0)

Generate uniformly random solutions to Count solutions bottom-up

Generate random solution top-downby setting the ‘s one after the otherwith the corresponding

probabilitiesExampleSample a uniformly randomindependent set of

Example

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#

ind.sets

=

792.070.839.848.372.253.127

Solve overall with

Bottom-up

dynamic program

Decorate BDD nodes

with

current

maximum

Example

Compute a maximum weight independent set of the USA graph,

vertex weights = postal codes

┴

┬

1

0

→ result

i

j

k

i

i+1

i+1

sums of skipped

positive weights

#

ind.sets

= 211.954.906

etc.

maximum weight =

492

(

2

optimal solutions)

Building BDDs

┴

┬

┴

┬

┴

┬

┴

┬

┴

┬

Atomic BDDs as

building blocks

Operations

on BDDs

i

┴

┬

Single

variable

BDDs

BDDs for symmetric functions

iff

n

n

n

n

n

1

2

2

3

3

3

+reduce

0

1

┴

┴

┴

┬

┴

┴

┴

┴

┴

┬

┬

┬

┬

┴

┴

┬

┬

┬

Building graph BDDs

Independent sets of a graph on verticesSimilar: Colorings of a

graphMatchings of a graph on edges

Somewhat more complicated:

Connected

spanning

subgraphs

Spanning

trees

Hamilton

paths

/

cycles

forbid adjacent vertices

enforce degree <=2

enforce m-1 edges

enforce degree = 2

enforce degree <= 2

Example: USA graphBDD size

CountIndependent setsB(IND) = 428

211.954.9064-colorings

B(4COL) = 25579

25.623.183.458.304

Matchings

B

(MAT) = 3376

105.349.480.877.850

Connected

spanning

subgraphs

B

(CONN) =

43109

393.352.428.583.002.517.479.225.491.456

Spanning

trees

188.245.384.298.803.435.554.555

Hamilton

pathsB(HAMPATHS) = 23385

26.652.195Hamilton cyclesB(HAMCYCLES) = 1

0

via

g.f.

┴

Size of BDDs

ordering: 1,2,3,4,5,6,7,8B(f)=10

ordering: 1,3,5,7,2,4,6,8B(f)=32

Construction of BDDs for a

fixed

variable

ordering

Different

ordering

→ different BDDs, with

possibly

very

different

size

Example

Theorem

[Bollig, Wegener ‘96

]

Computing

an optimal variable

ordering

is NP-hard.

Size of BDDsFew interactions between variables → small BDD

Theorem [McMillan ‘92]If can be computed by such a network, then .

Exampleindependent set BDD of 4x4 grid

B(f)=240

B(f)=92

B(f)=80

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Thank you!Questions?