minato robdd lecture 2

8/9/2019 Minato ROBDD lecture 2

1/31

Algorithm-Tokuron(1st part, lecture-4)

Shin-ichi Minato

Division of Computer Science,Graduate School of Information Science and Technology,

Hokkaido University


2/31


3/31

2012.05.01 Algorithm-tokuron 3

Today’s topic

• Basic data structure

– Shannon’s expansion, Binary tree and BDD, reduction rules.• Properties of BDDs – Comparison with truth-tables and SOP forms. – Uniqueness, compactness, fast logic operations. – BDDs for typical Boolean functions.

– Impact of variable ordering to BDD size• Algorithms for BDD construction

– BDD construction for a given Boolean expression. – Binary logic operation algorithms. – Finding satisfiable/minimal solutions, truth-table density.

• Improvement of BDDs – Shared BDDs for multiple functions. – Negative edge.

BDD（分決定グラフ）


4/31


BDD（Binary Decision Diagram)

（分決定グラフ）

a

b b

c c c c

0 0 01 1 1 1 1 0 1

a

b

c

10

1

0

1

0

0 1

a

b c

10

1

0

10

b1

c

0

10

Binary Decision Tree（真理値表と等価なBDD）

Reduced Ordered BDD（既約な順序付きBDD） Unordered BDD（既約でも順序付きでもないBDD）


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Nodes（節点）and edges（枝）in BDDs

• Decision node（分岐節点） • Terminal node（終端節点）

• 0-edge（0-枝）

• 1-edge（1-枝）

• Sub-graph（部分グラフ）

0 1

a b c

1

0

F0 F1

v10

F

Shannon’s expansion（シャノンの展開） (Boole’s expansion)

F (v, X ) = ~v F (0, X ) + v F (1, X )


6/31


ROBDD (Reduced Ordered BDD)

• In general, BDD is not unique for a Boolean function.

• Ordered BDD：

– A fixed total order relation is defined for the variables.

– On each path from the root node to a terminal node,

any pair of variables follows the total order relation.

• Reduced BDD

– A BDD form where the two reduction rules are applied as much

as possible.

• ROBDD has the most important properties. – In this lecture, we may use “ BDD” for discussing ROBDD.


7/31


BDD reduction rules

(a) Eliminate all redundant nodes.(b) Share all equivalent nodes.

x

f f

(jump)

x

f 0 f 1

x x

f 0 f 1

(share)

Reduced BDD.

(cf.)

When using the rule (b) only,

we can obtaina “ quasi-reduced” BDD.

（準規約なBDD）

(a)

(b)


8/31


Example of BDD reduction

• Final BDD form is anyway unique.

• A unique (canonical) form for a Boolean function.

– However, different variable orders may produce different forms.

(occasionally same but in general different.)

a

b b

c c c c

0 0 01 1 1 1 1

a

b b

c c

01 1

a

b

c

01 1


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Properties of BDDs

• Canonical form for a Boolean functions. – Easy to equivalence checking

• Compact for many practical Boolean functions. – Linear size for n-bit parity functions and n-bit full-adder func.

– More than 100 inputs can be handled for good cases.

• Binary logic operations between two BDDs can be

performed in almost l inear t ime. – NOT operation is also easily done.

• BDDs cannot be compact in some Boolean functions. – Exponential size for n-bit multiplier functions.

• BDDs becomes large if variable ordering is poor. – NP complete problem to get an optimal variable order.(Some heuristic methods have been developed, however.)


10/31


Size of BDDs

• Total number of n input Boolean functions: 22 n

To distinguish all the functions, at least 2

n bit needed.

• Also for BDDs, the worst case of BDD size is:

O(2 n / n) nodes.

– O( n) bit for each node, so O(2 n) bit in total.

• However, for many practical Boolean functions,

the number of BDD nodes are within a polynomial of n.


11/31


n-input AND, OR, EXOR functions

AND（論理積） OR（論理和） EXOR (parity)

（排他的論理和）

• Number of BDD nodes is linear for n: O( n)

• Swapping 0- and 1-terminal nodes gives NAND/NOR/XNOR.

0 1

x1

x2

x3

x4

10

x1

x2

x3

x4

10

x1

x2

x3

x4

x2

x3

x4


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n-bit binary-coded arithmetic addition

• Increasing n, BDD grows vertically but the width is bounded. BDD size: O( n) nodes.

• Subtraction function is also O( n) nodes. • Equality/inequality function is also O( n) nodes.

S0

S1

S2 CBDD for the 3-bit adder function.

(Variable order: a2 , b2 , a1 , b1 , a0 , b0 , c0 )


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n-input data selector

• Boolean function selecting

one data from n data inputs.

• [log2 n] control input specifies

which data input is selected.

• BDD size: O( n

) nodes.


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Symmetric function （対称関数）

• Symmetric functions:

– Exchanging variable order

has no effect to outputs.

– Output value only depends

on the number of “1” in the

n-inputs.

• Each level of the BDDshows the number of “1”

in the upper k -inputs.

– BDD width is up to n.

– Total BDD size: O( n2) nodes.

example of 9-inputsymmetric function.


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Effect of variable ordering (1)

• BDD size may greatly depend on the variable order.

x1 x2 + x3 x4 + x5 x6 + x7 x8 x1 x5 + x2 x6 + x3 x7 + x4 x8

O( n) O(2 n)

Eff t f i bl d i (2)


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• 8-input data selectorControl inputs higher than data inputs.

O( n) O(2 n)

Data inputs higher than control inputs.


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• Binary-coded multiplier

is a weak point. – Proved that O(2 n) nodes

needed in any variable

ordering.

– Division function also

becomes an

exponential size.

5×5-bit multiplier(5th output function)

(a)Easy for any order.

(b)Easy/hard may changedepending on the order.

(c) Hard for any order.

Patterns of variable ordering.



18/31


BDD construction algorithm

• Reduction from “binary decision tree” to BDD

always requires exponential time and space.

we use the algorithm [Bryant1986] for directlyconstructing BDDs from given Boolean expressions.

a

b b

c c c c

0 0 01 1 1 1 1 0 1

a

b

c

10

1

0

1

0

reduction

(compress)

F = a b + ~cdirect construction

BDD t ti f B l i


19/31

• Any BDD can be generated by

repeating binary logic operations

between two BDDs.


BDD construction from Boolean expressions

0 1

a

b

c

10

1

0

1

0

0 1

a

b

10

0

1

0 1

a

10

0 1

b10

0 1

c 10

1 0

c10

a

b

c~c

a b

a b + ~c AND

NOT

OR


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Binary logic operation （項論理演算） algorithm

• Recursive expansion assigning 0/1 into top variable v.

• Eventually trivial operation appears and result obtained.

(in OR operation) F+1=1, F+0= F, F+ F= F …

• Unifying all results of sub-operations into one BDD.

H = ( F [op] G)

v10

H (v=0) = ( F(v=0) [op] G(v=0)) H (v=1) = ( F(v=1) [op] G(v=1))


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Execution of binary logic operation

0 1

a

b

c

10

1

0

1

0

d d 11

00

0 1

b

c1

0

1

0

d10

F

G N 1

N 3 N 2

N 4 N 5

N 6 N 7 N 8

N 1+ N 3

N 2+ N 3 N 4+ N 3

N 6+ N 8 N 4+ N 5 N 4+ N 8 N 4+ N 5

F + G

a=0 a=1

b=0 b=1 b=0 b=1

0 + N 7 N 6+ N 8 0 + N 8 N 6+ N 8

c=0 c=1

0 + 1 1 + 0

d =0 d =1

1

N 7 N 8


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Computation time for binary logic operation

• We denote |F| as the number of nodes in BDD F,

then, in the worst case | H | = | F| |G| for H = F [op] G.

• This means that BDD construction for n-input

Boolean expression requires O(2 n) time.

– However, using techniques such as hash tables, we may

execute in O( | F| + |G| + | H | ) time for many practical cases.

• There are cases where | F|, |G| are very large but | H | is very small.

Opposite cases are also possible.

– If BDD size is relatively small, logic operation becomes

very fast, even for a large n.

– if BDD grows exponentially large (e.g. multiplier),

almost no advantage of using BDDs.


23/31


NOT operation algorithm

• Just copying the BDD, and

swapping 0- and 1-terminalnodes.

• Linear time for BDD size.

• Using “negative edges”

(explain later), we can

execute in a constant time.

0 1

a

b

c

10

1

0

1

0

10

a

b

c

10

1

0

1

0

F ~F

Finding satisfiable/minimal solutions


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Finding satisfiable/minimal solutions

• If a BDD is already reduced,satisfiability checking can be done immediately.

• If satisfiable, a solution (0/1 assignment into each input)can be found easily. – Choosing decisions not connecting to the 0-terminal node,

then eventually we must reach the 1-terminal node.

– Linear t ime for the number of inputs. (less than BDD size)

• If the cost of assigning 1 to each input is defined,we can search the minimal-cost satisfiable solutions. – Linear time for BDD size.

• Truth-table density (ratio of “1” in the table) andprobability of satisfiability can be computed efficiently. – Linear time for BDD size.


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Improvements of BDDs

• Various improvement techniques for efficient BDDmanipulation have been developed.

• The two techniques are important for practical use.

1. Shared BDDs for multiple functions.

2. Negative edges.

These two techniques are implemented in

most of BDD manipulation programs.

Sh d BDD BDD


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Shared BDD（複数BDDの有化）

• Use a uniform variable order.

• All BDDs are shared into a

multi-rooted graph.

0 1

a

b

c

10

1

0

1

0

0 1

a

b

10

0

1

0 1

a10

0 1

b10

0 1

c10

1 0

c10

a b c ~c

a b

a b + ~c

0 1

a

c c

b b

a a

a b c ~c

a b a b + ~c

Logic operation in shared BDDs


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Logic operation in shared BDDs

0 1

a

b

a a

b

c c

a

b

c ~c

a b

a b + ~c

AND

NOT

OR

Shared BDD

manipulator

• Shared BDD growswhen executinglogic operations.

• Equivalencechecking can bedone just bycomparing pointers.(constant time)

N ti d


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Negative edge（否定枝）

• Attaching a mark of NOT

operation into an edge to

a BDD, then complimentBDDs can be shared into

one BDD.

• NOT operation can be

done in a constant time.

– Just switch on/off of the

edge at the root node.

• Constraints in using

negative edges to keep

uniqueness of BDDs. – Don’t use 1-terninal node.

– 0-edges never be negative.

1

a

b

c

10

1

0

1

0

0

a

b

c

10

1

0

1

0

F ~F

1

0

a

b

c

10

0

1

0

F ~F

1 0 x x

10 10

BDD package


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BDD package

• BDD manipulation programs have been developed

actively in 1990s.

– Some of them are public domain software as “BDD package.”

• In many cases, implemented as a C or C++ library.

– We call a library function with pointers to BDDs as

function parameters, then new BDD is constructed in the

memory, and a pointer to the new BDD is returned.

– User designs the main program to call BDD operations, and

then compile the program with the BDD package.

• Detailed implementation techniques of BDD package

will be shown in the next lecture.

S

BDD


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Summary

• Basic data structure – Shannon’s expansion, Binary tree and BDD, reduction rules.

• Properties of BDDs – Comparison with truth-tables and SOP forms. – Uniqueness, compactness, fast logic operations. – BDDs for typical Boolean functions. – Impact of variable ordering to BDD size

• Algorithms for BDD construction – BDD construction for a given Boolean expression. – Binary logic operation algorithms. – Finding satisfiable/minimal solutions, truth-table density.

• Improvement of BDDs – Shared BDDs for multiple functions. – Negative edge.

BDD（分決定グラフ）


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Exercises

1. Draw a structure of BDDfor an Achilles’s heel function:

F = x1 x2 x3 + x4 x5 x6 + x7 x8 x9 + … + x3 n-2 x3 n-1 x3 n

2. For the above function, if the worst variableorder is used, how is the BDD size?

minato robdd lecture 2

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