BDDsBDDs 11

Binary Tree RepresentationBinary Tree RepresentationThe recursive Shannon expansion corresponds to The recursive Shannon expansion corresponds to

a a binary treebinary tree

Example:Example:

Each path from the root to a leaf corresponds to Each path from the root to a leaf corresponds to a minterm.a minterm.

1111

11

00

00

00

xx

yyyy

( ) ( )x x

xy x y xy x y

f xf x f

x yf y f x yf y f

xyfx yfxyfx yf

BDDsBDDs 22

The root represents the original function The root represents the original function ff.. The two nodes immediately below the root The two nodes immediately below the root

represent the co-factors of represent the co-factors of ff.. As you go deeper, each node represents a co-As you go deeper, each node represents a co-

factor of the function represented by its parent.factor of the function represented by its parent. The leaves are the terminal cases representing 0 The leaves are the terminal cases representing 0

and 1 which have no co-factors.and 1 which have no co-factors.

BDDsBDDs 33

ExampleExample

1111

11

00

00

00

11

aa

cc bb

00 1100

11 11

11

00

00

00

11

aa

bb bb

00

1100

11

00

cccc

SplittinSplitting g variablevariable

f ab ac

00 11 00

11

BDDsBDDs 44

Implicit Enumeration -Implicit Enumeration - Branch and Bound Branch and Bound

Checking for tautology and many other Checking for tautology and many other theoretically intractable problems (co-NP theoretically intractable problems (co-NP complete) can be complete) can be effectivelyeffectively solved using solved using implicit enumeration:implicit enumeration:

use recursive Shannon expansion to explore Buse recursive Shannon expansion to explore Bn.n.

In (hopefully) large subspaces of BIn (hopefully) large subspaces of Bnn, prune the , prune the binary recursion tree bybinary recursion tree by exploiting properties of the node function fexploiting properties of the node function fcc(v)(v) exploiting heuristic bounding techniquesexploiting heuristic bounding techniques

Even though in the worst case the recursion tree Even though in the worst case the recursion tree may have 2may have 2nn nodes, in practice nodes, in practice (in many cases),(in many cases), we typically encounter a we typically encounter a linearlinear number of nodes. number of nodes.

BDDsBDDs 55

Implicit Enumeration -Implicit Enumeration - Branch and Bound Branch and Bound

Thus we say that the 2Thus we say that the 2nn minterms of f have minterms of f have been been implicitly enumeratedimplicitly enumerated

BDD’s (Binary Decision Diagrams) are BDD’s (Binary Decision Diagrams) are alternate representations in which implicit alternate representations in which implicit enumeration is performed enumeration is performed staticallystatically, and , and nodes with identical path cofactors are nodes with identical path cofactors are identifiedidentified

BDDsBDDs 66

ROBDDsROBDDs represents a logic function by a directed acyclic represents a logic function by a directed acyclic

graph (DAG).graph (DAG). (many logic functions can be (many logic functions can be represented compactly - usually better than represented compactly - usually better than SOP’s)SOP’s)

canonicalcanonical form ( form (importantimportant) ) (only canonical if an (only canonical if an ordering of the variables is given)ordering of the variables is given)

many logic operations can be performed many logic operations can be performed efficientlyefficiently on BDD’s on BDD’s (usually linear in size of result (usually linear in size of result - tautology and complement are constant time)- tautology and complement are constant time)

size of ROBDD critically dependent on variable size of ROBDD critically dependent on variable orderingordering

BDDsBDDs 77

ROBDDsROBDDs one root node per function, two terminals 0,

1 each node, two children, and a variable Shannon co-factoring tree, except reduced

and ordered (ROBDD)Reduced:

any node with two identical children is removed two nodes with isomorphic BDD’s are merged

Ordered: Co-factoring variables (splitting variables) always

follow the same order from a root to a terminalxi1

< xi2 < xi3

< … < xin

BDDsBDDs 88

OBDDOBDDOrdered BDDOrdered BDD ( (OBDDOBDD) ) Input variables are ordered - Input variables are ordered -

each path from root to sink visits nodes with labels each path from root to sink visits nodes with labels ((variablesvariables) in ascending order.) in ascending order.

aacc cc

bb

00 11

orderedorderedorder = a,c,border = a,c,b

aa

bb cc

cc

00 11

notnotorderedordered

bb

BDDsBDDs 99

ROBDDROBDD Reduced Ordered BDD - reduction rules:Reduced Ordered BDD - reduction rules:

if the two children of a node are the same, if the two children of a node are the same, the node is eliminated: f = cf + c’fthe node is eliminated: f = cf + c’f

if two nodes have isomorphic graphs, they if two nodes have isomorphic graphs, they are replaced by one of themare replaced by one of them

These two rules added to ordering mean These two rules added to ordering mean each node represents a distinct logic each node represents a distinct logic function.function.

BDDsBDDs 1010

By tracing paths to the 1 node, we get a By tracing paths to the 1 node, we get a covercover of pairwise of pairwise disjointdisjoint cubes. cubes.

The power of the BDD representation is that it does not explicitly enumerate all paths; rather it represents paths by a graph whose size is measures by its nodes and not paths.

A DAG can represent an exponential number A DAG can represent an exponential number of paths with a linear number of nodes.of paths with a linear number of nodes.

F = b’+a’c’ = ab’+a’cb’+a’c’ (all paths to the 1 node)F = b’+a’c’ = ab’+a’cb’+a’c’ (all paths to the 1 node)

aa

ccbb

00 11

1100

11

110000

ff

ffaa= b’= b’ffa a = cb’+c’= cb’+c’

BDDsBDDs 1111

ExamplesExamplesaa

bb

cc

dd

00 11

11

00

aa

bb

cc

dd

00 11

BDDsBDDs 1212

EXOREXORaa

bb

cc

dd

00 11

dd

cc

bba b c d = a’b’c’d +

a’b’cd’ +a’bc’d’ +ab’c’d’ +a’bcd +ab’cd +abc’d +abcd’

Terms in SOP = ???

Nodes in BDD = ???

BDDsBDDs 1313

The Ordering EffectThe Ordering Effect

Two different orderings, same function.Two different orderings, same function.

aa

bb bb

cc cc

dd

00 11

c+bdc+bd bb

root root nodenode

c+dc+dcc

dd

f = ab+a’c+bc’df = ab+a’c+bc’d aa

cc

dd

bb

00 11

c+bdc+bd

d+bd+b

bb

11

00

BDDsBDDs 1414

ExampleExamplef = ad + be + cff = ad + be + cf

There are n! orderings – in this example 6! = There are n! orderings – in this example 6! = 720720

Try orderings:Try orderings:

a,b,c,d,e,fa,b,c,d,e,f

a,d,b,e,c,fa,d,b,e,c,f

BDDsBDDs 1515

TheoremTheoremTheorem 1Theorem 1 ((Bryant – 1986 posted on course Bryant – 1986 posted on course

pagepage)) ROBDD’s are canonicalROBDD’s are canonical

Thus two functions are the Thus two functions are the samesame iff iff their ROBDD’s are equivalent graphs their ROBDD’s are equivalent graphs ((isomorphicisomorphic). Of course must use ). Of course must use same ordersame order for variables. for variables.

BDDsBDDs 1616

ImplementationImplementation Each node is a triple (v,gEach node is a triple (v,g00,g,g11) )

representing a functionrepresenting a function

g = v gg = v g00 + v g + v g11

In most implementations, gIn most implementations, g00 and g and g11 are are pointerspointers to other nodes. to other nodes.

BDDsBDDs 1717

ITE OperatorITE OperatorTableTable Subset Subset Expression Expression Equivalent FormEquivalent Form00000000 00 0 0 0000010001 AND(f, g)AND(f, g) fg fg ite(f, g, ite(f, g, 0)0)00100010 f > gf > g f fgg ite(f,ite(f,g, 0)g, 0)00110011 ff f f ff01000100 f < gf < g fgfg ite(f, 0, g)ite(f, 0, g)01010101 gg g g gg01100110 XOR(f, g)XOR(f, g) f f g g ite(f,ite(f,g, g, g)g)01110111 OR(f, g)OR(f, g) f + g f + g ite(f, 1, g)ite(f, 1, g)10001000 NOR(f, g)NOR(f, g) f + g f + g ite(f, ite(f, 0,0,g)g)1001 1001 XNOR(f, g)XNOR(f, g) f f g g ite(f, g,ite(f, g,g)g)10101010 NOT(g)NOT(g) gg ite(g, 0, 1)ite(g, 0, 1)10111011 f f g g f + f + gg ite(f, 1, ite(f, 1, g)g)11001100 NOT(f)NOT(f) ff ite(f, 0, 1)ite(f, 0, 1)11011101 f f g g f + gf + g ite(f, g, 1)ite(f, g, 1)11101110 NAND(f, g)NAND(f, g) fg fg ite(f, ite(f, g, 1)g, 1)11111111 11 1 1 11

BDDsBDDs 1818

Efficient ImplementationEfficient ImplementationStrong canonical form:Strong canonical form: A “unique-id” is A “unique-id” is

associated (through a hash table) uniquely associated (through a hash table) uniquely with each element in a with each element in a setset..

With BDD’s the With BDD’s the setset is the set of all logic functions. A BDD is the set of all logic functions. A BDD node is a function. Thus each function has a unique-id node is a function. Thus each function has a unique-id in memory.in memory.

What is a good hash function?What is a good hash function?

vv00 11

ff

ffvv ffvv

BDDsBDDs 1919

Unique Table - Hash TableUnique Table - Hash Table

Before a node (Before a node (v, g, h v, g, h ) is added to BDD data base, it is looked up in the ) is added to BDD data base, it is looked up in the “unique-table”.“unique-table”. If it is there, then existing pointer to node is used to If it is there, then existing pointer to node is used to represent the logic function. Otherwise, a new node is added to the represent the logic function. Otherwise, a new node is added to the unique-table and the new pointer returned.unique-table and the new pointer returned.

Thus a Thus a strong canonical formstrong canonical form is maintained. The node for is maintained. The node for ff = ( = (v, g, h v, g, h ) ) exists exists iffiff((v, g, h v, g, h ) is in the unique-table. There is only one pointer for ) is in the unique-table. There is only one pointer for ((v, g, h v, g, h ) and that is the address to the unique-table entry.) and that is the address to the unique-table entry.

Unique-table allows single multi-rooted DAG to represent all users’ Unique-table allows single multi-rooted DAG to represent all users’ functions:functions:

hash valuehash valueof keyof key

collisioncollisionchainchain

BDDsBDDs 2020

Recursive Formulation of Recursive Formulation of ITEITE

vv = top-most variable among the = top-most variable among the three BDD’s three BDD’s f, g, hf, g, h

( , , )

( ) ( )

( ) ( )

( , ( , , ), ( , , ))

( , ( , , ), ( , , ))

( , , )

v v

v v v v v v v v

v v v v v v

v v v v v v

ite f g h fg f h

v fg f h v fg f h

v f g f h v f g f h

ite v ite f g h ite f g h

v ite f g h ite f g h

v f g

BDDsBDDs 2121

Recursive Formulation of Recursive Formulation of ITEITE

iteite((f, g, hf, g, h) ) if(if(terminalterminal casecase) {) { return result;return result;} else if(} else if(computed-tablecomputed-table has entry ( has entry (ff, , gg, , h h )) {)) { return result;return result;} else {} else { let let vv be the top variable of ( be the top variable of (ff, , gg, , h h );); f <- f <- iteite((ffvv , g , gvv , , hhv v );); g <- g <- iteite((ffvv , g , gvv , , hhv v );); if( f equals g ) return g;if( f equals g ) return g; R <- R <- find_or_add_unique_tablefind_or_add_unique_table(v, f, g );(v, f, g ); insert_computed_tableinsert_computed_table( {( {ff, , gg, , h h }, R);}, R); return R; } }return R; } }

BDDsBDDs 2222

Terminal casesTerminal cases: :

(0,(0, f, g f, g) = ) = gg (1,(1, f f,, g g) =) = f f ite ite ((ff, , gg, , gg) =) = g g

Unique tableUnique table: Used to ensure one : Used to ensure one physical representation for any given physical representation for any given function.function.

Computed tableComputed table: Used to cache results : Used to cache results to avoid recomputation.to avoid recomputation.

BDDsBDDs 2323

ExampleExample

II = = ite ite ((FF,, G G,, H H) ) = (= (aa, , ite ite ((FFaa ,, G Ga a , , HHa a ), ), ite ite ((FFa a ,, G Ga a , , HHa a ))))= (= (aa, , ite ite (1,(1, C C , , HH ), ), iteite((BB,, 0, 0, H H ))))= (= (aa, , CC, (, (bb ,, ite ite ((BBb b , 0, 0b b , , HHb b ), ), ite ite ((BBb b ,, 00b b , , HHbb )) )) = (= (aa, , CC, (, (bb , , ite ite (1,(1, 0, 1), 0, 1), ite ite (0, 0, (0, 0, DD))) ))) = (= (aa, , CC, (, (bb ,, 0, 0, DD))))== ((aa, , CC, , JJ))

F,G,H,I,J,B,C,DF,G,H,I,J,B,C,Dare pointersare pointers

bb11

11

aa

00

11 00

11 00

FF

BB1111

aa

00

11 00

00

GG

cc 00CC

11

bb

00

11 00

00

HH

dd DD

1111

00

aa11 00

00

II

bb JJ

11CC

DD

BDDsBDDs 2424

Computed TableComputed TableKeep a record of (Keep a record of (FF, , GG, , H H ) triplets already computed by ) triplets already computed by the the iteite operator in a operator in a hash-based cachehash-based cache ( ( “cache” table“cache” table). ). This means that the collision chain is This means that the collision chain is not usednot used ( (if if collision, old entry thrown away collision, old entry thrown away ).).

The above structure is wasteful since the BDD nodes and The above structure is wasteful since the BDD nodes and collision chain can be merged.collision chain can be merged.

BDDsBDDs 2525

Better ImplementationBetter Implementation

BDDsBDDs 2626

Edge ComplementationEdge Complementation

Can combine by making Can combine by making complement edgescomplement edges

00 11

GG

00 11

two differenttwo differentDAGsDAGs

00 11

GGG’G’

only one DAGonly one DAGusing complementusing complementpointerpointer

G’G’

BDDsBDDs 2727

Extension - Complement Extension - Complement EdgesEdges

To maintain the strong canonical form, we need to resolve 4 equivalences:To maintain the strong canonical form, we need to resolve 4 equivalences:

VV VV VV VV

VVVV VVVV

SolutionSolution: Always choose the one on the : Always choose the one on the leftleft, i.e. , i.e. the “1” edge must the “1” edge must not have a complementnot have a complement..

BDDsBDDs 2828

EXOR (revisited)EXOR (revisited)aa

bb

cc

dd

00 11

dd

cc

bb

aa

bb

cc

dd

11