dominator-based algorithms in logic synthesis and verification

8/13/2019 Dominator-Based Algorithms in Logic Synthesis and Verification

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f = ( x1 x2) (x1 x3) (x2 x3) v fanin (v) v fanin + (v)

v fanin (v) v fanout (v) v fanout + (v)

v fanout (v) x1 x2

x1 x4 x2 x3

x1 x4 x2 x3

f (x0, x1) = 7 (x0 x1) 3 (x0 x1) ( 2) (x0 x1) 0 (x0 x1)

g f g + f g f

o1 D(o1) o2 D(o2) D (h)

DR (h)

v V D (i)

l l j k


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l l j k

T ( j ) D ( j )

U v U e e U e e U e

U v ={a, b} U v = {c} U v = {a} U e = { a, c } U v = {c} U e = U v = {b,c,d,e,f } U e = { a, h , g, l } U v = {b,i ,j }

U e = { a, h , g, l } U v = {h,i ,l } U e = U v = {k, l} U e =

DR (k) DR (k) DER (k)

DR (h)

f =a(b + c) f = ab + ac


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C a1 , . . . , a n

DR (m).

A

C a1, . . . , a n

a b c D R (f ) DER (f )

{d, e}


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f = ( x1 x2) (x1 x3) (x2 x3)

f

f

B F

B ={0, 1} F

v V j k


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n

N Mdom D ER (root ) tD ER (root )

DER (root ) maxN DDt DER (root )


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A B

A B


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{a,b,c,d} {,, , } A

|A| a A a A

A B a A a B

A B A B A = B

a A a / A a U U

C = A B a C a A a B

C = A B a C a A a B

C = A B C = A B A B

A B = {(x, y) : x A and y B}.

A B A B A = {a, b} B = {, }

A B = {(a, ), (a, ), (b, ), (b, )}.


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a + b (a, b) (a, b)

a b

a a = a + a = a

a b = b a a + b = b + a

a + a b = a a (a + b) = a

a (b c) = ( a b) c a + ( b + c) = ( a + b) + c

a + 0 = a a 0 = 0 a 1 = a a + 1 = 1

a a a a = 0 a + a = 1

a + ( b c) = ( a + b) (a + c) a (b + c) = ( a b) + ( a c)

(A, + , )

a b = b a a + b = b + a

a + 0 = a a 1 = a

a a a a = 0 a + a = 1

a + ( b c) = ( a + b) (a + c) a (b + c) = ( a b) + ( a c)

B = {0, 1}

n x1, x2, . . . , x n B = {0, 1}

n Bn |B |n = 2 n f


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f : Bn B Bn B f

f x1 x2 x3

f

f = ( x1 x2) (x1 x3) (x2 x3)

f = x1 x2 f = ( x1 x2) (x1 x2) =(x1 x2) (x1 x2) (x1 x2) = ( x1 x2) (x1 x2) . . . (x1 x2)

f

1 2 3x /\ x /\ x

1 2 3x /\ x /\ x

1 2 3x /\ x /\ x

1 2 3x /\ x /\ x

1 2 3x /\ x /\ x

1 2 3x /\ x /\ x 1 2 3x /\ x /\ x

1 2 3x /\ x /\ x

f = ( x1 x2) (x1 x3) (x2 x3)

n 2n 22n

{AND,NOT } {OR,NOT }


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x1 x2 x3 f

f = ( x1 x2) (x1 x3) (x2 x3)

x1 x2 f 0 f 1 f 2 f 3 f 4 f 5 f 6 f 7

x1 x2 f 8 f 9 f 10 f 11 f 12 f 13 f 14 f 15

f

f


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f

f (x1, x2, x3) = ( x1 x2)(x1 x3) (x2 x3)

f

x1 x2 x3 f M 1 = x1 x2 x3 M 2 = x1 x2 x3 M 3 = x1 x2 x3 m1 = x1 x2 x3 M 4 = x1 x2 x3 m2 = x1 x2 x3

m3 = x1 x2 x3 m4 = x1 x2 x3

f

f

f = m1 m2 m3 m4 (DN F )

f = M 1 M 2 M 3 M 4 (CNF )

f

f = x f |x =1 x f |x =0 .

f |x =1 f |x =0 f x

C C = ( V,E, I ,O )

V E V V v1, v2 E v1 v2

v1 , . . . , vn vi , vi+1 E, 1 i < n

I V O V v V fanin (v) =

{v V : v , v E } v fanin + (v)


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v1, . . . , vn , v1 I vn fanin (v) v fanin (v)

v1, . . . , vn , v1 I vn = v fanin (v) ins (v) v I

fanin (v) = fanin + (v) = fanin (v) = v

v / I fanin (v) V

000000000111111111000000000000111111111111 0000000001111111110000000011111111

0000000011111111000000000111111111000000111111

cba

v 000000111111000000000111111111000000111111000000000111111111

v v

v fanin (v) v fanin + (v) v fanin (v)

v V fanout (v) ={v V : v, v E } v fanout + (v)

v1, . . . , vn , v1 fanout (v) vn O v fanout (v)

v1, . . . , vn , v1 = v vn O v O fanout (v) = fanout + (v) = fanout (v) =

v

v V v1, . . . , vn v1 I vn O

C m n {0, 1}n

{0, 1} n m

x0, x1, . . . xa , xb, . . .

xa xb (xa xb) xa xb


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vv

cba

000000000000111111111111 00000000011111111100000000111111110000000011111111000000111111000000111111000000111111

0000000011111111000000000

0001111111111110000000011111111000000000111111111000000000111111111000000111111000000000111111111

000000000111111111v

v fanout (v) v fanout + (v) v fanout (v)

e = v, v E inv (e) { 0, 1} inv (e) = 1 inv (e) = 0 v

x1 x2

x

x

1

2

x1 x

2


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1 01 01 001

x3

x 1

x 2

x 3 x 3

x 4 x 4

x 2 x /\ x2 3

x x4 4

4 3x \/ x

1 4 3x /\ x \/ x /\ x2

4x \/ x /\ x2 3

x1 x4 x2 x3

x4

f f f (x0, . . . ) =

(x0 f |x 0 =1 ) (x0 f |x 0 =0 ) f |x 0=1 = f |x 0 =0 f |x 0 =1 f |x 0 =0

n 2n 1 n


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0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

1 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 1

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

1 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 1

1 01 01 001

x 1

x 2

x 3 x 3

x 4 x 4

x 2

0

x 1

x 2

x 3 x 3

x 2

x 4

1

(a) (b)

0 x

f f

x 0

f


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0

x 1

x 2

x 3 x 3

x 2

x 4

1 0

x 1

1

x 2

x 4

x 3

x1 x4 x2 x3

G f |G| G


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O(|G| log|G|) f 1 < op > f 2 O(|G1 | | G2 |)

f |x i = b, b { 0, 1} O(|G|) f 1|x i = f 2 O(|G1 |2 |G2 |)

B n F B = {0, 1} F F 2n

n F

f F x0 x1 x0 x1 0 7

f (x0, x1) = 7 (x0 x1) 3 (x0 x1) 2 (x0 x1) 0 (x0 x1) x

x0 x1 f (x0, x1)

f B F B = {0, 1} F

g(x0 , x1) = 3 x0 0 (x0 x1) ( 2) (x0, x1)

f (x0, x1) = 5 (x0 x1) ( 1) (x0 x1) 7 (x0 , x1) 0 (x0, x1)

g + f g f

g + f = 8 (x0 x1) 2 (x0 x1) 7 (x0, x1) ( 2) (x0 , x1)


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x x

x 0

1 1

072 3

f (x0, x1) = 7 (x0 x1) 3 (x0 x1) ( 2) (x0 x1) 0 (x0 x1)

g f = 15 (x0 x1) ( 3) (x0 x1) 0 (x0)

x0 x1 g f g + f g f

3

x

x 0

1

0 2

x x

x 0

1 1

075 1 0

x

x 0

1

15 3

g f g + f g f

A B A B

A B A B

A B = xi (A|x i =1 B|x i =1 ) xi (A|x i =0 B|x i =0 ) xi


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O(n4) n


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O(m (m, n )) m n

C (V,E, I ,O )

v V w V z V w z v

b b e o1 o1 Doms (v, z ) v

z Dominated (v, z ) u V v z b

o2 b o2 v V

w V iDoms (v, w) z iDoms (v, w) Doms (v, w)

I O z iDoms (v, w) : z Doms (v, w) z / I O

o

o

2

1

a

b

c

d

e

f

g

h

i

o

o

2

1

a

b

c

d

e

f

g

h

i

(b)(a)

v w = v z v = idom(w, z ) v w v Doms (w, z )


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w v Doms (w, z ) u = v : u Doms (w, z ) v /Doms (u, z ) u Doms (v, z )

{ idom (w, z ), w : w fanin + (z)} z D(z)

D (o1) D(o2)

o

o

2

1

a

b

c

d

e

f

g

h

i

o2

h

i

b

c

fd

ea

c

b o1

a

e

g

h

(b)(a) (c)

o1 D(o1) o2 D(o2)

D R (z) D (z)

v V DR (z) v z u DR (z) v = idom(u, z )

D R (z)

z D R (z) v D(z)

v

u DR (z) v = idom(u, z )


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b

a

c

d

e

li

j

k

n

m

h

a

b

c

d

n

e e

a

b

c

dj,k

h,i

n

b

a

c

d

e

n

a

b

c

d

ej,k

h,i

l,m n

(a) (c)

(b)

m

li

k

j

f

g

h

(d)

(e)

f

g

D (h) D R (h)

u v z u

v v Doms (v, z ) v

v Doms (v, z ) = {v} p fanout (v) Doms ( p, z)

C = ( V,E,I , {root }) V E V V root V

(V,E,I , {root })Doms (root,root ) = {root }

v V { root }Doms (v) = {root,v 1, v2 , . . . , v |V | 1}

v V { root }Doms (v,root ) = {v} p fanout (v) Doms ( p, root )

v V


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D (i) Doms (v, i ) v V { i} V

i Doms (i, i ) = {i} i

Doms (v, i ) D(i) Doms (v, i ) i v D(i)

a

b

c

e

d

f

g

hi b

d

g

hi

e

f

a

c

(a) (b)

D (i)

{i} Doms (i, i ) = {i} {g} Doms (i, i ) Doms (g, i) = {g, i}

{h} Doms (i, i ) Doms (h, i ) = {h, i } {e} Doms (g, i) Doms (h, i ) Doms (e, i ) = {e, i}

{f } Doms (g, i) Doms (f, i ) = {f ,g , i } {a} Doms (e, i ) Doms (a, i ) = {a,e,i }

{b} Doms (e, i ) Doms (f, i ) Doms (b, i) = {b, i} {c} Doms (f, i ) Doms (c, i) = {c,f ,g , i }

{d} Doms (h, i ) Doms (d, i ) = {d,h,i }

T O(m logn)

m n

O(m (m, n )) (m, n )


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a

b

c

e

d

f

g

hi

u V { root }

sdom (w) w u sdom(u,root ) T (root )

idom(u, root ) = sdom(u, root ) sdom(u,root ) = sdom(w, root ),

idom (w,root ) otherwise.

C root

n n 1

u

u A fanout (u) a Adfs a < dfs u

dfs u

T (root ) w w f anout (u) dfs u < dfs w


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u sdom(u,root ) T (root ) u sdom(u, root ) sdom (u, root ) sdom(u,root )

u idom(u,root )

w T (root )

dfs sdom (w,root ) dfs sdom (u,root ) w w

idom (u, root ) = idom(w,root )

u sdom (u,root )

u dfs sdom (u,root )

u sdom(u,root ) = idom(u, root ) u sdom(u,root ) = idom(u, root )

v w sdom(w, root )

u sdom(u,root ) u sdom(u,root ) root

idom (w,root ) u


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C (V,E, I ,O ) w V v V w

v o O vw absDoms (v, O ) : o (fanout + (v) O) w Doms (v, o)

g d f b

v V C T (r ) o O (o)

v V O o O v (v, T (o))

v r v

o

T (o)

w v v

o O w v


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(V, E , I , O )

o O

T (o) = (o) v V O

absDoms (v, O ) =o fanout + (v ) O (v, T (o))

c

f

g

i

h

k

l

j ha

c

d

jf

l

k

i

g

e

b

a

b

c

d g

f h

j

b

c

f

g

e

i

d k

(a) (b)

(c) (d)

e

a

b

d

l l j k


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O fanout + (v)

C C

T (l) C v V C T (l) v l

T (l) C (V,E, I ,O )

O = {r 1, r 2} C C (V , E , I , O ) O = {o} V = V { o}

E = E { r 1 , o , r 2 , o }

l j k

C T (l) C

C C w V { o}

v C v v C

v V { r 1, r 2}

r1

r 2 v V : v fanin + (r 1) v fanin + (r 2) v V { r 1, r 2} r1 r2

v V : v fanin + (r 1) v fanin + (r 2)

v v r r1 v r1

C v r1 C v C v C

v

v C v C r1 r2 u C v r1

r 2 v o C

P v o C r 1 r2 w


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v o w Doms (v, o) v r1 v r2 v

o C v r1 r2 C

u v r1 r2 C u v r C C v o

u C v o r1 r2 v u

r 1 r2 u v r1 r 2 u v o C

(O) l j k l

(l)

v V O absDoms (v, O ) T (l)

l l

(V, E , I , O )l = (O)T (l) = (l)

v V O absDoms (v, O ) =

(v, T (l)) { l}


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absDoms (v, O ) = absDoms (v, O ) =v Doms (v, j ) Doms (v, k ) o O fanout + ( v ) Doms (v, o ) Doms (v, l ) { l}

{a,h, j } {a,h, j } {a,h, j } {b,f,h , j } {b , f , i ,k } {b, f } {b, f }

{c, j } {c, j } {c} {c}

{d,g, j } {d,g, i ,k } {d, g } {d, g } {e, k } {e, k } {e, k }

{f ,h , j } {f , i ,k } {f } {f } {g, j } {g , i ,k } {g} {g}

{h, j } {h, j } {h, j } {i, k } {i, k } {i, k }

{ j } { j } { j } {k} {k} {k}

v V j k

l

j

k

T (l) T ( j )

T (k)

T ( j ) T (k) T (l) v V l j

v o T (o) v o T (o) v

l k

T (l) C

l b j k Doms (b, j ) =

{b,f,h,j } Doms (b, k) = {b,f,i ,k } {b, f } = Doms (b, j ) Doms (b, k)

absDoms (v, R ) {b, f } b k b j

b f


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s p e e

d u p =

1 0

absolute dominator computation

a l g o r i

t h m

b y K i r k l a n

d a n

d M e r c e r

new algorithm

s p e e d u p = 1 0 0 0

runtime in seconds

0.0001

0.001

0.01

0.1

1

0.0001 0.001 0.01 0.1


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number of absolute dominatorsper vertex

p141k

n u m

b e r o f v e r t

i c e s


p267k

n u m

b e r o f v e r t

i c e s


p269k

n u m

b e r o f v e r t

i c e s


p279k

n u

m b e r o f v e r t

i c e s


n u m

b e r o f v e r t

i c e s

p330k


n u m

b e r o f v e r t

i c e s

p388k


n u m

b e r o f v e r t

i c e s

p286k


n u m

b e r o f v e r t

i c e s

p418k


p951k

n u m

b e r o f v e r t

i c e s

1

10

100

1000

10000

100000

0 2 4 6 8 10 1 2 1 4 1 6

1

10

100

1000

10000

100000

0 2 4 6 8 10 12 1 4 16

1

10

100

1000

10000

100000

0 2 4 6 8 10 12 14 16

1

10

100

1000

10000

100000

0 5 10 15 20 25

1

10

100

1000

10000

100000

0 5 10 15 20 25

1

10

100

1000

10000

100000

0 5 10 15 20

1

10

100

1000

10000

100000

0 5 10 15 20

1

10

100

1000

10000

100000

0 5 10 15 20 25 30 1

10

100

1000

10000

100000

0 5 10 15 20 25


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l l

C (V,E, I ,O ) C (V , E , I , O ) O = {l} V = V { l}

E = o O { o, l } E P O C p v V idom (v, l) = l

{ p, q } P : idom(v, p) = idom(v, q )idom (v, p), p P

v l l P v

v p P l

p P l

v

v idom(v, l) = l idom(v, l) absDoms (v, O)

idom (v, l) = l { p, q } P : idom(v, p) = idom(v, q ) w v l

v p P idom(v, l ) = l p P idom (v, p) = idom(v, l)

C C P O

v V O v l l { p, q } P

idom(v, p) = idom(v, q )

idom(v, l ) = l w = l C v l


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C C E = E o O { o, l } w V

v p P C

v p P

{ p, q } P idom (v, p) = idom(v, q )

idom(v, l ) = l { p, q } : idom(v, p) =idom(v, q )

idom(v, l ) = l p P idom (v, l ) =idom (v, p) idom(v, l) =l { p, q } : idom(v, p) = idom(v, q ) idom (v, l) = l v P idom (v, p) = idom(v, l )

C C P

v V O C v l C l { p, q } P

idom (v, p) = idom(v, q )

{ p, q } P idom(v, p) = idom(v, q ) idom (v, l ) = l

u = idom(v, p) u v p fanin ( p)

idom (v, p) = idom(v, q ) v idom(v, p) v idom(v, q )

fanin ( p) fanin (q )

fanin (q ) fanin ( p)

v p q q p


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C v u v

u

v

u w v u v / fanin (u) w fanout (v) : w /

fanin (u)

v

p q

l P

l

{ p, q } P

idom(v, p) =idom (v, q ) v C l

C C P

v V O C idom(v, l) = l p P idom (v, p) = idom(v, l)

v V O idom(v, l ) = w = l E = E o O { o, l } v

p P C C v l C v p P C

idom(v, l ) = l p P idom (v, p) = idom(v, l)

l j k

D (l) (l)


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c

f

g

i

h

k

l

j ha

c

d

jf

lk

i

g

e

b

a

b

c

d g

f h

j

b

c

f

g

e

i

d k

(a) (b)

(c) (d)

e

a

b

d

l l j k

o O o T (o) (o)

v v (v, o)

v o

v v l l l = idom(v, l )

u v o u Doms (v, o)

u w f anout (v) o w o u

v o u w f anout (v) : u Doms (w, o) u Doms (v, o)


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C o o u

v = u u Doms (v, o) { v} v

v = o v oP = v = w0, . . . , wk = o wi , i = {1, . . . , k }

dfs w i dfs v u = v v v o u u

P u v v

v w o v = idom(w, o) v w

v = idom(w, o) v Doms (w, o) u Doms (w, o) {w, v} u Doms (v, o)

u w v v v = idom(w, z ) v

w

v, w1, . . . , wk = u u v u w1 , . . . , wk

u wi 1 i k o wi o u

v o u u / Doms (v, o) u Doms (v, o) wi 1 i k u Doms (wi , o)

v w ov = idom(w, o)

Z z x f anout (w)

z x fanout (w) z Doms (x, o)


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z w dfs u < dfs w

z x w dfs x = max (dfs u : u

Doms (x, o) dfs u < dfs w ) D (o) x o

v Doms (x, o) z D (o) x o

z dfs z < dfs w (x,w,o ) x fanout (w) x T (o)

z = (x,w,o ) x fanout (w) zmin

P w wanc zmin T (o) x f anout (w) x x f anout (w) u P : u Doms (x, o)

wanc zmin v

P w v w v Doms (w, o) D(o)

wanc o v dfs v dfs zmin wanc v x f anout (w)

v v w o v = idom(w, o)

a

b

c

d

f

e

g

h

i

jh

7

2

g10 i

e4

3a

5b

c

9d

1j8f6

a

b

d

e h

i

j

gf

c

(a) (c)(b)

T ( j ) D ( j )

c D ( j )

c {e,f ,g } T ( j )

c zmin

e j (x,c,j ) x f anout (c) dfs e = 4


62/165

dfs j = 1 zmin = j v c e j zmin

c j j = idom(c, j ) j e dfs j dfs zmin

(V, E , I , O )l = fake node(O)

(l) o O

T (o) = (o) w T (o) o in BFS order

idom (w, l ) = l idom (w, o) = idom(w, l )

Z = ; x f anout (w) x T (o)

Z = Z { (w,v,r )};zmin = z : min (dfs (z, o), z Z )idom (w, o) = ( (w, o), zmin , o);

(x,w,o) (x, o) (w, o)

return x ;

return (idom (x, o), w, o);

l j k

D ( j ) j

j h a f b c g d h

h,g,a,f ,b,c,d h,a,b d l

D (l) D ( j )

g


63/165


64/165

0 1 2 3 4

S p e e

d u p

Number of primary outputs 10 10 10 10

10

0.1

1

10

N u m

b e r o

f e x a m p

l e s

Percentage of maintained dominators

0

5

10

15

20

25

30

20% 40% 60% 80%


65/165


66/165


67/165

v V { root } O(k 2k |n | + |n |k )

k n


68/165

v u v u

C O(|n | | m |)

u Mdom (u, z ) V z

Mdom (u, z ) V u V Mdom (u, z ) z

u z v Mdom(u, z )

v Mdom(u, z ) u z v w Mdom(u, z )

u z Imdom (u, z ) u

Mdom (u, z ) u Imdom (u, z ) fanout (u)

U V Cmdom (U, z) Cmdom (U, z) V u U Mdom(u, z ) Cmdom (U, z)

D ER (z) D R (z)

DR (n) I DR (n)

D R (n) {h, i } { j, k } DR (n)

DER (n) DER (n)

{l, m }


69/165

Imdom (v,root ) v root v

v Imdom (v,root ) = fanout (v) u fanout (v) Imdom (v,root )

fanout (v) {u} u u fanout (v) Mdom (u, root ) fanout (v) { u} u / Imdom (v,root )

v u fanout (v)

v v v

|fanout (v)| = 1 f anout (v) = Imdom (v,root )

u Imdom (v,root ) z Imdom (v,root )

v z Imdom (v,root )

Imdom (v,root ) { z} |Imdom (v,root )| z Imdom (v,root )

v Mdoms (v,root )

v V V { root } V

V =

u

Imdom (v,root )

U Mdoms (u, root ) U Imdom (v,root ) { u}

Imdom (v,root ) Mdoms (u,root )

u Imdom (v,root ) v v V


70/165

u Imdom (v,root ) v V

v Mdom(v,root ) u Mdom(v,root ) W

u W Mdoms (u,root ) z Mdom(v,root ) W

O(|V |n ) n

a

bc

e

df

a d e

c

d

e

d

e c

fanout (c) = Imdom (c, f ) c d e b w f anout (b)

u f anout (b) v fanout (b)

{d, e} c b {c,d,e} {d, e} c c

b Imdom (b, f ) = {d, e}

C (V,E,I , {root }) x V v V U V { x} root

x u U U fanout (v) x v

v V {root } root


71/165


72/165

D R (root ) D ER (root )

DER (root )

C (V,E, I ,O ) o O i I

o O v fanin + (o) : v V I

U v V (I O)

u U v w U v w U v u U v : w = u u

fanout + (w) fanin + (w)

u U v v V I u U v :v fanin (u) fanout (u)

u U v

U e

U e E p1, p2 U e p1 fanin + (u) p2 fanout + (w) w, u U v

w = u

U v U e U v U e U v = {e, f } U e =

{ a, g , d, h } U v U e

P C u U v w U v P P in C : u P, u U v w P :

w = u w U v

w P C u u u

w P : w = u P w fanout + (u) w fanin + (u)

U v u U v w U v u


73/165

u U v P P w U v

P C e U e d U e P

v u u w v

w v fanin + (u) u fanin + (w) v fanin + (w)

v u u w v

w v fanout+

(u) u f anout+

(w) v f anout+

(w) P e = p1, p2 U e v1, . . . , v i = p1, vi+1 =

p2, . . . , vn p1 fanin + (u) p2 f anout + (w) w, u U v vy 1 y i

U v vy i + 1 y n U v

e U e P d U e P

a

c

d

f

e

h

gb

U v U e

U v

U e

C (V,E, I ,O )

P i I o O u U v p1, p2 U e P in C ( 1u U v : u P 1e

U e : e P ) C i I o O P C


74/165

P P C ( 1u U v : u P 1e U e : e P ) P P C ( u U v : u P e U e : e P )

u U v P P C P e U e P P C :u P u U v e P e U e

P P C : u P u U v e P e U e e P e U e u P u U v

e P e U e u P u U v u P u U v e P e U e

P P C : u P u U v e P e U e u P u U v e P e U e

C i I o O u U v

P C P C v P C e P P C v u U v P C e

P C e = P C P C v

P P C v e P e U e P P C e e P e U e

P P C v u U v v1 I, . . . , v i =u , . . . , v n O vy 1 y < i

u vy i < y n u

e U e P P C v u U v

P P C u U v

P P C e p1, p2

p1 fanin+

(u) p2 fanin+

(u) : u U v p1 fanin+

(u) p2 fanout + (w) : u, w U v p1 f anout + (u) p2 fanout + (u) : u U v

P P C e i I o O e = p1, p2 P

p1 fanin + (u) p2 f anout + (w) : u, w U v e U e


75/165

P C u U v e U e

C U e p1, p2 p1, px , p2 fanin ( px ) = { p1} fanout ( px ) = { p2}

px u U v

a, g d, h a, x 1 , g d, x2, h

a

c

d

f

e

h

gbx

x2

1

e U e

U c u U v px p1, px , p2 p1, p2 U e

U c

u U c w U c

i I o O u U c

U c

U c I o o O I o fanin + (o)

{x1, e} {a,b,c} g {f, x 2} {c, d} h

U c C

U v U e

U e U c


76/165

U v S U v U e S

i I o O U v = U e = u S

u u S u U v S

u U v S w S w

u v U v u

c

b

a

d g

f

eh

i

j

c

b

a

d g

f

eh

i

j

c

b

a

dg

f

eh

i

j

c

b

a

dg

f

eh

i

j

(a) (b)

(c) (d)

e U e

U v U e u S U v

e = p1, p2 p1 fanin + (v) p2 fanout + (u) p1 fanin + (u) p2 f anout + (v) v U v e U e

U v U e

U e U v

S S e

e h {e, h}


77/165

S a h U e

f S f S S

g U v U e g S U v U v = {e,f ,g }

U e = { a, h }

S U v U e

a, h a, h a, h

V E I V O V S U v

U e i I fanout + (i) o O f anin + (o)

S U u U v U v :u fanin (v) fanout (v) (I O) S U = U v U = S = i I f anout

+ (i) o O fanin + (o) S S = U = i I f anout

+ (i) o O f anin + (o)

U v S

S U e

v U v S U v U v U e

|U v | + |U e |

a, b U v

fanout (a) = {c} P = b = v1 , . . . , c = vn

vi P 1 < i < n vi / fanout + (u), u U v {b} p1, p2 U e : p1 = vi

vi 1 i < n


78/165

(V, E , I , O )S = i I f anout

+ (i) o O fanin + (o)U v = U e =

S =

v = (S )S = S (fanin (v) fanin (v))U v = U v { v}

p1 , p2 E : p1 I p2 f anout + (v) U e = U e { p1 , p2 } u U v

p1 , p2 E : p1 fanin + (v) p2 f anout + (u) p1 , p2 E : p1 fanin + (u) p2 f anout + (v)

U e = U e { p1 , p2 }U c = (U v , U e )

U c

U v = ( U v { a, b}) { c}

U e = U e { a, c } U v = ( U v { b}) { c} b U v a, c U e

P = b = v1 , . . . , c = vn

vi P 1 < i < n vi / fanout+

(u), u U v {b} p1, p2 U e : p1 = vi vi 1 i < n

p1, p2 U e p1, px , p2

{c, d} {e, f } a, h b g, j j

{h, i } U c = {k, l}


79/165

(b)

ac

a

bbc

(c)

le

a

c

n

b

d

fg

h

i

j

k

m

(d)

le

a

c

n

b

d

fg

h

i

j

k

m

(f)

le

a

c

n

b

d

fg

h

i

j

k

m

(e)

le

a

c

n

b

d

fg

h

i

j

k

m

b

(a)

bc

ac

a

U v ={a, b} U v = {c} U v = {a} U e = { a, c } U v = {c} U e =

U v = {b,c,d,e,f } U e = { a, h , g, l } U v = {b,i,j } U e ={ a, h , g, l } U v = {h,i,l } U e = U v = {k, l} U e =


80/165

U c

D ER (root ) U i

U i DER (root )

v U i y U i

U i = {root }

v DER (root ) v

i U i

U i U i = {g, e} U o = { j } DR (k)

U i root (V,E,U i , U o) U i U o root U c

(V,E,U i , U o) U i U o U c = {i, h } U c

{e, g} j

{e, g} k

D R (k)

D ER (k) U i U o

U c i U i U o i, U c U c , U o DER (k) DER (k)

U i U o

C (V,E,I , {root }) V E I root

D R (root ) C

D R (root ) DER (root ) v V


81/165


82/165


83/165

|V | |V |

|V | |V | e U e

O(|V |2) O(|V |3)

v V


84/165


85/165


86/165


87/165


88/165

f

f

f (X ) g h

f (X ) = h(g(Y ), Z )

Y Z X ={x1, x2, . . . , x n } f g h

Y f

g Y

f(X)

Y

Z

g(Y)

h(g(Y), Z)


89/165

f (I ) C = ( V,E,I , {root }) v V

g(Y ) f (I ) = h(g(Y ), Z )

|I | |Y | 1 + |Z |

Pcut (root ) C root

v V root v Pcut (root ) v Pcut (root ) i

I fanin (v) v Doms (i,root )

v V root v

v DR (v)

v Pcut (root ) DR (root ) v Pcut (root ) v DR (root )

v V

v v V i I : i fanin (v)

v Pcut (root )

DR (root ) v P cut (root )

DR (root )

Pcut (root )

C (V,E,I , {root }) Rpath (v) v V i I v

z Rpath (v) z Doms (i,root )

v V i I fanin (v) 1z Rpath (v) : z Doms (i,root ).


90/165

Rpath (v) |I | DR (root )

Pcut (root ) Pcut (root ) = Rpath (v) v V Rpath (v) =

v V v

Rpath (v) Rpath (w) w fanin (v) v DR (root ) v

Rpath (v) v Rpath (v) w V Rpath (v)

D R (root ) v Pcut (root ) DR (root )

Rpath (v) DR (root )

Rpath (v) v V

i I fanin (v) 1z Rpath (v) : z Doms (i, root )

i I fanin (v) z Rpath (v) : z Doms (i,root )

i I i I fanin (i) =

i I DR (root )

i I Rpath (i) = {i}

C i I

v V w fanin (v) i I fanin (w) z Rpath (w) : z Doms (i,root )

i I fanin (v) z Rpath (v) : z Doms (i,root )

fanin (v) = {w1 , w2} w fanin (v) i

I fanin (w) z Rpath (w) : z Doms (i,root )

i I fanin (w1) z Rpath (w1) : z Doms (i,root ) i I fanin (w2) z Rpath (w2) : z Doms (i, root )

u1 P 1 u2 P 2 u1 u2 P 1 P 2 i I (fanin (w1) fanin (w2)) z Rpath (w1) : z

Doms (i,root ) z Rpath (w2) : z Doms (i, root )


91/165

u1 P 1 u2 P 1 u1 u2 P 1 i I (fanin (w1) fanin (w2)) z (Rpath (w1) Rpath (w2)) : z

Doms (i,root )

Rpath (v) = w fanin (v) Rpath (w)

i I (fanin (w1) fanin (w2)) z Rpath (v) : z Doms (i,root )

v V I fanin (v) = I fanin + (v)

fanin (v) = {w1, w2} fanin + (v) = fanin (w1)fanin (w2)

I fanin (v) = I (fanin (w1) fanin (w2)) v V

v i I fanin (v) z Rpath (v) : z Doms (i, root )

i I fanin (v) ( z Rpath (v) : z Doms (i, root )) ( u Rpath (v) u = z u / Doms (i, root ))

i I Rpath (i) = {i}

i I 1z Rpath (i) : z Doms (i, root )

v V I fanin (v) = {w1 , w2}, w1 = w2 i I v i I fanin + (v)

i w1 w2 i fanin (w1) i fanin (w2)

i w1 w2i fanin (w1) fanin (w2)

i fanin (w1) Rpath (w1) Rpath (v)


92/165

x1 Rpath (w1) : x1 Doms (i,root ) x2 Rpath (w2) : x2 Doms (i,root ) x1 = x2

fanin (v) = {w1 , w2}, w1 = w2 i x1 x2

x1 x2 i root x1 = x2

w1 w2 v

Rpath (v) v V

D R (root ) v DR (root ) DR (root )

v V Rpath (v) = {v}

v

v DR (root )Rpath (v) = ( Rpath (v) Dominated (v,root )) { v}

Rpath (v) v v v V : v Pcut (root ) Rpath (v) = {v}

Rpath (v) v V |I | O(|V | | I |)

Rpath (v) = w fanin (v) Rpath (w) |V | 1 Rpath (w) w fanin (v) O(|I |2 |V |)

v V O(|I |2 |V |2)

DR (h) Rpath (v)


93/165

(V,E,I , {root })DR (root ) = (V,E,I,root )Pcut (root ) =

v V Rpath (v) =

v V Rpath (v) = w fanin (v) Rpath (w) v DR (root )

Rpath (v) = ( Rpath (v) Dominated (v,root )) { v} v DR (root )

Rpath (v) = {v} Pcut (z) = P cut (z) { v}

Pcut (z)

v Pcut (h) v V DR (h) Rpath (v) =

(Rpath (v) Dominated (v,root )) { v} Rpath (v) v V Pcut (h)

a b c d i I Rpath (i) ={i} Pcut (h) = {a,b,c,d}

e Rpath (b) Rpath (c) {b, c} e b c Rpath (e) = {e} f

f e e Rpath (f ) Rpath (f ) ={e, f } f g Rpath (g)

{e, g} g Pcut (h) h V h

Pcut (h)


94/165

b

c

d

a

e

f

g h

b

c

d

a

ef

gh

(a) (b)

DR (h)

v Dominated (v, h ) Rpath (v) Pcut (h) {a} {a} {a}

{b} {b} {a, b} {c} {c} {a,b,c}

{d} {d} {a,b,c,d} {b,c,e} {e} {a,b,c,d,e}

{a, f } {e, f } {a,b,c,d,e } {g, d} {e, g} {a,b,c,d,e}

{e,f ,g ,h } {h} {a,b,c,d,e,h }

N DO


95/165

f = a(b + c) {b, c}

f

f (X ) = h(g(Y ), Z )

f : {0, 1}n {0, 1} g : {0, 1}|Y | {0, 1,...,m 1} h : {0, 1,...,m 1} {0, 1}|Z | {0, 1} m h

k = log2 m g1, g2 , . . . , gk

f (X ) = h(g1(Y ), g2(Y ), . . . , gk (Y ), Z )

f (X ) = h(g1(Y ), g2(Y ), Z )


96/165

N N DO N DO N DO


97/165

(a)

N u m

b e r o

f c

i r c u

i t s

41010 10 10

0 1 210

3

Sum of all proper cuts in a circuit graph

(b)30 40 50 60 70 80 9010 20

N u m

b e r o

f o u

t p u

t s

respect to all inputs in the circuit graphPercentage of inputs in a proper cut with

2

6

8

10

12

14

16

0

4

20

18

0

2000

3000

4000

5000

6000

1000

(a)

a

bc

f

(b)

b

a

c

f

f =a(b + c) f = ab + ac

f(X)

Y

Z

g (Y)1

2g (Y)

h(g (Y), g (Y), Z)1 2


98/165

4 3 2 1

2

1

0

1

2

0 2

B D D

b a s e

d a p p r o a c

h

10

10

10

10

10

10

10

3

4

10 10 10 10 10 10 101

Combined approach

Time in seconds

u v v o

u u v o


99/165

c, g g Doms (g, k) = {g,j ,k }

k

b

a

c

d

e

f

gh

ij

kxstuckat fault

B (v) v i I B(i) B(i) = Doms (i, o)

B (v) v V I B (w) w f anout (v) v B(v) v v

w


100/165

(V,E,I,root ) i I

B (i) = Doms (i,root ) v V I

B (v) = w fanin (v) B (w) v V v B (v)

v

w w w

w w

k p g

j l m i l m

o1

o2

o1

o2

c

b

d

h

k

e

g

da

bf

i

(a) (b)

b

dk

e

g

c

a

bf

h

i

j

mn ppnl


101/165

1/true 0/false

true false

false

true

false

a = 0 b = 1 c = 0


102/165


103/165

x1 = ( z + d)x2 = ( a + z + e)x3 = ( e+ e+ f )x4 = ( b+ f + g)x5 = ( c+ f + h)x6 = ( h + g)

x1

2x

x3

x5

x4

x6

d3

d2d1

d4

a = 0/1

f = 0/4z = 1/4

d = 0/4

e = 1/4

c = 0/3h = 0/4

b = 1/2g = 1/4

k (conflict)

I


104/165


105/165

A B

A B


106/165

A B oA oB

oA oB

oA oB


107/165

C


108/165


109/165

(C 1, C 2) i I 1 I 2

ROBDD (i) = () v V 1 I 1

ROBDD (v) = vw fanin (v) ROBDD (w) v V 2 I 2 ROBDD (v) =

vw fanin (v) ROBDD (w)

ROBDD (root 1) == ROBDD (root 2) C 1 C 2

C 1 C 2

v v v


110/165

bdx

adb

x

f1f2

f2

f1

(a) (b)

(d)(c)

adbb

ca

bd

b

ca xx

f 1 f 2

x x x

f 1 f 2 f 1 f 2 f 1 f 2

f 1 f 2 f 1 f 2

f 1 f 2


111/165

A B oA oB

oA oB

A

B


112/165

f (a,b,c) = ab+ bc

fb

a

c


113/165

X

1X

X

1X

X

0

1 X

0

1

1

10

1

10

X

X0

X

X0

X

0X

X

00

X

X1 1

1

1

h e g g

e g f e b

c b c f

f b = c =

c = b = b = c = b = c =


114/165

g

e

1

0X

1

X

0 1

10

1

1i

hf

c

b

a

d

0

X

X

(b)

g

e

1

00

X

1

1

X

0 1

10

1

1i

hf

c

b

a

d

0

(c)

g

e

ih

fc

b

a

d(d)

0

g

e

1

0X

X

X

1

10

1

1i

hf

c

b

a

d

0

(a)

X

X

X


115/165

2n n


116/165


117/165


118/165


119/165

Z p p + x1 , x2, . . . xa , xb, . . .


120/165

x1 x2 x1 x2

F = ( f 0f 1 . . . f 2n 1) f T n 2n 2n

T 1 df = 1 0 1 1 ,

T n df = T 1 T n 1,

C f = T n F C f = ( c0c1 . . . c2n 1)

A[f ] : Z n p Z p n f

A[f ] C f T

n A[f ]

A[f ] =2n 1

i=0ci

i1x1 inxn

(i1 , . . . , i n ) i ij

x j

i jx j =

1 i j = 0x j i j = 1,

T 2 F f = x1 x2 C f c0 = 0 c1 = 1 c2 = 1 c3 = 2

A[f ](x1 , x2) = 0 1 1 + 1 x1 1 + 1 1 x2 + ( 2) x1 x2 A[f ](x1, x2) = x1 + x2 2x1x2

T 2 F C f x1x2

1 0 0 0 1 1 0 0 1 0 1 0

1 1 1 1

0110

=

011

2

0 00 11 01 1

F C f F = C f [T n ] 1


121/165

[T 1] 1 df = 1 01 1 ,

[T n ] 1 df = [T 1] 1 [T n 1] 1.

[T 2] 1 C f F x1x2

1 0 0 01 1 0 01 0 1 01 1 1 1

011

2

=

0110

0 00 11 01 1

x = y

f A[f ]x 1 x

x y x yx y x + y x yx y x + y 2 x y

f = x f |x =1 xf |x =0 A[f ] xii { 1, . . . , n }

A[f ] = (1 x i ) A[f |x i =0 ] + x i A[f |x i =1 ].

(a1, . . . , a n )


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A[f g] =A[f ] A[g], A[f ] = 1 A[f ]

f g A[f g] = A[f ] A[g]

xki A[f ] xi

d e A[d] = a b A[e] = a + b a b f A[f ] = A[A[e]A[c]] = a c+ bc a bc

g a b d f

A[g] A[g] = A[A[d]+ A[f ] A[d]A[f ]] = a b+ a c+ bc a bc a2 bc a b2 c+ a2 b2 c

A[g] = a b+ a c+ bc a bc a bc a bc+ a bc A[g] = a b + a c + b c 2 a b c

ab

c

e

d

f

g

O(|V | 2| I | ) V I

v V 2| I |


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(V,E,I,root,a 1 , . . . , a n )DR (root ) = (V,E,I,root )

v V v I

A[v] = a i

A[v] = (v,fanin (v)) v DR (root )

u DR (root ) : v = idom(u,root ) x[u] = A[u]

A[v] = (A[v], A[u], x[u]) (v DR (root ) | fanout (v)| > 1)

x[v] = ;

x[v] = A[v] A[root ]

C a1 , . . . , a n

v V A[v] v x[v]

v i I A[i] ai Z p

x[u] u fanin (v) (v,fanin (v))

v u fanin (v) A[u] v 1 u {u} = fanin (v) A[v]

v A[x x] = A[x] A[x]

x xk k > 1 x

i p1, p2 P DR (root ) i I

root i p1 p2


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p1, p2 P DR (root ) P = i I , . . . , root x p1 p1

|fanout ( p1)| > 1 A[ p2] A[i] A[v] v V

|fanout ( p1)| = 1 p1 p2 A[i] A[ p2]

|fanout ( p1)| > 1 p1 p2

A[i] x p

1 p1 A[ p1]

p2 A[ p1]

A[ p2]|x p 1 = A [ p1 ] = (1 A[ p1]) A[ p2|x p 1 =0 ] + A[ p1] A[ p2 |x p 1 =1 ].

p2 p1 x p1 p2 x p1 A[v] v fanout + ( p2)

v V i root p1, p2

A[i]

v V i root

v DR (root ) A[i] i I v V

i I P i , . . . , root e = p1 , p2

P p2 = idom( p1,root ) v DR (root ) root e D R (root )

e = p1 = v, p2

|fanout (root )| = 0 root


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v DR (root )

i I

A[i] i I v V

v DR (root )

v V A[v] x[v]

x[c] x[d] c d

g xc xd A[g]

g|x c = A [c]|x d = A [d] xc xd A[d] xc xc

xd

A[v] xv xw w

v

b

c

ae

fgd

b

c

ae

gd

(a) (b)

d h A[d]

aa aa A[h]


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v A[v] x[v]a a a aab ab abc ac xcd ab xc xde aa xd aa xdf x c xd xc xdg aa xc xd aa xc xd

g|x c = A [c] aa ac xd aa ac xdg|x c = A [c]|x d = A [d] aa ac ab xc aa ac ab xc

g|x c = A [c]|x d = A [d]|x c = A [c] aa ab ac aa ab acg|x d = A [d] aa xc ab aa xc ab

g|x d = A [d]|x c = A [c] aa ab ac aa ab ac

d

j f

c

e

gb

a

i

k

h

(a)

a

c

h

k

i

(b)

d

g

b

fj

e

xb h

F = Z p a e g A[d] = 1 3xb

A[i] = 1 5(1 3xb) = 4 + 15xb A[ j ] = 1 1(1 3xb) = 3 xb A[h] = 1 ( 4 + 15xb)(3xb) = 1 + 12 xb 45x2b = 1 33xb

h A[h] = 1 3xb aa

d h xd d

A[d] = 1 3xb h

b xd d A[i] = 1 5xd


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A[ j ] = 1 xd A[h] = 1 (1 5xd )(1 xd ) = 6 xd 5x2d = xd xd A[d] = 1 3xb A[h] = 1 3xb

A[f ] c b A[k] = 1 2xb A[h] = 1 3xb

A[f ] = 1 (1 2xb)(1 3xb) = xb xb A[b] = 4 A[f ] = 4

(aa , a b, a c , a e , a g ) = (3 , 4, 2, 5, 1)

A[v]

Z p

Z p

D R

n n = 2 i , i = 6, . . . , 16


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(a)

m

j

i

d

c

b

a

(b)

k

h

a

b

d

c

e

f

g

i

jl

m

(c)

a

b

dc

i

j

m

DR (m).


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n

BDDbasedMTBDDbasedSATbased

runtime in ms

1

0

2

3

4

5

m i l l i s e c o n

d s

1

number of inputs

1 2 3 4 5

10

10

10

10

10

10

10 10 10 10 1010


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MTBDDbasedBDDbased

SATbased

number of inputs

r u n

t i m e

d e c e

l e r a

t i o n w

i t h r e s p e c

t t o

c i r c u

i t s

i z e

d u p

l i c a

t i o n

1

1.5

2

2.5

3

3.5

4

4.5

0 10000 20000 30000 40000 50000 7000060000

|fanout (v)| > 1


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132/165

10 10 10 10 10 10 10 10 104 3 2 1 0 1 2 3 4

Number of dominators per input

N u m

b e r o

f o u

t p u

t s

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

10 10 10 10 10 101010

10

10

10

10

10

10

10

10

65432101

1

0

1

2

3

4

5

6

BDDbased algorithm

D o m

i n a

t o r

b a s e

d a

l g o r i t h

m

Maximum number of DD nodes


133/165

410

10 3

102

101

100

101

102

10 5

102

101

100

101

102

103

104

105

Time in seconds

BDDbased algorithm

D o m

i n a

t o r

b a s e

d a

l g o r i

t h m

A


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135/165

t t maxN DD maxN DD


136/165

t t maxN DD maxN DD


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xa xb a b i

A[f (i) = a b] = xa + xb 2 xa xb c d

xa c xb d A[f ( j ) = c d] = xa + xb 2 xa xb

xc xd c d A[f ( j ) = c d] = xc + xd 2 xc xd


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1 2 3 4 5

number of inputs

runtime in ms

BDDbased

MTBDDbasedSATbasedMTBDDbased, variablerecycling

1

1

0

2

3

4

5

m i l l i s e c o n

d s

10

10

10 10 1010 10

10

10

10

10

10

10

BDDbasedMTBDDbased

MTBDDbased, variable recycling

number of inputs

n u m

b e r o

f B D D n o

d e s

0

1e+06

1.5e+06

2e+06

2.5e+06

3e+06

0 20000 40000 60000

500000


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number of inputs

c i r c u

i t s

i z e

d u p

l i c a

t i o n

r u n

t i m e

d e c e

l e r a

t i o n w

i t h r e s p e c

t t o

MTBDDbasedBDDbased

SATbasedMTBDDbased, variable recycling

1

1.5

2

2.5

3

3.5

4

4.5

0 10000 20000 30000 40000 50000 60000 70000

A[f ]

k v1, . . . , vk V u DER (root )

k x1, . . . , x k v1, . . . , vk

x1, . . . , x k

A[v1 , . . . , vk ](i1, . . . , i k ) =k

j =1

i jA [v j ]


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(V,E,I,root,a 1 , . . . , a n )DER (root ) = (V,E,I,root )

v V v I

A[v] = a i

A[v] = (v,fanin (v)) v DER (root )

u DER (root ) : v = idom(u,root ) x[u] = A[u]

A[v] = (A[v], A[u], x[u]) (v DER (root ) | fanout (v)| > 1)

z V |v Mdom (z,root ) x[v] = ;

x[v] = A[v] A[root ]

C a1, . . . , a n

i = {0, . . . , 2k 1} (i1 , . . . , i k ) i i jA [v j ]

i jA [v j ] =

1 A[v j ] i j = 0A[v j ] i j = 1,

j { 1, . . . , k } w

v1 , . . . , vk A[w] A[u |(v1 ,...,v k )=( i1 ,...,i k )] u

x1, . . . , x k v1 , . . . , v k

A[u] =2k 1

i=0(A[v1, . . . , vk ](i1 , . . . , i k ) A[u |(v1 ,...,v k )=( i1 ,...,i k ) ])

(i1 , . . . , i k ) i

DR (f ) D ER (f ) {d, e}


141/165

a

b fd,e

a

b

ce

df

a

b

ce

df

c

(a) (b) (c)

a b c DR (f ) DER (f )

{d, e}

f (f ) = a b c A[a b

c] = aa ab ac

{d, e} b DER (f ) |fanout (b)| > 1 xb

b d e A[d] = aa xb A[e] = ac xb

2k

A[d, e](0, 0) = (1 A[d]) (1 A[e]) = (1 aa xb) (1 ac xb)

A[d, e](0, 1) = (1 A[d]) A[e] = (1 aa xb) (ac xb)

A[d, e](1, 0) = A[d] (1 A[e]) = ( aa xb) (1 ac xb)

A[d, e](1, 1) = A[d] A[e] = (aa xb) (ac xb)

xb

A[d, e](0, 0) = 1 aa ab ab ac + aa ab ac

A[d, e](0, 1) = ab ac aa ab ac

A[d, e](1, 0) = aa ab aa ab ac

A[d, e](1, 1) = aa ab ac


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{d, e} f xd xe e d

A[f ] =A[d, e](0, 0) A[f |x d =0 ,x e =0 ] + A[d, e](0, 1) A[f |x d =0 ,x e =1 ]+A[d, e](1, 0) A[f |x d =1 ,x e =0 ] + A[d, e](1, 1) A[f |x d =1 ,x e =1 ] =(1 aa ab ab ac + aa ab ac) 0 + ( ab ac aa ab ac) 0+(aa ab aa ab ac) 0 + ( aa ab ac) 1 =aa ab ac

DER (root )


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t D ER ( root )

t

t

N Mdom maxN DD maxN DD

N Mdom DER (root ) tD ER (root )

D ER (root ) maxN DD t DER (root )


144/165

A D D

A D D

(a) (b)

BDD BDD

Number of lookups Runtime in sec

101

100

102

103

104

105

106

107

101

102

103

104

105

106

100

105

104

103

102

101

100

105

104

103

102

101

100


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146/165

(a1, . . . , a n ) H 1 H 2

f 1 = f 2 H 1 = H 2

x1, . . . , x n Z |Z |

( |Z | 1|Z | )n

Z

= 1 ( |Z | 1|Z | )n k k

f 1 f 2


147/165

f 1 f 2

f 1 = f 2 f 1 = f 2 f 1 = f 2 f 1 = f 2 f 1 = f 2

f 1 = f 2 f 1 = f 2 f 1 = f 2 f 1 = f 2 f 1 = f 2


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O(|V | | E |)


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1020


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dominator-based algorithms in logic synthesis and verification

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