synchronous networks an interconnection structure with an - - PowerPoint PPT Presentation

synchronous networks

an interconnection structure with

• boolean nodes, primary inputs and primary outputs

ti d l i i t d ith h b l d an interconnection structure with

• a propagation delay is associated with each boolean node;

all input-output delays of a node are equal

• single-phase edge-triggered memory elements on the edges;

g p g gg y g all memory elements are triggered by the same clock

• cycles have at least one memory element

a d v b c d e

D D

i

D

e u

D D

i

D

retiming

i i ddi d l i i l k d l retiming = adding, deleting, moving clocked memory elements, neither changing the topology of the network nor the boolean operations in the node nor th oo an op rat ons n th no

retiming

i i l fi filt

• riginal fir-filter:

clock time

• 2
• 1

1

• utputs of

flipflops

• utput

input

retiming

clock

retimed fir-filter:

p

flipflop delay adder delay

clock time

• 2
• 1

1

input

• utputs of

flipflops flipflops

• utput

adder delay flipflop delay

retiming changes

flipflop delay

m g g the temporal behavior of the network, and not the sequential behavior

leiserson's correlator

g e f

+

host g

+

e

+

f

=

host a

= =

c d

=

b

a finite directed multigraph,

7 e g 7 f 7

vertex weighted (delay ≥ 0), and arc weighted

3 d h

(number of memory elements)

3 1 c 3 1 a 3 b 1 1

)

retiming steps

b shifting clocked memory elements: D b clock D a c l k a D c clock model:

a

d

b

d b

a

d

b

b

d

c

d

a c

c

d

a c retiming = adding, deleting, moving clocked memory elements,

= delay of boolean node

• d

neither changing the topology of the network nor the boolean operations in the node

retiming model

retiming = an integer labeling r of vertices. interpretation: interpretation:

r < 0 r > 0 r elements removed r elements added r elements added r elements removed

a retiming is legal if the retimed network has no negative weights

w r

netwerk, retimed by : all “equivalent” synchronous networks can be obtained from an initial network by has no negative weights

wij i j rj ri

be obtained from an initial network by legal retiming

[Leiserson 1983]

i j ri rj wij+ rj- ri

feasible clock periods

d(p)=path delay= d(p)=path delay= sum of all vertex-weights (delays) on a directed path ( ) h i h w(p)=path weight= sum of all arc-weights ( # memory elements) on a directed path properties under retiming: condition: all cycle weights must be positive! properties under retiming:

changes in path weights depend only on end vertices cycle weights are invariant

a clock period is feasible if it exceeds any path delay on a zero-weight path y p y g p the minimum feasible clock period Φ = max {d(p) | w(p) = 0}

a retiming problem formulation

problem statement for clock period minimization : given G (V, E, d, w), find a legal retiming r so that

Φ { d( ) | ( ) 0 }

is minimized

Φ = max { d(p) | w(p) = 0 }

example : smaller correlator

7 a a h c

+

host a 3 2 b c

= =

host b c 3 b

two auxiliary matrices

p p

W(u,v) = min {w(p) | u → v}: the minimum number of m.e. on any path from u to v D(u v) = max {d(p) | u → v and w(p) = W(u v)}:

l b φ i f ibl l k i d fif D( ) φ W( ) 0

D(u,v) max {d(p) | u → v and w(p) W(u,v)}: the maximum path delay over a "critical" path

• ther facts:

a real number φ is a feasible clock period fif D(u,v) >φ → W(u,v)>0

the set of "critical" paths invariant under retiming Wr(u,v) := W(u,v) + r(v) - r(u) D (u v) := D(u v) Dr(u,v) := D(u,v) Φ, the minimum feasible clock period, is equal to some D(u,v)

→ W and D are useful quantities!

all-pairs shortest paths

4 3 2 input: 3 8 ∞

• 4

4 8 1 2 1 3 ∞ ∞ 2 3 4 ∞ 8 ∞

• 5

1 ∞ 4 7 ∞ ∞ M = 1 7 2 6

• 4

4 5

• 5

∞ ∞ ∞ 6

• utput:

3 1

• 3
• 4

2 1

• 4
• 1

N 4 3 N 4 4 5 2 1 1 7 2 8 4

• 1

5

• 5

1 5 6 3

• 2

C = 4 4 4 3 3 3 N 4 4 2 N 5 1 1 N Π =

floyd-warshall

assume V = {1,2,…,n} consider the subset V(k) = {1 2 k} for some k ≤ n consider the subset V(k) = {1,2,…,k} for some k ≤ n let i and j be two vertices consider the set P(i,j,k) of paths from i to j ith i t m di t d s f m V(k) with intermediate nodes from V(k) let p be a shortest path in P(i,j,k) if k is not an intermediate node of p then if k is not an intermediate node of p, then if k is an intermediate node of p, then p is a shortest path in P(i,j,k-1) i h h h i P(i k k 1) p=qr with q a shortest path in P(i,k,k-1) and r a shortest path in P(k,j,k-1) recursively formulated: let be the weight of the shortest path in P(i,j,k)

(k) ij

c

ij

m

if k = 0

(k)

c =

y

(k 1) (k 1) (k 1) ij ik kj

min(c ,c c )

− − −

+ if k > 0

j ij

c =

desired answer:

( )

(n) (n) ij

C c =

bottom-up computation

4 3 2 floyd-warshall(W) 1 n ← rows[W] 8 1 2 4 1 3 5 [ ] 2

(0)

C W ← 3 for k ← 1 to n 4 do for i ← 1 to n 7 6

• 4

4 5

• 5

(k) (k 1) (k 1) (k 1) ij ij ik kj

c min(c ,c c )

− − −

← +

5 do for j ← 1 to n 6 7 return

(n)

C

3 8 ∞

• 4

3 8 ∞

• 4

3 8 ∞

• 4

) d d , d min(

) ( 11 ) ( 41 ) ( 41

+

C

∞ ∞ 2 3 4 ∞ 8 ∞

• 5

1 ∞ 4 7 ∞ ∞ D =

(0)

∞ ∞ 2 3 4 ∞ 8 ∞

• 5

1 ∞ 4 7 ∞ ∞ C =

(0)

C =

(1)

3 8 4 ∞ ∞ 1 7 ∞ 4 ∞ ∞ ∞ ∞ ∞ 6 ∞ ∞ ∞ 6

bottom-up computation

4 3 2 floyd-warshall(W) 1 n ← rows[W] 8 1 2 4 1 3 5 [ ] 2

(0)

C W ← 3 for k ← 1 to n 4 do for i ← 1 to n 7 6

• 4

4 5

• 5

(k) (k 1) (k 1) (k 1) ij ij ik kj

c min(c ,c c )

− − −

← +

5 do for j ← 1 to n 6 7 return

(n)

C

3 8 ∞

• 4

3 8 ∞

• 4

3 8 ∞

• 4

) d d , d min(

) ( 12 ) ( 41 ) ( 42

+

C

∞ ∞ 2 3 4 ∞ 8 ∞

• 5

1 ∞ 4 7 ∞ ∞ D =

(0)

∞ ∞ 2 3 4 ∞ 8 ∞

• 5

1 ∞ 4 7 ∞ ∞ C =

(0)

C =

(1)

3 8 4 ∞ ∞ 1 7 ∞ 4 ∞ ∞ 2 ∞ ∞ ∞ 6 ∞ ∞ ∞ 6

bottom-up computation

4 3 2 floyd-warshall(W) 1 n ← rows[W] 8 1 2 4 1 3 5 [ ] 2

(0)

C W ← 3 for k ← 1 to n 4 do for i ← 1 to n 7 6

• 4

4 5

• 5

(k) (k 1) (k 1) (k 1) ij ij ik kj

c min(c ,c c )

− − −

← +

5 do for j ← 1 to n 6 7 return

(n)

C

3 8 ∞

• 4

3 8 ∞

• 4

3 8 ∞

• 4

C

1

• 3

2

• 4

∞ ∞ 2 3 4 ∞ 8 ∞

• 5

1 ∞ 4 7 ∞ ∞ D =

(0)

∞ ∞ 2 4 ∞ ∞

• 5

1 ∞ 7 ∞ ∞ C =

(0)

C =

(1)

3 8 4 ∞ ∞ 1 7 ∞ 4 ∞ ∞ 2 5

• 5
• 2

C =

(n)

3 7 2 4

• 1
• 4
• 5

1 5

• 1

3

• 2

∞ ∞ ∞ 6 ∞ ∞ ∞ 6 ∞ ∞ ∞ 6 8 5 1 6

construction of shortest paths

) k ( ij

π

let be the predecessor of j on a shortest path in P(i,j,k)

= π

) ( ij

NIL

if i = j or

∞ =

ij

w

i if i ≠ j and

∞ <

ij

w

) 1 k ( kj ) 1 k ( ik ) 1 k ( ij ) 1 k ( kj

d d d if

− − − −

+ > π = π

) k ( ij

(k > 0)

) 1 k ( kj ) 1 k ( ik ) 1 k ( ij ) 1 k ( ij

d d d if

− − − −

+ ≤ π

desired answer:

( )

) n ( ij ) n (

π = Π

kj ik ij kj

d d d if + > π

remarks

floyd-warshall(W) 1 n ← rows[W] [ ] 2

(0)

C W ← 3 for k ← 1 to n 4 do for i ← 1 to n

(k) (k 1) (k 1) (k 1) ij ij ik kj

c min(c ,c c )

− − −

← +

5 do for j ← 1 to n 6 7 return

(n)

C

• detection of negative-weight cycles

(a negative number on the diagonal of )

• running time

) (V3 O

C

(n)

C

(a negative number on the diagonal of )

• space-optimization possible
• works for any (min,+)algebra that is a semiring

( )

C

calculating W and D

• reweight each edge with
• calculate all-pair shortest paths with:

( )

ij, i

w d − – additions must be done as two-component vectors – comparisons must be done lexicographically, p g p y, i.e. for and if or ( and )

1 1 1

c (a ,b ) =

2 2 2

c (a ,b ) =

1 2

c c <

1 2

a a <

1 2

a a =

1 2

b b < f ( ) – if then and

1 2 1 2 1 2 1 2 (n) ij ij ij

c (x ,y ) =

ij ij

w x =

ij j ij

d d y = −

checking clock feasibility

p p p

W(u,v) = min {w(p) | u → v}: the minimum number of m.e. on any path from u to v D(u v) = max {d(p) | u → v and w(p) = W(u v)}:

retiming r allows a clock period φ fif

( ) ( ) ( ) [ ]

v u w v r u r

E e

→ ≤ − ∀ ∈

e

l b φ i f ibl l k i d fif D( ) φ W( ) 0

p

D(u,v) max {d(p) | u → v and w(p) W(u,v)}: the maximum path delay over a "critical" path

( ) ( ) ( ) [ ]

1 v , u W v r u r ] v , u [ D

v u,

− ≤ − ⇒ φ > ∀

a real number φ is a feasible clock period fif D(u,v)>φ → W(u,v)>0

7 h b 2 c 2 a 2 W h

2 ) b ( r ) h ( r ≤

7 a 2 2 2 2 2 2 h b c a

) a ( r r(b) ) c ( r r(b) 2 ) b ( r ) h ( r ≤ − ≤ − ≤ −

3 2 h c h 13 b 3 3 c 6 6 a 13 13 D h b

) h ( r ) a ( r ) a ( r r(c) ≤ − ≤ −

3 2 b 13 10 7 3 13 10 6 3 13 13 10 7 b c a

checking clock feasibility

p p p

W(u,v) = min {w(p) | u → v}: the minimum number of m.e. on any path from u to v D(u v) = max {d(p) | u → v and w(p) = W(u v)}:

retiming r allows a clock period φ fif

( ) ( ) ( ) [ ]

v u w v r u r

E e

→ ≤ − ∀ ∈

e

l b φ i f ibl l k i d fif D( ) φ W( ) 0

p

D(u,v) max {d(p) | u → v and w(p) W(u,v)}: the maximum path delay over a "critical" path

( ) ( ) ( ) [ ]

1 v , u W v r u r ] v , u [ D

v u,

− ≤ − ⇒ φ > ∀

a real number φ is a feasible clock period fif D(u,v)>φ → W(u,v)>0

7 h b 2 c 2 a 2 W h

2 ) b ( r ) h ( r ≤ 1 ) a ( r ) h ( r ≤ −

7 a 2 2 2 2 2 2 h b c a

) a ( r r(b) ) c ( r r(b) 2 ) b ( r ) h ( r ≤ − ≤ − ≤ − 1 ) h ( ( ) 1 ) a ( r r(b) 1 ) h ( r r(b) 1 ) a ( r ) h ( r − ≤ − − ≤ − ≤

3 2 h c h 13 b 3 3 c 6 6 a 13 13 D h b

) h ( r ) a ( r ) a ( r r(c) ≤ − ≤ − 1 ) a ( r r(c) 1 ) b ( r ) c ( r 1 ) h ( r r(c) − ≤ − ≤ − − ≤ −

3 2 b 13 10 7 3 13 10 6 3 13 13 10 7 b c a

7 = φ

1 ) c ( r ) a ( r 1 ) b ( r ) a ( r ) ( ( ) ≤ − ≤ −

linear programming

find an n-vector x

p g g

maximizing the objective function

∑ =

n 1 i i ix

c

subject to the constraints subject to the constraints A x ≤ b where A is an m×n matrix and b an m-vector where A is an m×n matrix and b an m vector important facts: any such linear programming problem can be solved in polynomial time y p g g p p y any problem that can be formulated as LP belongs to P the widely used simplex-algorithm does not run in polynomial time ! ! feasibility: find an n-vector x subject to the constraints A x ≤ b j where A is an m×n matrix and b an m-vector

difference constraints

• feasibility problem (the constraints of an LP: )

A x ≤ b

• each row in A has only 0's except for one 1 and one -1

1 -1 0 0 0 x x

2 1

≤ −

• feasibility problem (the constraints of an LP: )

A x ≤ b

2 1

x x x 1 1 0 0 0 1 0 0 0 -1 0 1 0 0 -1

• 1 0 1 0 0

≤

• 1

1 5 1 x x

5 1

− ≤ − 1 x x

2 5

≤ − 5 x x

1 3

≤ −

2 1 5 4 3

x x x

• 1 0 0 1 0

0 0 -1 1 0 0 0 -1 0 1 0 0 0 1 1 ≤ 4

• 1
• 3

3 4 x x

1 4

≤ − 3 x x

3 5

− ≤ − 3 ≤ 1 x x

3 4

− ≤ − 0 0 0 -1 1

• 3

3 x x

4 5

− ≤ −

• ne solution: (-5,-3,0,-1,-4),

but also: (0,2,5,4,1), or (100,102,105,104,101) property: for any solution x of a system of difference constraints, x+d with d some constant is a solution as well

constraint graphs

1

v 5 ≤

• 5 - -3

5

v

• 1

1

• 4
• 3
• 5

≤

• 5 - -4
• 1

≤

• 3 - -4

1 etc 1 1 0 0 0

2

v v 5 4

• 3
• 1
• 3
• 1

etc

2 1

x x 1 -1 0 0 0 1 0 0 0 -1 0 1 0 0 -1

• 1 0 1 0 0
• 1

1 5

4

v

3

v 1

5 4 3

x x x 1 0 1 0 0

• 1 0 0 1 0

0 0 -1 1 0 0 0 -1 0 1 ≤ 5 4

• 1
• 3

assigning shortest distance δ(v0 vi) claim: 0 0 0 -1 1

• 3

assigning shortest distance δ(v0,vi)

• f vi from v0 to variable to xi

yields a feasible solution

constraint graphs

for every arc (vi,vj) in E we have

1

v

) v , v ( w ) v , v ( ) v , v (

j i i

• j
• +

δ ≤ δ

and so:

5

v

• 1

1

) v , v ( w ) v , v ( ) v , v (

j i i

• j
• ≤

δ − δ

and therefore:

2

v v 5 4

• 3
• 1
• 3

) v , v ( w x x

j i i j

≤ −

f l ( ) claim

4

v

3

v for every cycle (va,vb, . . . ,vk) we have

) v , v ( w x x

b a a b

≤ − ) v , v ( w x x

c b b c

≤ −

assigning shortest distance δ(v0 vi) claim:

) v , v ( w x x

a k k a

≤ −

assigning shortest distance δ(v0,vi)

• f vi from v0 to variable to xi

yields a feasible solution ) cycle ( w 0 ≤ negative weight cycle infeasible

checking clock feasibility

p p p

W(u,v) = min {w(p) | u → v}: the minimum number of m.e. on any path from u to v D(u v) = max {d(p) | u → v and w(p) = W(u v)}:

retiming r allows a clock period φ fif

( ) ( ) ( ) [ ]

v u w v r u r

E e

→ ≤ − ∀ ∈

e

l b φ i f ibl l k i d fif D( ) φ W( ) 0

p

D(u,v) max {d(p) | u → v and w(p) W(u,v)}: the maximum path delay over a "critical" path

( ) ( ) ( ) [ ]

1 v , u W v r u r ] v , u [ D

v u,

− ≤ − ⇒ φ > ∀

a real number φ is a feasible clock period fif D(u,v)>φ → W(u,v)>0

7 h b 2 c 2 a 2 W h

2 ) b ( r ) h ( r ≤ 1 ) a ( r ) h ( r ≤ −

7 a 2 2 2 2 2 2 h b c a

) a ( r r(b) ) c ( r r(b) 2 ) b ( r ) h ( r ≤ − ≤ − ≤ − 1 ) h ( ( ) 1 ) a ( r r(b) 1 ) h ( r r(b) 1 ) a ( r ) h ( r − ≤ − − ≤ − ≤

3 2 h c h 13 b 3 3 c 6 6 a 13 13 D h b

) h ( r ) a ( r ) a ( r r(c) ≤ − ≤ − 1 ) a ( r r(c) 1 ) b ( r ) c ( r 1 ) h ( r r(c) − ≤ − ≤ − − ≤ −

3 2 b 13 10 7 3 13 10 6 3 13 13 10 7 b c a

7 = φ

1 ) c ( r ) a ( r 1 ) b ( r ) a ( r ) ( ( ) ≤ − ≤ −

checking clock feasibility

retiming r allows a clock period φ fif

( ) ( ) ( ) [ ]

v u w v r u r

E e

→ ≤ − ∀ ∈

e

( ) ( ) ( ) [ ]

1 v , u W v r u r ] v , u [ D

v u,

− ≤ − ⇒ φ > ∀

2

1 ) a ( r ) h ( r ≤ − 2 ) b ( r ) h ( r ≤ −

• 1

1 r(b) r(b) r(h) r(h)

• 1

1

1 ) h ( r r(c) 1 ) a ( r r(b) 1 ) h ( r r(b) − ≤ − − ≤ − − ≤ − ) a ( r r(c) ) a ( r r(b) ) c ( r r(b) ≤ − ≤ − ≤ −

r(b) r(b) r(h) r(h) 1 1 1

1 ) b ( ) ( 1 ) a ( r r(c) 1 ) b ( r ) c ( r 1 ) h ( r r(c) − ≤ − ≤ − ≤ ) h ( r ) a ( r ) a ( r r(c) ≤ − ≤

r(a) r(a) r(c) r(c) 1

• 1
• 1

1

• 1

1

1 ) c ( r ) a ( r 1 ) b ( r ) a ( r ≤ − ≤ −

• 1

1 ( ) ( ) ( ) ( ) constraint graph constraint graph 1

checking clock feasibility

retiming r allows a clock period φ fif

( ) ( ) ( ) [ ]

v u w v r u r

E e

→ ≤ − ∀ ∈

e

( ) ( ) ( ) [ ]

1 v , u W v r u r ] v , u [ D

v u,

− ≤ − ⇒ φ > ∀

2

1 ) a ( r ) h ( r ≤ − 2 ) b ( r ) h ( r ≤ −

• 1

1 r(b) r(b) r(h) r(h)

• 1

1

, 1 , 1 , ] a , c , b , h [ r − − = r 1 ) h ( r r(c) 1 ) a ( r r(b) 1 ) h ( r r(b) − ≤ − − ≤ − − ≤ − ) a ( r r(c) ) a ( r r(b) ) c ( r r(b) ≤ − ≤ − ≤ −

r(b) r(b) r(h) r(h) 1 1 1

1 ) b ( ) ( 1 ) a ( r r(c) 1 ) b ( r ) c ( r 1 ) h ( r r(c) − ≤ − ≤ − ≤ ) h ( r ) a ( r ) a ( r r(c) ≤ − ≤

r(a) r(a) r(c) r(c) 1

• 1
• 1

1

• 1

1

1 ) c ( r ) a ( r 1 ) b ( r ) a ( r ≤ − ≤ −

• 1

1 ( ) ( ) ( ) ( ) constraint graph constraint graph 1

the smaller correlator

7

13 Φ

+

a 7 a

13 = Φ

= =

host b c 3 2 h c

= =

3 2 b 7 a

, 1 , 1 , ] a , c , b , h [ r − − = r

1 a

+

h a 3 1 h c 1

= =

host b c 3 b

= =

7 = Φ

checking clock feasibility

retiming r allows a clock period φ fif

( ) ( ) ( ) [ ]

v u w v r u r

E e

→ ≤ − ∀ ∈

e

a real number φ is a feasible clock period fif D(u,v)>φ → W(u,v)>0

( ) ( ) ( ) [ ]

1 v , u W v r u r ] v , u [ D

v u,

− ≤ − ⇒ φ > ∀

7 h b 2 c 2 a 2 W h

2 ) b ( r ) h ( r ≤ 1 ) a ( r ) h ( r 1 ) h ( r ) a ( r 1 ) a ( r ) a ( r ≤ − ≤ − − ≤ −

7 a 2 2 2 2 2 2 h b c a

) a ( r r(b) ) c ( r r(b) 2 ) b ( r ) h ( r ≤ − ≤ − ≤ − 1 ) a ( r r(b) 1 ) h ( r r(b) 1 ) a ( r ) h ( r − ≤ − − ≤ − ≤ −

3 2 h c h 13 b 3 3 c 6 6 a 13 13 D h b

) h ( r ) a ( r ) a ( r r(c) ≤ − ≤ − 1 ) a ( r r(c) 1 ) b ( r ) c ( r 1 ) h ( r r(c) − ≤ − ≤ − − ≤ −

3 2 b 13 10 7 3 13 10 6 3 13 13 10 7 b c a

6 = φ

1 ) c ( r ) a ( r 1 ) b ( r ) a ( r 1 ) a ( r r(c) ≤ − ≤ − ≤

checking clock feasibility

retiming r allows a clock period φ fif

( ) ( ) ( ) [ ]

v u w v r u r

E e

→ ≤ − ∀ ∈

e

a real number φ is a feasible clock period fif D(u,v)>φ → W(u,v)>0

( ) ( ) ( ) [ ]

1 v , u W v r u r ] v , u [ D

v u,

− ≤ − ⇒ φ > ∀

2 ) b ( r ) h ( r ≤

2 r(b) r(b) r(h) r(h)

• 1

1

1 ) a ( r ) h ( r 1 ) h ( r ) a ( r 1 ) a ( r ) a ( r ≤ − ≤ − − ≤ − ) a ( r r(b) ) c ( r r(b) 2 ) b ( r ) h ( r ≤ − ≤ − ≤ −

r(b) r(b) r(h) r(h) 1 1 1

• 1

1 ) a ( r r(b) 1 ) h ( r r(b) 1 ) a ( r ) h ( r − ≤ − − ≤ − ≤ −

negative weight cycle:

φ = 6 is infeasible ) h ( r ) a ( r ) a ( r r(c) ≤ − ≤ −

r(a) r(a) r(c) r(c) 1

• 1
• 1

1

• 1

1 1

1 ) a ( r r(c) 1 ) b ( r ) c ( r 1 ) h ( r r(c) − ≤ − ≤ − − ≤ −

( ) ( ) ( ) ( ) constraint graph constraint graph 1

• 1

1

6 = φ

1 ) c ( r ) a ( r 1 ) b ( r ) a ( r 1 ) a ( r r(c) ≤ − ≤ − ≤

relaxation

shortest-path weight property: δ(s,v) ≤ δ(s,u) + w(u,v) for any edge (u,v) shortest-path estimates: d[v] for any vertex v initialize-single-source(G,s) 2 9 5 initialize single source(G,s) 1 for each vertex v in V[G] 2 do d[v] ← ∞ 3 π[v] ← NIL 9 5 Relax(u,v,w) 2 3 π[v] ← NIL 4 d[s] ← 0 relax(u,v,w) 2 7 5 2 6 5 ( ) 1 if d[v] > d[u] + w(u,v) 2 then d[v] ← d[u] + w(u,v) 3 π[v] ← u 6 5 Relax(u,v,w) 2 3 π[v] ← u 2 6 5 > d[v] ≤ d[u] + w(u,v)

the bellman-ford algorithm

1 initialize-single-source(G s) bellman-ford(G,w, s)

(V 1) E relaxations

∅

2 for i ← 1 to | V[G] | - 1 1 initialize-single-source(G,s) 4 do relax(u,v,w) 3 do for each edge (u,v) in E[G]

(V-1) E relaxations

7 then return false 5 for each edge (u,v) in E[G] 6 do if d[v] > d[u] + w(u,v)

negative-cycle detection

8 return true

relax(u,v,w) negative cycle detection 1 if d[v] > d[u] + w(u,v) 2 then d[v] ← d[u] + w(u,v) 3 π[v] ← u

the bellman-ford algorithm

Order: (u,v),(u,x),(u,y),(v,u),(x,v), (x,y),(y,v),(z,u),(z,x)

1 initialize-single-source(G s) bellman-ford(G,w,s)

6

• 2

u v 5 ∞ ∞ 6 11 4 2 1 2 3 4

∅

2 for i ← 1 to | V[G] | - 1 1 initialize-single-source(G,s) 4 do relax(u,v,w) 3 do for each edge (u,v) in E[G]

6 8

• 3

2 7 z

• 4

7 then return false 6 do if d[v] > d[u] + w(u,v) 5 for each edge (u,v) in E[G]

7 2 9 x y ∞ ∞ 7 2

• 2

8 return true

relax(u,v,w) running time: O(VE) 1 if d[v] > d[u] + w(u,v) 2 then d[v] ← d[u] + w(u,v) 3 π[v] ← u

correctness

given: system of difference constraints A x ≤ b theorem: given:

• system of difference constraints A x ≤ b
• constraint graph G

if G has no negative-weight cycles, then theorem:

)) v , v ( ),..., v , v ( ), v , v ( ( x

n 2 1 1

δ δ δ =

is a feasible solution for the system if G has negative-weight cycles, g g y , then the system has no feasible solution proof:

k 1

v v =

w(c) < 0

j

v ) v , v ( w ) v , v ( ) v , v (

j i i j

+ δ ≤ δ ) v , v ( w ) v , v ( ) v , v (

j i i j

≤ δ − δ

c

) v , v ( w

2 1

≤

1 2

x x −

2 3

x x − ) v , v ( w

3 2

≤

i

v

1 k

v −

2 3

) , (

3 2

. . .

1 k 1

x x

−

− ) v , v ( w

1 1 k−

≤ ) v , v ( w x x

j i i j

≤ −

+ ≤ w(c)< 0 ≤ w(c)< 0

bellman-ford: correctness

assumption: no reachable negative-weight cycles property: upon termination, d[v] = δ(s,v) for all v in V proof: 1. v not reachable invariant: d[v] = ∞ = δ(s v) invariant: d[v] = ∞ = δ(s,v)

• 2. v reachable

let p = v0 v1 … vk be a shortest path from s (=v0) to v (=vk) th k | V | then, k < | V | induction: d[v0] = δ(s,v0) = δ(s,s) = 0 is an invariant basis: d[vi] = δ(s,vi) after the i-th pass d ( ) f h ( ) h step: assume d[vi-1] = δ(s,vi-1) after the (i-1)-th pass edge (vi-1,vi) is relaxed in the i-th pass d[vi] ≤ d[vi-1]+w(vi-1,vi) = δ(s,vi-1)+w(vi-1,vi) = δ(s,vi d[vi] = δ(s,vi)

bellman-ford: correctness

t : property: no reachable negative-weight cycles fif bellman-ford returns true ti h bl ti i ht l to prove: for all (u,v) in E, d[v] ≤ d[u] + w(u,v) assumption: no reachable negative-weight cycles consider (u,v): d[v] = δ(s,v) ≤ δ(s,u) + w(u,v) = d[u] + w(u,v) assumption: bellman ford returns true assumption: bellman-ford returns true to prove: no reachable negative-weight cycles assume there is a negative-weight cycle c = v0 v1 … vk (=v0)

∑

< =

k

) vi 1 vi ( w ) c ( w

∑ =

< − =

1 i

) vi , 1 vi ( w ) c ( w

∑ ∑ ∑

= = =

− + − ≤

k 1 i k 1 i k 1 i

) vi , 1 vi ( w ] 1 vi [ d ] vi [ d

∑ ∑

− =

k k

] 1 vi [ d ] vi [ d

∑ ∑

= = 1 i 1 i

] 1 vi [ d ] vi [ d

6 do if d[v] > d[u] + w(u,v) 5 for each edge (u,v) in E[G]

∑ =

≥ −

k 1 i

) vi , 1 vi ( w

7 then return false

[ ] [ ] ( , )

clock period minimization

ll h h

( )

3

O

1) compute all W and D : all pairs shortest paths 2) sort all entries of D :

( )

3

V

O

( )

V ln V2

O

) 3) find the minimum feasible clock period Φ : b h

• by testing entries D

( )

O

( )

V ln V

O

binary search manner shortest path in constraint graph

• by testing entries D
• check feasibility

for a clock period of dij

( )

V ln

O

( )

E V

O

4) determine r for Φ in constraint graph shortest path

( )

E V •

O

( )

E V

O

in constraint graph

( )

E V •

O

complexity:

( )

V ln V

3

• O

leiserson’s correlator

+

g

+

e

+

f

Φ 24

+

host g

+ +

Φ=24

=

a

= =

c d

=

b adder 7 f a r delay comparator delay =7 =3

2

7 e g 7 f 7 delay

rg=0 rf=-1 re=-2 rh=0 rd=-2

3 1 d 1 h

rh 0

d

3 1 c 3 1 a 3 b 1 1

ra=-1 rb=-1 rc=-2 wab:=1-1+1=1 wbc:=1-2+1=0

leiserson’s correlator

+

g

+

e

+

f

Φ 13

+

host

+ +

Φ=13

=

a

= =

c d

=

b f

2 wef:=0-1+2=1 wfg:=0-0+1=1

7 e g 7 f 7

rg=0 rf=-1 re=-2 rh=0 rd=-2 wde:= +2=0 +1=0 +1=1 wgh:= 0-0+0=0

3 1 d 1 h

rh 0

d

wcd:=

de

0-2+2=0 wce:=0-2+ wbf:=0-1+ wag:=0-0+ wah:=

3 c 3 a 3 b 1 1

ra=-1 rb=-1 rc=-2

cd

1-2+2=1 w w wab:=1-1+1=1 wbc:=1-2+1=0 w

ah

1-1+0=0