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ELEC 7770 Advanced VLSI Design Spring 2014 Constraint Graph and Retiming SolutionPowerPoint Presentation

ELEC 7770 Advanced VLSI Design Spring 2014 Constraint Graph and Retiming Solution

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### ELEC 7770Advanced VLSI DesignSpring 2014Constraint Graph and Retiming Solution

Vishwani D. Agrawal

James J. Danaher Professor

ECE Department, Auburn University

Auburn, AL 36849

http://www.eng.auburn.edu/~vagrawal/COURSE/E7770_Spr14/course.html

ELEC 7770: Advanced VLSI Design (Agrawal)

Retiming Theorem

- Given a network G(V, E, W) and a cycle time T, (r1, . . . ) is a feasible retiming if and only if:
- ri – rj≤ wij for all edges (vi,vj) ε E
- ri – rj ≤ W(vi,vj) – 1 for all node-pairs vi, vj such that
D(vi,vj) > T

Where,

W(vi,vj) is the minimum weight path between vi and vj

D(vi,vj) is the maximum delay among all minimum weight paths between vi and vj

ELEC 7770: Advanced VLSI Design (Agrawal)

Retiming Theorem Explained

- Condition 1, ri – rj≤ wij, is related to edge weight:
- Original circuit is feasible original weight wij is positive
- Originally, ri = rj = 0
- Retiming, rj flip-flops added to eij, ri flip-flops removed from eij, net reduction ri – rj must be less than wij to leave the retimed weight of eij positive.

- Condition 2, ri – rj ≤ W(vi,vj) – 1 is related to path delays between node pairs being less than clock period T whenever path weight is 0.

ELEC 7770: Advanced VLSI Design (Agrawal)

Examine Condition 2

W1, D1

rj

ri

vj

W2, D2

vi

W3, D3

W1 = W2 < W3, W(vi, vj) = W1 = W2, minimum weight among paths

D1 > D2, therefore D(vi, vj) = D1, maximum delay of a minimum weigh path

If D1 ≤ T, there is no requirement on ri, rj

If D1 > T, Retimed weight W1’ = W1 – ri + rj ≥ 1 (at least 1 FF on path)

or ri – rj ≤ W1 – 1

ELEC 7770: Advanced VLSI Design (Agrawal)

Timing Optimization

- Find the clock period (T) by path analysis.
- Set clock period to T/2 and find a feasible retiming.
- If feasible, further reduce the clock period to half.
- If not feasible, increase clock period.
- Do a binary search for optimum clock period.
- Retime the circuit.

ELEC 7770: Advanced VLSI Design (Agrawal)

Representing a Constraint

ri – rj ≤ wij or rj ≥ ri – wij

– wij

rj

ri

ELEC 7770: Advanced VLSI Design (Agrawal)

Constraint Graph

-6

r1 ≥ r0 + 3

r1 ≥ r2 + 1

r2 ≥ r0 + 1

r2 ≥ r1 – 1

r3 ≥ r1 + 1

r3 ≥ r2 + 4

r0 ≥ r3 – 6

r1

3

1

r0

r3

-1

1

1

4

r2

ELEC 7770: Advanced VLSI Design (Agrawal)

Feasibility Condition

- A set of values for variables can be found if and only if the constraint graph has no positive cycles.
- This is also the condition for the solvability of the longest path problem, which provides a solution to the set of constraints.

ELEC 7770: Advanced VLSI Design (Agrawal)

Example: Infeasible Constraints

x2

x1 ≥ x2 + 6

x2 ≥ x1 – 3

6

x2 ≥ x1 – 3

x1

x2

3

-3

Positive cycle mean no

longest path can be found.

x1 ≥ x2 + 6

x1

0

3

6

ELEC 7770: Advanced VLSI Design (Agrawal)

Solving a Constraint Set

-6

r1 ≥ r0 + 3

r1 ≥ r2 + 1

r2 ≥ r0 + 1

r2 ≥ r1 – 1

r3 ≥ r1 + 1

r3 ≥ r2 + 4

r0 ≥ r3 – 6

r1

3

1

r0

r3

-1

1

Longest paths from source r0

to r0, r1, r2, r3

Path lengths: s0=0, s1=3, s2=2, s3=6

Solution: r0=0, r1=3, r2=2, r3=6

1

4

r2

ELEC 7770: Advanced VLSI Design (Agrawal)

The General Path Problem

- Find the shortest (or longest) path in a graph from a source vertex to all other vertices.
- Graph has vertices and directed edges:
- Edge weights can be positive or negative
- Graph can be cyclic
- Single source vertex – a vertex with 0 in-degree (not a necessary condition)

- Inconsistent problems
- Negative weight cycles for shortest path
- Positive weight cycles for longest path

ELEC 7770: Advanced VLSI Design (Agrawal)

Dijkstra’s Shortest Path Algorithm

- Greedy algorithm.
- Applies to directed acyclic graphs (DAG) with positive edge weights.
- Computational complexity
O(|E| + |V| log |V|) ≤ O(n2)

- References:
- A. Aho, J. Hopcroft and J. Ullman, Data Structures and Algorithms, Reading, Massachusetts: Addison-Wesley, 1983.
- T. Cormen, C. Leiserson and R. Rivest, Introduction to Algorithms, New York: McGraw-Hill, 1990.

ELEC 7770: Advanced VLSI Design (Agrawal)

Dijkstra’s Shortest Path Algorithm Example 1

v1

w01=15

3

v0

v3

10

source

2

6

v2

si = path weight (v0, vi)

Each step marks the path with smallest weight and updates the unmarked

path weights.

ELEC 7770: Advanced VLSI Design (Agrawal)

Dijkstra’s Shortest Path Algorithm Example 2

v1

w01=15

3

v0

v3

6

source

2

10

v2

si = path weight (v0, vi)

Each step marks the path with smallest weight and updates the unmarked

path weights.

ELEC 7770: Advanced VLSI Design (Agrawal)

Dijkstra’s Algorithm, G(V, E, W)

s0(1) = 0 initialize source

for ( i = 1 to n ) initialize path weights, n=|V| –1

si(1) = w0i

repeat {

Select an unmarked vertex vq such that sq is minimal

Mark vq

foreach ( unmarked vertex vi )

si = min { si, sq + wqi }

}

until (all vertices are marked)

ELEC 7770: Advanced VLSI Design (Agrawal)

Try Dijkstra’s Algorithm for Your Graph

http://www.dgp.toronto.edu/people/JamesStewart/270/9798s/Laffra/DijkstraApplet.html

ELEC 7770: Advanced VLSI Design (Agrawal)

Dijkstra’s Longest Path Algorithm

v1

Either change min to max

Or change all positive weights to negatives

w01=15

3

v0

v3

10

source

2

6

v2

v1

w01= -15

-3

v0

v3

-10

source

-2

-6

v2

si = path length (v0, vi)

ELEC 7770: Advanced VLSI Design (Agrawal)

Dijkstra’s Alg. Does Not Work for Cycles, Mixed Weights

-2

v1

w01=15

3

v0

v3

5

source

2

4

v2

si = path weight (v0, vi)

Algorithm stops because all vertices are marked.

But, there exists a v0 to v3 path of length 5

ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman’s Equations – Shortest Path

vj

vk

wki

wji

For all vertices:

si = min (sq + wqi)

vq ε pred(vi)

vi

vm

wmi

wni

vn

sq = minimum path weight between

source and vq

ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman-Ford Algorithm, G(V, E, W)

Bellman-Ford {

s0(1) = 0 initialize source

for ( i = 1 to n ) initialize path weights, n = |V| – 1

si(1) = w0i

for ( j = 1 to n ) n iterations

for ( i = 1 to n ) n nodes

si(j+1) = min { si(j), sk(j) + wki }

vkεpred(vi)

}

if ( si(j+1) == si(j) i ) return (true)

}

return (false)

Complexity = O(|V||E|) ≤ O(n3)

ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman-Ford Shortest Path

n = 3

v1

w01=15

3

v0

v3

10

source

2

6

v2

si = path weight (v0, vi)

ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman-Ford Longest Path

Reverse the sign of weights and solve shortest path problem.

(Alternative: keep original weights and change min operator in

algorithm to max.)

n = 3 (shortest path)

Weights reversed

v1

w01= -15

-3

v0

v3

-10

source

-2

-6

v2

si = path weight (v0, vi)

ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman’s Equations – Longest Path

vj

vk

wki

wji

For all vertices:

si = max (sq + wqi)

vq ε pred(vi)

vi

vm

wmi

wni

vn

sq = maximum path weight between

source and vq

ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman-Ford for Cycles, Neg. Weights

n = 3 (shortest path)

-2

v1

w01=15

3

v0

v3

5

source

2

4

v2

si = path weight (v0, vi)

This was incorrect with Dijkstra’s shortest path algorithm

ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman-Ford for Negative Cycle

n = 3 (shortest path)

2

v1

w01=15

-3

v0

v3

5

source

2

4

v2

si = path weight (v0, vi)

Values not stabilized after n iterations.

Inconsistent problem: negative cycle.

ELEC 7770: Advanced VLSI Design (Agrawal)

Retiming Graph

FF

a

b

c

10

5

5

1

0

0

h

0

a

10

b

5

1

c

5

Critical path = 15

It is the longest path consisting only of zero weight edges.

ELEC 7770: Advanced VLSI Design (Agrawal)

Feasibility Constraints (Condition 1)

FF

a

b

c

10

5

5

1

rb FFs

0

0

h

0

a

10

b

5

1

c

5

rc FFs

rh – ra ≤ 0

ra – rb ≤ 0

rb – rc ≤ 1

rc – rh ≤ 1

ri – rj ≤ wij edges i → j

Retiming should not cause

negative edge weights.

ELEC 7770: Advanced VLSI Design (Agrawal)

Constraint Graph

FF

a

b

c

10

5

5

-1

0

0

rh

0

ra

10

rb

5

-1

rc

5

rh – ra ≤ 0 rh – 0 ≤ ra

ra – rb ≤ 0 ra – 0 ≤ rb Constraints for

rb – rc ≤ 1 rb – 1 ≤ rc Condition 1

rc – rh ≤ 1 rc – 0 ≤ rh

ri – rj ≤ wij edges i → j

Retiming should not cause

negative edge weights.

Observation: Constraint graph has the same structure as the original

retiming graph, with signs of weights reversed. Vertex labels are the

retiming integer variables.

ELEC 7770: Advanced VLSI Design (Agrawal)

Max Delay for Min Weight Paths

1

0

0

h

0

a

10

b

5

1

c

5

T = 15

W(b,c) = 1 D(b,c) = 10

W(b,h) = 2 D(b,h) = 10

W(b,a) = 2 D(b,a) = 20

W(c,h) = 1 D(c,h) = 5

W(c,a) = 1 D(c,a) = 15

W(c,b) = 1 D(c,b) = 20

W(h,a) = 0 D(h,a) = 10

W(h,b) = 0 D(h,b) = 15

W(h,c) = 1 D(h,c) = 20

W(a,b) = 0 D(a,b) = 15

W(a,c) = 1 D(a,c) = 20

W(a,h) = 2 D(a,h) = 20

ELEC 7770: Advanced VLSI Design (Agrawal)

Timing Optimization, T = 7.5?

-1

Constraint graph

(feasibility)

0

0

rh

0

ra

10

rb

5

-1

rc

5

Add constraints for

Condition 2: ri – rj ≤ W(I,j) – 1 paths (i,j) with D(i,j) > 7.5

W(b,c) = 1 D(b,c) = 10

W(b,h) = 2 D(b,h) = 10

W(b,a) = 2 D(b,a) = 20

W(c,h) = 1 D(c,h) = 5

W(c,a) = 1 D(c,a) = 15

W(c,b) = 1 D(c,b) = 20

W(h,a) = 0 D(h,a) = 10

W(h,b) = 0 D(h,b) = 15

W(h,c) = 1 D(h,c) = 20

W(a,b) = 0 D(a,b) = 15

W(a,c) = 1 D(a,c) = 20

W(a,h) = 2 D(a,h) = 20

ELEC 7770: Advanced VLSI Design (Agrawal)

Timing Optimization, T = 7.5?

Positive cycle;

no solution for

longest path

-1

0

1

1

0

1

0

0

rh

0

ra

10

rb

5

-1

rc

5

-1

-1

0

0

-1

0

W(b,c) = 1 D(b,c) = 10

W(b,h) = 2 D(b,h) = 10

W(b,a) = 2 D(b,a) = 20

W(c,h) = 1 D(c,h) = 5

W(c,a) = 1 D(c,a) = 15

W(c,b) = 1 D(c,b) = 20

W(h,a) = 0 D(h,a) = 10

W(h,b) = 0 D(h,b) = 15

W(h,c) = 1 D(h,c) = 20

W(a,b) = 0 D(a,b) = 15

W(a,c) = 1 D(a,c) = 20

W(a,h) = 2 D(a,h) = 20

ELEC 7770: Advanced VLSI Design (Agrawal)

Timing Optimization, T = 11.25?

-1

rh = 0

rb = 1

rc = 0

ra = 0

0

1

1

0

0

0

rh

0

ra

10

rb

5

-1

rc

5

-1

-1

0

0

W(b,c) = 1 D(b,c) = 10

W(b,h) = 2 D(b,h) = 10

W(b,a) = 2 D(b,a) = 20

W(c,h) = 1 D(c,h) = 5

W(c,a) = 1 D(c,a) = 15

W(c,b) = 1 D(c,b) = 20

W(h,a) = 0 D(h,a) = 10

W(h,b) = 0 D(h,b) = 15

W(h,c) = 1 D(h,c) = 20

W(a,b) = 0 D(a,b) = 15

W(a,c) = 1 D(a,c) = 20

W(a,h) = 2 D(a,h) = 20

ELEC 7770: Advanced VLSI Design (Agrawal)

Retiming Graph

FF

a

b

c

10

5

5

1

0

0

h

0

a

10

b

5

1

c

5

1

0

rc = 0

rh = 0

ra = 0

rb = 1

wij_retimed = wij + rj – ri

ELEC 7770: Advanced VLSI Design (Agrawal)

Retimed Circuit

FF

a

c

b

10

5

5

Logic optimization

will remove these.

1

1

0

0

h

0

a

10

b

5

c

5

rc = 0

rh = 0

ra = 0

rb = 1

Critical Path = 10

ELEC 7770: Advanced VLSI Design (Agrawal)

Correlator Circuit

Critical path delay = 24

f

g

e

0

0

7

7

7

re=0

0

rf=0

rg=0

0

0

0

0

0

h

rh=0

1

3

3

3

3

1

1

1

rd=0

rb=0

rc=0

ra=0

a

c

b

d

Initial retiming vector = {0,0,0,0,0,0,0,0}

ELEC 7770: Advanced VLSI Design (Agrawal)

Retiming Optimization

ELEC 7770: Advanced VLSI Design (Agrawal)

Retiming of Correlator Circuit

Critical path delay = 13.5

f

g

e

0→1

0→2→1

7

7

7

re= -2

0

rf= -1

rg=0

0→2→0

0

0→1→0

0→1

0→2→0

h

rh=0

1→2→0

1→0

1→3→1

3

3

3

3

1→2→1

rd= -2

rb= -1

rc= -2

ra= -1

a

c

b

d

retiming vector = {-1,-1,-2,-2,-2,-1,0,0}

ELEC 7770: Advanced VLSI Design (Agrawal)

Retimed Correlator Circuit

Critical path delay = 13

f

g

e

1

1

7

7

7

re= -2

0

rf= -1

rg=0

0

0

0

0→1

0

h

rh=0

0

0

1

3

3

3

3

1

rd= -2

rb= -1

rc= -2

ra= -1

a

c

b

d

retiming vector = {-1,-1,-2,-2,-2,-1,0,0}

ELEC 7770: Advanced VLSI Design (Agrawal)

References

C. E. Leiserson, F. Rose and J. B. Saxe, “Optimizing Synchronous Circuits by Retiming,” Proc. 3rd Caltech Conf. on VLSI, 1983, pp. 87-116.

C. E. Leiserson and J. B. Saxe, “Retiming Synchronous Circuitry,” Algorithmica, vol. 6, pp. 5-35, 1991.

G. De Micheli, Synthesis and Optimization of Digital Circuits, New York: McGraw-Hill, 1994, Section 9.3.1.

N. Maheshwari and S. S. Sapatnekar, Timing Analysis and Optimization of Sequential Circuits, Boston: Springer, 1999, Chapter 4.

ELEC 7770: Advanced VLSI Design (Agrawal)

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