ELEC 7770 Advanced VLSI Design Spring 2012 Constraint Graph and Retiming Solution Vishwani D. Agrawal James J. Danaher Professor ECE Department, Auburn University Auburn, AL 36849 vagrawal@eng.auburn.edu http://www.eng.auburn.edu/~vagrawal/COURSE/E7770_Spr10/course.html Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

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 Spring 2015, Feb 15 . . . 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. Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Examine Condition 2 W1, D1 rj ri vj vi W2, D2 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 Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

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. Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Representing a Constraint ri – rj ≤ wij or rj ≥ ri – wij rj – wij ri Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

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 -1 1 r3 1 4 r2 Spring 2015, Feb 15 . . . 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. Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Example: Infeasible Constraints 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 3 6 Spring 2015, Feb 15 . . . 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 -1 1 r3 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 Spring 2015, Feb 15 . . . 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 Spring 2015, Feb 15 . . . 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. Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Dijkstra’s Shortest Path Algorithm Example 1 v1 Alg. steps s0 s1 s2 s3 Initially: mark v0 15 2  Step 1: mark v2 12 8 Step 2: mark v3 11 Step 3: mark 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. Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Dijkstra’s Shortest Path Algorithm Example 2 v1 Alg. steps s0 s1 s2 s3 Initially: mark v0 15 2  Step 1: mark v2 8 12 Step 2: mark v1 Step 3: mark v3 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. Spring 2015, Feb 15 . . . 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) Spring 2015, Feb 15 . . . 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 Spring 2015, Feb 15 . . . 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 Alg. steps s0 s1 s2 s3 Initially -15 -2  Step 1: mark v1 Step 2: mark v2 -8 Step 3: mark v3 source 2 6 v2 v1 w01= -15 -3 v0 v3 -10 source -2 -6 v2 si = path length (v0, vi) Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Dijkstra’s Alg. Does Not Work for Cycles, Mixed Weights Alg. steps s0 s1 s2 s3 Initially: mark v0 15 2  Step 1: mark v2 7 6 Step 2: mark v3 Step 3: mark v1 6? -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 Spring 2015, Feb 15 . . . 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 Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman-Ford Algorithm, G(V, E, W) 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 } vk ε pred(vi) } if ( si(j+1) == si(j) i ) return (true) return (false) Complexity = O(|V||E|) ≤ O(n3) Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman-Ford Shortest Path v1 w01=15 Alg. steps s0 s1 s2 s3 Initially 15 2  Iteration 1 12 8 Iteration 2 11 Iteration 3 3 v0 v3 10 source 2 6 v2 si = path weight (v0, vi) Spring 2015, Feb 15 . . . 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 Alg. steps s0 s1 s2 s3 Initially -15 -2  Iteration 1 -8 Iteration 2 v1 w01= -15 -3 v0 v3 -10 source -2 -6 v2 si = path weight (v0, vi) Spring 2015, Feb 15 . . . 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 Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman-Ford for Cycles, Neg. Weights -2 n = 3 (shortest path) v1 Alg. steps s0 s1 s2 s3 Initially 15 2  Iteration 1 7 6 Iteration 2 5 Iteration 3 w01=15 3 v0 v3 5 source 2 4 v2 si = path weight (v0, vi) This was incorrect with Dijkstra’s shortest path algorithm Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Bellman-Ford for Negative Cycle 2 n = 3 (shortest path) v1 Alg. steps s0 s1 s2 s3 Initially 15 2  Iteration 1 7 6 Iteration 2 3 Iteration 3 5 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. Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Retiming Example FF a b c 10 5 5 Delay Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Retiming Graph FF a b c 10 5 5 1 h a 10 b 5 1 c 5 Critical path = 15 a 10 b 5 1 c 5 Critical path = 15 It is the longest path consisting only of zero weight edges. Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Feasibility Constraints (Condition 1) FF a b c 10 5 5 1 h a 10 b 5 1 c 5 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. Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Constraint Graph FF a b c 10 5 5 -1 rh ra 10 rb 5 -1 rc 5 rh – ra ≤ 0 ra – rb ≤ 0 Constraints for rb – rc ≤ 1 Condition 1 rc – rh ≤ 1 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. Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Max Delay for Min Weight Paths 1 h a 10 b 5 1 c 5 T = 15 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 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 Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Timing Optimization, T = 7.5? -1 Constraint graph (feasibility) rh ra 10 rb 5 -1 rc 5 Add constraints for Condition 2: ri – rj ≤ W(I,j) – 1  paths (i,j) such that D(i,j) > 7.5 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 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 Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Timing Optimization, T = 7.5? Positive cycle; no solution for longest path -1 1 1 1 rh ra 10 rb 5 -1 rc 5 -1 -1 -1 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 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 Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

Timing Optimization, T = 11.25? -1 rh = 0 rb = 1 rc = 0 ra = 0 1 1 rh ra 10 rb 5 -1 rc 5 -1 -1 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 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 Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Retiming Graph FF a b c 10 5 5 1 h a 10 b 5 1 c 5 1 rh = 0 ra = 0 rb = 1 rc = 0 wij_retimed = wij + rj – ri Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Retimed Circuit FF a b c 10 5 5 Logic optimization will remove these. 1 h a 10 1 b 5 c 5 rh = 0 ra = 0 rb = 1 rc = 0 Critical Path = 10 Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)

ELEC 7770: Advanced VLSI Design (Agrawal) Reference G. De Micheli, Synthesis and Optimization of Digital Circuits, New York: McGraw-Hill, 1994. Spring 2015, Feb 15 . . . ELEC 7770: Advanced VLSI Design (Agrawal)