1. ICCAD Nov-2000 Timing Driven Gate Duplication: Complexity Issues and Algorithms Ankur Srivastava, Ryan Kastner and Majid Sarrafzadeh Embedded & Reconfigurable System Design ER-Group UCLA Ankur Srivastava, Ryan Kastner and Majid Sarrafzadeh Embedded & Reconfigurable System Design ER-Group UCLA

2. ICCAD Nov-2000MotivationMotivation Need for new methodologies of delay improvement in the light of the stringent timing constraint that designers have Gate duplication has been studied primarily for cut-set minimization. Applicability of this method for improving delay has not been studied by the research community Need for new methodologies of delay improvement in the light of the stringent timing constraint that designers have Gate duplication has been studied primarily for cut-set minimization. Applicability of this method for improving delay has not been studied by the research community

3. ICCAD Nov-2000 Load Dependent Delay Model (LDDM)  i  i  j  j i j ii jj  (i) =  i +  i * COUT wire-delays are assumed to be zero

4. ICCAD Nov-2000 Gate Duplication for Delay Improvement A B C D E r = 2  = 5 r = -14  = 1  = 1  = 0.1 r = -15.1 r = Input pin required time = required time at O/P - gate delay C D = 15 C E = 0.1

5. ICCAD Nov-2000 Gate Duplication for Delay Improvement r = -9 B C E r = 2  = 5 r = -10.2 D D’ A  = 1  = 1  = 0.1 C D = 10 C D’ = 5 C E = 0.2

6. ICCAD Nov-2000 Complexity Issues Theorem: Global Gate Duplication is NP-Complete in LDDM MONO3SAT gets transformed to an instance of the global problem Theorem: Local Gate Duplication is NP- Complete PARTITION problem gets transformed to an instance of the local problem Theorem: Global Gate Duplication is NP-Complete in LDDM MONO3SAT gets transformed to an instance of the global problem Theorem: Local Gate Duplication is NP- Complete PARTITION problem gets transformed to an instance of the local problem

7. ICCAD Nov-2000 Complexity Issues (Comparison with Buffer Insertion) Local Buffer Insertion Problem: Polynomially Solvable if the net topology is fixed. Global Buffer Insertion Problem: Polynomially solvable if the delay model has same pin to pin parameters Situations in which buffer insertion is polynomially solvable, Gate Duplication becomes NP-Complete Local Buffer Insertion Problem: Polynomially Solvable if the net topology is fixed. Global Buffer Insertion Problem: Polynomially solvable if the delay model has same pin to pin parameters Situations in which buffer insertion is polynomially solvable, Gate Duplication becomes NP-Complete

8. ICCAD Nov-2000 Algorithm for Gate Duplication Based on the structure of dynamic programming Applies duplication to all the gates in the circuit. Hence works in the pro- active mode Assumption: The circuit has only single output combinational gates. Based on the structure of dynamic programming Applies duplication to all the gates in the circuit. Hence works in the pro- active mode Assumption: The circuit has only single output combinational gates.

9. ICCAD Nov-2000 Algorithm for Gate Duplication Stage1: Traverse the network from POs to PIs in the topological order evaluating tuples at every step Stage2: Now traverse the network from PI to PO in topological order deciding the gates to be duplicated Stage3: Traverse the network from PO to PI physically duplicating the gates Stage1: Traverse the network from POs to PIs in the topological order evaluating tuples at every step Stage2: Now traverse the network from PI to PO in topological order deciding the gates to be duplicated Stage3: Traverse the network from PO to PI physically duplicating the gates

10. ICCAD Nov-2000 Stage 1: Need to find the best duplication strategy of the fanouts such that the input pin required time is maximized g i tup(i,g).dup.r_small tup(i,g).dup.r_large g g’ tup(i,g).nodup i’ i

11. ICCAD Nov-2000 Stage 1: Need to find the best duplication strategy of the fanouts and the best fanout partitioning between g and g’ such that the input pin required time is maximized g i tup(i,g).dup.r_small tup(i,g).dup.r_large g g’ tup(i,g).nodup i’ i

12. ICCAD Nov-2000 Stage 1: NODUP: Sort the fanouts and duplicate in that order. (total n+1 duplication strategies) RESULT: This Algorithm is optimal g g

13. ICCAD Nov-2000 Stage 1: DUP: g g’ g

14. ICCAD Nov-2000 1 0 1 1 0 1 Stage 2: Stage2: Forward traversal in topo sorted order

15. ICCAD Nov-2000 Stage 3: Stage 3: Traverse the circuit backwards from PO to PI, physically duplicating the gates

16. ICCAD Nov-2000 Experimental Results The circuit was first optimized using script.rugged of SIS followed by speed_up Results obtained in two categories, one with minimum delay technology mapping map -n 1, other with minimum delay technology mapping with fanout optimization map -n 1 - AFG The circuit was first optimized using script.rugged of SIS followed by speed_up Results obtained in two categories, one with minimum delay technology mapping map -n 1, other with minimum delay technology mapping with fanout optimization map -n 1 - AFG

17. ICCAD Nov-2000 Experimental Results (map -n 1)

18. ICCAD Nov-2000 Experimental Results (map -n 1 - AFG)

19. ICCAD Nov-2000ConclusionConclusion We presented an algorithm for gate duplication and showed it’s effectiveness in reducing circuit delay, both with and without buffer insertion We proved the local problem NP- Complete The future work would include the extension of this algorithm in a layout driven framework. We presented an algorithm for gate duplication and showed it’s effectiveness in reducing circuit delay, both with and without buffer insertion We proved the local problem NP- Complete The future work would include the extension of this algorithm in a layout driven framework.

20. ICCAD Nov-2000 Timing Driven Gate Duplication: Complexity Issues and Algorithms Ankur Srivastava, Ryan Kastner and Majid Sarrafzadeh Embedded & Reconfigurable System Design ER-Group UCLA Ankur Srivastava, Ryan Kastner and Majid Sarrafzadeh Embedded & Reconfigurable System Design ER-Group UCLA