Incremental Layer Assignment for Critical Path Timing

Transcription

1 Incremental Layer Assignment for Critical Path Timing Derong Liu 1, Bei Yu 2, Salim Chowdhury 3, and David Z. Pan 1 1 ECE Department, University of Texas at Austin, Austin, TX, USA 2 CSE Department, Chinese University of Hong Kong, Hong Kong 3 Oracle Corp., Austin, TX, USA 1

2 Introduction t As VLSI technology scales to deep submicron Interconnect dominates timing issues Global routing integral part of timing convergence flow 2D Global Routing Layer Assignment 3D Global Routing Global Routing Flow 2

3 Previous Works on GR and LA t Example papers on global routing and layer assignment: Timing-driven GR [Liu et al. TCAD 13] Via count and overflow minimization during layer assignment - NVM [Liu+, ASPDAC 11] Delay-driven layer assignment [Yu+, ICCAD 15] t Limitations of previous layer assignment: Net-by-net method may lead to sub-optimal results Focus on sum of net delays Lack global optimization Linear approximation of via delays influences accuracy. 3

4 Contributions of this Work t A novel incremental layer assignment framework for critical path timing with via delays t Semidefinite programming (SDP) modeling for better optimal solutions t Self-adaptive partitioning methodology based on K*K partitions for speed-up 4

5 Problem Formulation t Critical Path Layer Assignment (CPLA) Given a 3-D grid graph, edge and layer information, initial layer assignment solution and set of critical nets Minimize: critical path timing (Elmore delay) Subject to: edge capacity constraints Non-Critical Nets: n1 n2 ; Critical Net: n3 n3 n2 n1 n3 n1 n2 (a) (b) 5

6 CPLA Algorithm t Mathematical Formulation Segment costs Via costs t Constraints: Each segment should be assigned on one and only one layer Edge capacity constraint: Via capacity constraint: 6

7 Self-adaptive Quadruple Partition t K x K division [Yu+, ICCAD 15] Unbalanced computation overhead t Limit the number of segments in each partition Each thread deals with workload in a well-balanced manner 7

8 Semidefinite Programming t Input: parameter matrix T t Output: variable matrix X Segment costs Via costs 8

9 Example t Segments S1 and S2 t 4 layers: Layer 1 and 3 forx-dimension Layer 2 and 4 fory-dimension S1 T(M1): T(M3): S2 T(M2): T(M4): t Post mapping strategy provides integer solutions 9

10 Experimental Results t Implemented the framework in C++ t ILP solver -- GUROBI t SDP solver CSDP t Parallel implementation OpenMP t Evaluation on ISPD 08 global routing benchmarks t Performance Metrics Average Critical Path Timing Worst Critical Path Timing # of Via capacity violation # of Via 10

11 Evaluation on ISPD 08 Benchmarks t Initial global routing input: Generated by NCTU-GR 2.0 [Liu et al. TCAD 13] t Initial layer assignment: From NVM [Liu et al. ASPDAC 11] Targeting via number and overflow minimization t Wire resistance and capacitance values obtained from industry settings t Release 0.5% critical and non-critical nets t Compared with TILA [Yu et al. ICCAD 15] 11

12 Delay Comparison Results t ISPD 08 Global Routing Benchmarks 14% improvement in Avg(T cp ) 4% improvement in Max(T cp ) 12

13 Via Comparison Results t ISPD 08 Global Routing Benchmarks Via Overflow (#OV) decreases by 10% Similar number of vias 13

14 Conclusion t Propose Incremental Layer Assignment for Critical Path Timing (CPLA) algorithm Self-adaptive partition provides balanced workload for multiple threads and potential speed-up Semidefinite programming (SDP) relaxation Post mapping satisfies constraints t CPLA suitable for future heterogeneous layer structures 14

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16 Overview t Introduction t Problem Formulation t Algorithms t Experimental Results t Conclusion 16

17 Model Description t Elmore timing model: Consider both segment delay and via delay s4 s1 s2 s3 v1 Maximum Sink Path Delay: t(s1) + t(s2) + t(s3) + t(s4) + t(v1) 17

18 Partition Size Impact t Different number of segments in each partition Impact on Avg(T cp ); Impact on Max(T cp ); Impact on runtime 18

19 Mapping Algorithm t Provide integer solutions For each edge with critical nets: Find the current highest layer j to assign segments cap(j) = allowable number of segments to assign Select cap(j) segments with highest solutions Assign these segments on layer j of edge Finish all layers? N Exit Y 19