VLSI Placement and Global Routing Using Simulated Annealing

From my B.E.E degree at the University of Minnesota and right through my S.M. degree at M.I.T., I had specialized in solid state devices and microelectronics. I made the decision to switch to computer-aided design (CAD) in 1981, only a year or so prior to the introduction of the simulated annealing algorithm by Scott Kirkpatrick, Dan Gelatt, and Mario Vecchi of the IBM Thomas 1. Watson Research Center. Because Prof. Alberto Sangiovanni-Vincentelli, my UC Berkeley advisor, had been a consultant at IBM, I re­ ceived a copy of the original IBM internal report on simulated annealing approximately the day of its release. Given my background in statistical mechanics and solid state physics, I was immediately impressed by this new combinatorial optimization technique. As Prof. Sangiovanni-Vincentelli had suggested I work in the areas of placement and routing, it was in these realms that I sought to explore this new algorithm. My flJ'St implementation of simulated annealing was for an island-style gate array placement problem. This work is presented in the Appendix of this book. I was quite struck by the effect of a nonzero temperature on what otherwise appears to be a random in­ terchange algorithm.

Contenu

1 Introduction.- 1.1 Placement and Global Routing of Integrated Circuits.- 1.2.1 The gate array placement and global routing problem.- 1.2.2 The standard cell placement and global routing problem.- 1.2.3 The macro/custom cell placement and global routing problem.- 1.3 Previous Approaches to Placement and Global Routing.- 1.3.1 Previous placement methods.- 1.3.2 Previous global routing methods.- 1.4 A New Approach to Cell-Based Placement and Global Routing.- 2 The Simulated Annealing Algorithm.- 2.1 Introduction.- 2.2 The Basic Simulated Annealing Algorithm.- 2.3 Theoretical Investigations of the Simulated Annealing Algorithm.- 2.4 Overview of Work on General Annealing Schedules.- 2.4.1 The initial temperature.- 2.4.2 The temperature decrement.- 2.4.3 The equilibrium condition.- 2.4.4 The stopping, or convergence, criterion.- 2.5 Implementations of Simulated Annealing for Placement and Global Routing.- 2.6 The Function f().- 2.7 Fast Evaluation of the Exponential Function.- 3 Placement and Global Routing of Standard Cell Integrated Circuits.- 3.1 Introduction.- 3.2 The General TimberWolfSC Methodology.- 3.2.1 Finding the optimal target row lengths.- 3.2.2 Critical-net weighting.- 3.3 The Algorithm for Stage 1 of TimberWolfSC.- 3.3.1 The cost function.- 3.3.1.1 The first term in the cost function.- 3.3.1.2 The second term in the cost function.- 3.3.1.3 The third term in the cost function.- 3.3.2 An alternative objective function.- 3.3.3 The generation of new states function.- 3.3.4 The inner loop criterion.- 3.3.5 The range limiter.- 3.3.6 The control of T.- 3.3.7 The effects of net weighting.- 3.4 The Algorithms for Stage 2 of TimberWolfSC.- 3.4.1 Implementation of the stage 2 simulated annealing functions.- 3.4.2 The first phase of the global router.- 3.4.3 The second phase of the global router.- 3.5 The Algorithm for Stage 3 of TimberWolfSC.- 3.6 TimberWolfSC Results.- 3.6.1 Comparisons taken at the end of stage 1.- 3.6.2 The effectiveness of the global router.- 3.6.3 The effectiveness of stage 3 of TimberWolfSC.- 3.6.4 TimberWolfSC comparisons including stage 3.- 4 Macro/Custom Cell Chip-Planning, Placement, and Global Routing.- 4.1 Introduction.- 4.2 The General TimberWolfMC Methodology.- 4.2.1 Algorithms for handling rectilinear ceils.- 4.2.1.1 The bust() algorithm.- 4.2.1.2 The unbust() algorithm.- 4.2.2 Generating the initial placement configuration.- 4.2.3 Custom-cell pin placement.- 4.2.3.1 Introduction to the TimberWolfMC pin site methodology.- 4.3 The Algorithm for Stage 1 of TimberWolfMC.- 4.3.1 The cost function.- 4.3.1.1 The first term in the cost function.- 4.3.1.2 The second term in the cost function.- 4.3.1.3 The third term in the cost function.- 4.3.2 The generate() function.- 4.3.2.1 Introduction.- 4.3.2.2 The Range Limiter.- 4.3.2.3 Single-cell displacement-point selection.- 4.3.3 Additional stage 1 simulated annealing algorithmic details.- 4.4 The Algorithms for Stage 2 of TimberWolfMC.- 4.4.1 Channel generation.- 4.4.2 Global routing.- 4.4.3 Placement refinement.- 4.5 TimberWolfMC Results.- 4.6 Conclusion.- 5 Average Interconnection Length Estimation.- 5.1 Introduction.- 5.2 The Placement Model.- 5.3 Previous Approaches.- 5.4 Average Interconnection Length for Random Placements under the Assumption of Two-Pin Nets.- 5.4.1 Practical considerations.- 5.5 Average Interconnection Length for Random Placements Having Nets of Arbitrary Pin Counts.- 5.5.1 Results.- 5.6 A Model for Optimized Placement.- 5.6.1 The average number of other cells connected to a cell.- 5.6.1.1 The new method.- 5.6.1.2 Practical considerations.- 5.6.1.3 Results.- 5.6.2 A notion of optimized placement.- 5.6.3 The enclosing Cm × Cs rectangles.- 5.7 Results.- 6 Interconnect-Area Estimation for Macro Cell Placements.- 6.1 Introduction.- 6.2 Interconnect-Area Estimation Based on Average Net Traffic.- 6.3 Baseline Channel Width Modulation Based on Channel Position.- 6.4 Associating the Estimated Interconnect Area with the Cell Edges.- 6.5 Interconnect-Area Estimation as a Function of Relative Pin Density.- 6.6 The Implementation of the Dynamic Interconnect-Area Estimator.- 6.7 Results.- 7 An Edge-Based Channel Definition Algorithm for Rectilinear Cells.- 7.1 Introduction.- 7.2 The Basic Channel Definition Algorithm.- 7.2.1 Identifying critical cell-edge pairs.- 7.2.2 Characterization of fixed cell edges.- 7.2.3 An algorithm for finding critical regions.- 7.3 The Generation of the Channel Graph.- 7.4 The Generation of the Channel Routing Order.- 8 A Graph-Based Global Router Algorithm.- 8.1 Introduction.- 8.2 Basic Graph Algorithms Used by the Global Router.- 8.2.1 Prim's algorithm for the minimum spanning tree problem.- 8.2.2 Dijkstra's algorithm for the shortest path problem.- 8.2.3 Lawler's algorithm for finding the M-shortest paths.- 8.3 The Algorithm for Generating M-Shortest Routes for a Net.- 8.4 The Second Phase of the Global Router Algorithm.- 8.5 Results.- 9 Conclusion.- 9.1 Summary.- 9.2 Future Work.- 9.2.1 Simulated annealing.- 9.2.2 Row-based cell placement.- 9.2.3 Row-based global routing.- 9.2.4 Macro/custom cell placement.- 9.2.5 Interconnection length estimation.- 9.2.6 Channel definition.- 9.2.7 Graph-based global routing.- Appendix Island-Style Gate Array Placement.- A.1 Introduction.- A.2 The Implementation of the Simulated Annealing Functions.- A.2.1 The generation of new states.- A.2.2 The cost function.- A.2.2.1 The first cost function.- A.2.2.2 The second cost function.- A.2.3 The inner loop criterion.- A.2.5 The stopping criterion.- A.3 Results.- A.3.1 Performance comparison of the two cost functions.- A.3.2 Performance comparison on benchmark problems.