Porosity Aware Buffered Steiner Tree Construction C. Alpert G. Gandham S. Quay IBM Corp M. Hrkic Univ Illinois Chicago J. Hu Texas A&M Univ

Outline Introduction and Previous work Problem formulation Algorithm Experimental results Conclusion

Buffer Insertion Improve timing –Drive long wire –Shield load from critical path Van Ginnekens Algorithm –Given tree topology fixed –Find optimal solution at fast speed Slack

If There Are Big Blockages

Previous Works Simultaneous tree construction and buffer insertion –Buffer blockage driven Recursively Merging and Embedding [Cong and Yuan, DAC 00] Graph-based[Tang, et al., ICCAD 01] –General purpose SP-Tree [Hrkic and Lillis, ISPD 02] –Excellent solution quality –High complexity Sequential tree construction + buffer insertion –Adaptive blockage avoidance [Hu, et al., ISPD 02] –Very good solution quality –Practical computation speed

If There Are Many Small Blockages

Porosity Has to Be Considered Handling small blockages will slow down computation Buffers in dense region may be spiraled away No previous work handles porosity directly

Express Porosity through Tile Graph For a tile g A(g): tile area a(g): usage area d(g) = a(g)/A(g) Porosity cost is d 2 (g), if a buffer is placed in g

Problem Formulation Porosity-aware Buffered Steiner Tree Problem: Given –A net N = {v 0, v 1, …, v n } –Load capacitance c(v i ) and required arrival time q(v i ) –Tile graph G(V G, E G ) Construct a Steiner tree T(V,E), such that –Required arrival time q(v i ) are satisfied –Total porosity cost is minimized

Observation Easy to deal with node-to-node path –Congestion can be avoided by rerouting without affecting timing Hard to deal with Steiner nodes –Moving Steiner nodes may degrade timing

Basic Strategy Construct a timing driven Steiner tree regardless porosity Adjust Steiner nodes simultaneously with length-based buffer insertion –Adjustment range need to be restrained –A Steiner node is moved only when buffer is needed there

Length-based Buffer Insertion Simple buffering following rule of thumb –Capacitance load of driver/buffer bound L Dynamic programming based Candidate solutions are propagated bottom- up Solution is characterized by load capacitance and porosity cost A solution with greater load and cost will be pruned L=2

Plate: Adjustment Range

Plate-based Adjustment Integrate Steiner node adjustment with length-based buffer insertion Solutions are propagated to and merged at each tile of plate Merged solutions at each tile are further propagated toward root Alternative topologies are generated A candidate topology is selected only when it is a part of min cost solution at the root

Example of Plate-based Adjustment

Methodology Flow 1.Timing-driven Steiner tree ( C-Tree ) 2.Plate-based adjustment 3.Local blockage avoidance If a wire overlaps with blockage, it is rerouted within its local tiles 4.Van Ginneken style buffer insertion

Experiment Setup Integrated into industrial physical synthesis tool Three testcases –155K, 334K and 293K cells –209, 848 and 18 blockages FOM(Figure of Merit): cumulative negative slacks

Experimental Result on FOM

Resource Consumption Wirelength increase is negligible CPU time is increased significantly –Plate-based adjustment –More candidate buffer locations enabled

Result Regardless Porosity

Result Considering Porosity

Conclusion Porosity need to be considered in buffered Steiner tree construction A plate-based adjustment in a four- stage flow is proposed as a solution Experiments with industrial physical synthesis system show encouraging results

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