1. Hypergraph Partitioning With Fixed Vertices Andrew E. Caldwell, Andrew B. Kahng and Igor L. Markov UCLA Computer Science Department {caldwell,abk,imarkov}@cs.ucla.edu Supported by Cadence Design Systems, Inc.

2. Outline F Hypergraph partitioning –abstract: free hypergraph context  no terminals –practical: top-down placement context  terminals F Terminals change the partitioning problem –empirical study of effects on FM performance F New heuristics needed that exploit terminals –early pass termination in FM F Open issues

3. Partitioning in the Research Literature  “Given hypergraph H = (V,E), partition V into V 1 and V 2 with |V 1 |  |V 2 | so as to minimize the number of cut hyperedges…” –balance constraints  NP-hard –pass-based KLFM variants most successful F Benchmark-driven research –partitioning benchmarks have no fixed-terminal information F Entire literature is on “free hypergraphs”

4. Partitioning in Top-Down Placement F Global placement –map cells of netlist into layout area –satisfy performance constraints, minimize area F Top-down divide-and-conquer approach F Divide step: hypergraph partitioning –connections among blocks modeled as fixed vertices (terminals) in the partitioning instance

5. Placement Blocks: Many Terminals  Rent’s rule: #terminals = k  (#cells) p F For given Rent parameter value p, below what #cells will more than y% of vertices be terminals?

6. Disconnect! F Top-down placement always generates instances with fixed terminals F Partitioning research has focused on instances without fixed terminals F Obvious questions –is effect of terminals on algorithm performance sufficient to require new techniques? –can we exploit, rather than tolerate, terminals?

7. Demonstration: Effects of Terminals F Experiment with well-assigned terminals –find “good solution”: best of 100 partitioner runs –make increasing % of nodes into terminals fixed as in good solution –“good solution” cost - an upper bound for min cost of all instances (by construction) –run partitioner again - how does it do? F Expectations –problem gets easier as more terminals are fixed –smaller runtime, better average quality

8. Expectations Partly Wrong F “Well-assigned” terminals can hurt! –good solutions are harder to find –spike at 5%

9. Presence of Terminals is Significant F Interpretation of the spike –failure of FM –other heuristics may be more successful

10. Can We Exploit Terminals? F Best for free hypergraphs  best with terminals –need methods specifically to exploit terminals –different trade-offs/tunings of traditional heuristics F Example: shorter passes in FM –terminals shorten the useful part of the pass (find best sooner)

11. Terminals in FM Partitioning Moves Cut Moves Cut No Fixed Terminals Fixed Terminals

12. FM Partitioning With Pass Limits F Allow at most x% of nodes to be moved in a pass F Pass limits: –hurt results when there are no terminals –help given “sufficiently many“ terminals

13. Conclusions F Fixed terminals matter F Current methods not adequate F Better methods are possible F Many questions remain...

14. Open Directions F Quantify effects of terminals –interpret “sufficiently many terminals” (“# terminals” is meaningless in general!) F Explain nonmonotonicity for < 5% fixed terminals F Variant pass-limiting schemes F Stronger effects for multi-way? F New heuristics specialized for fixed terminals F Fixed-terminals benchmark suite (ISPD99) at http://vlsicad.cs.ucla.edu