Hypergraph Partitioning With Fixed Vertices

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Hypergraph Partitioning With Fixed Vertices

• Andrew E. Caldwell, Andrew B. Kahng and Igor L.
  Markov
• UCLA Computer Science Department
• caldwell,abk,imarkov_at_cs.ucla.edu
• Supported by Cadence Design Systems, Inc.

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Outline

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

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Partitioning in the Research Literature

• Given hypergraph H (V,E), partition V into V1
  and V2 with V1 V2 so as to minimize the
  number of cut hyperedges
• balance constraints NP-hard
• pass-based KLFM variants most successful
• Benchmark-driven research
• partitioning benchmarks have no fixed-terminal
  information
• Entire literature is on free hypergraphs

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Partitioning in Top-Down Placement

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

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Placement Blocks Many Terminals

• Rents rule terminals k (cells)p
• For given Rent parameter value p, below what
  cells will more than y of vertices be terminals?

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Disconnect!

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

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Demonstration Effects of Terminals

• 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?
• Expectations
• problem gets easier as more terminals are fixed
• smaller runtime, better average quality

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Expectations Partly Wrong

• Well-assigned terminals can hurt!
• good solutions are harder to find
• spike at 5

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Presence of Terminals is Significant

• Interpretation of the spike
• failure of FM
• other heuristics may be more successful

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Can We Exploit Terminals?

• Best for free hypergraphs ? best with terminals
• need methods specifically to exploit terminals
• different trade-offs/tunings of traditional
  heuristics
• Example shorter passes in FM
• terminals shorten the useful part of the pass
  (find best sooner)

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Terminals in FM Partitioning
Cut
No Fixed Terminals
Moves
Cut
Fixed Terminals
Moves
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FM Partitioning With Pass Limits

• Allow at most x of nodes to be moved in a pass
• Pass limits
• hurt results when there are no terminals
• help given sufficiently many terminals

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Conclusions

• Fixed terminals matter
• Current methods not adequate
• Better methods are possible
• Many questions remain...

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Open Directions

• Quantify effects of terminals
• interpret sufficiently many terminals
  ( terminals is meaningless
  in general!)
• Explain nonmonotonicity for lt 5 fixed terminals
• Variant pass-limiting schemes
• Stronger effects for multi-way?
• New heuristics specialized for fixed terminals
• Fixed-terminals benchmark suite (ISPD99) at
  http//vlsicad.cs.ucla.edu