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CS137: Electronic Design Automation

CS137: Electronic Design Automation. Day 15: November 4, 2005 SAT for Partitioning. Today. SAT Partitioning Devadas Wrighton. Partitioning Problem. Given: netlist of interconnect cells Graph G=(V,E) Partition into two equal halves (A,B) V=A B AB = {} |A|=|V|/2

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CS137: Electronic Design Automation

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  1. CS137:Electronic Design Automation Day 15: November 4, 2005 SAT for Partitioning

  2. Today • SAT Partitioning • Devadas • Wrighton

  3. Partitioning Problem • Given: netlist of interconnect cells • Graph G=(V,E) • Partition into two equal halves (A,B) • V=AB • AB = {} • |A|=|V|/2 • minimize the number of nets shared by halves • |e=(s,t)E and sA and tB or sB and tA |<C

  4. Constraining Partitions • Balance: • |V|  2P • P-long bit vector for each vV • Goal assign each v a unique bitvector • v0 v1 … v(p-1) • partition on v(p-1) [Devadas DAC1989]

  5. Balance • Balance Constraints: • Forall pairs vi,vj • vivj • vi0^vj0+vi1^vj1+…vi(p-1)^vj(p-1)=1 • |V|<2P • Assign 2P-|V| codes such that vi(p-1) will form partition • E.g. if |V|=6 • Assign 000 and 100 • vi0^0+vi1^0+vi2^0=1 • |V|(2P-|V|) clauses

  6. Cost • Decision problem • Can ask for a partition that satisfies a constraint • Optimization: search for min C will satisfy

  7. Partitioning Results • Devadas 1989

  8. Wrighton SAT Partitioning

  9. Constraint sets • Cardinality • Results • Variations • Replication • Soed

  10. Formulation • Acceptable Partition: • All Nodes Represented • Partitions Balanced • Cut Edges < Bound • For each variable, for each partition • Vi  True if V appears in partition i

  11. All Nodes Represented (A0A1A2) (B0B1B2) (C0C1C2)…

  12. Exclusivity • If do not want replication • Assert that node is in only one partition • E.g. /(A0&A1)+/(A1&A2)+(/A0&A2)

  13. Balance • Sum of nodes in each part less than target

  14. Cut Edges • Edge cut if source or any sinks in different partitions • Pick source as point of reference • Cut if any sink not in partition with source

  15. Bound on Edges Cut

  16. Counting • Can use boolean logic to create binary counters • Problem: don’t enable implications on partial assignments

  17. Alternate Counting • Unencoded counter representation • …b3b2b1b0 • Position i says: there are at least i things • Count: 00000 00001 00011 00111 … • Add: • XXX11 + XXXX1 = XX111

  18. Totalizers

  19. Size of Totalizers • N+N  2N totalizer • Ci=Ai+(Ai-1&B1)+…+(A1&Bi-1)+Bi • i clauses Totalizer Tree: 2N2 + 2(N2/2)+4(N2/8)+…  4N2

  20. Replication • Two partitions of size 60%

  21. Other Targets • Relatively easy to change constraints • Sum of external degrees • Sum of IO from all partitions • Maximum subdomain degree • IO from largest partition • Just need to add appropriate counters and constraints

  22. Sum of External Degrees 20% total extra space

  23. Maximum Subdomain Degree 20% total extra space

  24. Admin • Assignment 3B due today

  25. Big Ideas • Tools: SAT Reduction • Bounding  Enable pruning

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