Technology Mapping

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• Technology Mapping
• as a Binate Covering Problem

Slides adapted from A. Kuehlmann, UC Berkeley 2003
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Circuit as DAG (Subject Graph)

• A Directed Acyclic Graph, also called a DAG, is a
  directed graph with
• no directed cycles that is, for any vertex v,
  there is no nonempty
• directed path starting and ending on v.

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Pattern Graphs used for the cover
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Binate Covering Problem Example
Gate Cost Input Produces Covers
m1 inv 1 a g1 g1
m2 inv 1 b g2 g2
m3 nand2 2 g1, g2 g3 g3
m4 nand2 2 a, b g4 g4
m5 nand2 2 g3, g4 g5 g5
m6 inv 1 g4 g6 g6
m7 nand2 2 g6, c g7 g7
m8 nand3 3 a, b g7 g4, g6, g7
m9 xnor2 5 a, b g5 g1, g2, g3, g4, g5
m10 oai21 3 a, b, g4 g5 g1, g2, g3, g5
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Binate Covering Problem

• Compute all possible matches mk for each node.
• Using a variable mi for each match of a pattern
  graph in the subject graph, (mi 1 if match is
  chosen mi 0 otherwise)
• Write a clause for each node of the subject graph
  indicating which matches cover this node. Each
  node has to be covered.
• e.g., if a subject node is covered by matches
  m2, m5, m10 , then the clause would be (m2 m5
  m10).
• Repeat for each subject node and take the
  product over all subject nodes (CNF).

m1 m2 . . . mk
n1 n2 . . . nl
nodes
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Binate Covering Problem Example
m1 m2 m3 m4 m5 m6 m7 m8 m9 m10
n1 1 0 0 0 0 0 0 0 1 1
n2 0 1 0 0 0 0 0 0 1 1
n3 0 0 1 0 0 0 0 0 1 1
n4 0 0 0 1 0 0 0 1 1 0
n5 0 0 0 0 1 0 0 0 1 1
n6 0 0 0 0 0 1 0 1 0 0
n7 0 0 0 0 0 0 1 1 0 0

• Generate constraints that each node ni be covered
  by some match.(m1 m9 m10)(m2 m9 m10)(m3
  m9 m10)(m4 m8 m9)
• (m5 m9 m10)(m6 m8) (m7 m9)

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Binate Covering

• A Unate Covering Problem is a product-of-sums
  expression, containing Boolean propositions (i.e.
  variables which can have value 0 or 1).The
  propositions are all uncomplemented.
• For example
• P (P1 P3)(P1 P4 P5)(P6 P8)
• A BINATE COVERING PROBLEM is similar, but now the
  propositions can be both uncomplemented and
  complemented. For example
• P (P1' P3)(P1 P4' P5)(P2 P5)(P2'
  P3')

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Create Outputs for a Legal Cover

• Any satisfying assignment guarantees that all
  subject nodes are covered, but
• It does not guarantee that other matches create
  outputs needed as inputs for a given match (legal
  covering).
• Rectify this by adding additional clauses.

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Binate Covering Problem

• Let match mi have subject nodes si1,,sin as
  inputs. If mi is chosen, one of the matches that
  realizes sij must also be chosen for each
  intermediate input j
• Let Sij be the disjunctive expression in the
  variables mk giving the possible matches which
  realize sij as an output node. Selecting match mi
  implies satisfying each of the expressions Sij
  for j 1 n.This can be written as
• (mi ? (Si1 Sin ) )? (?mi (Si1 Sin ) ) ?
  ((?mi Si1) (?mi Sin ) )

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Binate Covering Problem Example

• For example, selecting m3 requires that
• a match be chosen which produces g2 as an output,
  and a match be chosen which produces g1 as an
  output.
• The only match which produces g1 is m1 same for
  g2 and m2, etc.
• So, two extra clauses are required

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Binate Covering Example

• Finally, we get
• seven additional clauses.
• Note
• A match which requires a primary input as an
  input is satisfied trivially.
• Matches m1,m2,m4,m8,m9 are driven only by
  primary inputs and do not require additional
  clauses

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Binate Covering Problem

• An assignment of values to variables mi that
  satisfies P, the above covering expression, is a
  legal graph cover.
• For area optimization, each match mi has a cost
  ci the area of the gate the match represents.
• The goal is a satisfying assignment with the
  least total cost.
• Find a least-cost prime
• if a variable mi 0 its cost is 0, else its
  cost is ci
• mi 1 means that match i is chosen
• mi 0 means that match i is not chosen

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Binate Covering Example

• The least cost prime implicant is
• This uses two gates for a cost of eight gate
  units, for matches
• m8 (nand3) and m9 (xnor2).

Note that the node g4 is covered by both matches
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Summary

• Given a subject graph, the binate covering
  provides the exact solution to the
  technology-mapping problem.
• Finding a feasible solution to a binate covering
  is NP-complete.
• Problems that are NP-complete can be solved by
  algorithms that run in exponential time. No
  polynomial time algorithms are known to exist for
  any of the NP-complete problems and it is very
  unlikely that polynomial time algorithms should
  indeed exist although nobody has yet been able to
  prove their non-existence.