Speaker: Danai Chasaki

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1995, O. Coudert, J. Madre (covering matrix reduction, pruning techniques) ... Best bet: Enhanced set of constraints / Restricted candidate set / Accept strategy ... – PowerPoint PPT presentation

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Title: Speaker: Danai Chasaki

1
Technology mapping using binate covering 1

Speaker Danai Chasaki
May 5th, 2009

1 Michal Z. SERVIT and Kang YI Technology
Mapping by Binate Covering
2
Technology mapping based on structural mapping

Input subject Boolean network (representation
of the network using a set of base functions
single logic cell from a given library)
Generate pattern graphs
Cover the subject graph with patterns
- DAG covering (tree covering/dynamic)
- binate covering
Binate Covering Problem extraction of a minimum
cost subset from a given set that satisfies
certain constraints expressed as a Boolean
formula in conjunctive normal form
NP-hard, heuristic methods used

3
Outline

Background
Problem formulation
Heuristics
Example
Comparison/Summary
References

4
Previous work on binate covering

Heuristic methods/ backtracking
- 1992, M. Servit, J. Zamazal (scoring function)
- 1995, R. Murgai, R. Brayton (partition
networks efficiently)
- 1995, O. Coudert, J. Madre (covering matrix
reduction, pruning techniques)
Branch and bound Thelen
Sort by length, literals, variables, reorder
BDD-based Lin and Somenzi

5
Binate covering problem formulation

Subject network
Nodes v1..vn (gates)
Set of clusters C1..Cn (subnets)
Nodes, clusters produce-consume a signal
v1 produces signal s1
v2 consumes signals s3 s4 sc
Constraints
Output
Implication
Task select optimum subset from all clusters C

P.O.
P.I.
With courtesy from reference paper by Servit,
1997
6
Product of SUMs Boolean formula
P.O.

Output constraints
primary output node consuming signal s
Signal s produced by set of clusters C1..Ck
SUM clause (x1 x2 xk)
Implication constraints
signal s consumed by cluster C0, produced by
clusters C1..Ck
SUM clause (x0 x1 x2xk)
Boolean formula T product of all SUM clauses

P.I.

T1 (x6x7),T2 (x3x4x5),T3(x3x1)
T4(x3x2), T5(x4x2), T6(x5x1)
T7 (x6 x3x4x5), T8 (x7x2)
T T1T2T3T4T5T6T7T8

7
Heuristic algorithm for binate covering

Problem Find assignment (set of clusters) that
satisfy T with the least sum of costs of all
selected clusters
Greedy algorithm combined with backtracking
Simplify formula T by reduction select a
variable xi (cluster Ci) and assign the value
TRUE (accept) or FALSE (reject) to it, then
simplify and repeat
Several possibilities
how to define a candidate set for variable
selection
how to select a variable from the candidate set
and
how to assign a value to the variable selected

8
Orthogonal set of heuristics

basic or enhanced set of constraints
Basic output, implication constraints
Enhanced covering constraints (each internal
node should be covered by at least one cluster)
complete or restricted candidate set
Restricted consider only variables that appear
in positively unate clauses in the basic set of
constraints
Motivation traversal from P.O. to P.I. (avoid
redundant solutions)
accept or reject or accept-or-reject strategy
Which variable out of the set to choose use
score functions
What value to assign true or false depending on
score

9
Score functions

Calculate weight of Tj - inversely proportional
to the number of literals it contains
Direct score of variable xi (DS) sum of weights
of all Ts that contain the literal xi
Indirect score of variable xi (IS) sum of
weights of all Ts that contain the literal xi
DIF DS IS

10
Value assignment

Value assignment based on scores
Accept strategy assigns xi TRUE to the
variable xi that has the highest value of DS
Reject strategy assigns xi FALSE to the
variable xi that has the highest value of IS
Accept-or-Reject strategy assigns a value to the
variable that has the highest value of DIF, xi
TRUE if DIF gt 0, otherwise
xi FALSE

11
Example CNF formulation
Covering constraints node 1 T8 (x1x6) node 2
T9 (x2x7) node 3 T10 (x3x6x7) node 4 T11
(x4x6x7) node 5 T12 (x5x6)
Implication constraints S1,C7 T2
(x7x1) S2,C6 T3 (x6x2) S4,C5 T4
(x5x4x7) S1,C4 T5 (x4x1) S3,C4 T6
(x4x3) S2,C3 T7 (x3x2)
Restricted candidate set x5, x6
Output constraints T1 (x5x6)
T T1T2T3T4T5T6T7T8T9T10T11T121
12
Example - Solution

Objective Assign values to variables xi so that
T is satisfied (TRUE)
When variable xi is assigned to TRUE it means
that cluster Ci is selected and inserted in the
final solution set
Look at restricted candidate set (x5, x6). Which
one do we pick?
DS(x6) gt DS(x5), so we pick x6. Which value do
we assign to x6?

Calculate weights of Ts w1w2w3w5w6w7w8w9
w121/2w4w10w111/3
Calculate DS, IS of x5, x6 DS(x5) w1w12 1,
IS(x5) w4 1/3 DS(x6) w1w8w10w11w12 7/3,
IS(x6)w31/2

Accept strategy x6TRUE
Notice It is sufficient to select just one of
the clusters producing the same signal gt
x5FALSE (from T1) and x2 TRUE (from T3) gt
x4x7FALSE (from T4) gt
x1FALSE (from T5) and x3FALSE (from T6)
Finally we have (x1, x2, x3, x4, x5, x6, x7)
(0,1,0,0,0,1,0) so clusters C2, C6 are selected
Solution set S C2, C6

13
Experimental results

Enhanced set of constraints gt basic
Restricted candidate set gt complete
Accept strategy gt accept-reject strategy gt reject
Few redundancies appeared only if the complete
candidate set and accept or accept-or-reject
strategy were used
Backtracking was not needed

14
Summary

Binate covering problem formulation CNF
Clauses for all constraints (output, implication,
covering)
Satisfy Boolean formula T, with minimum cost
Value assignment (accept/reject) based on score
functions
Best bet Enhanced set of constraints /
Restricted candidate set / Accept strategy

15
References

1 Michal Z. SERVIT and Kang YI Technology
Mapping by Binate Covering, 1997
2 M. Servit, J. Zamazal Heuristic Approach to
Binate Covering Problem, EDAC'92 Proc.,1992
3 R. Murgai, R. Brayton, A. Sangiovanni-Vincente
lli Logic Synthesis for Field-Programmable Gate
Arrays, 1995
4 O. Coudert, J. Madre New Ideas for Solving
Covering Problems, 31st DAC Proc., 1995
5 B.Lin, F.Somenzi Minimization of Symbolic
Relations, 1990
6 J. Bieganowski, A. Karatkevich, Heuristics
for Thelens Prime Implicant Method, 2004