1. 1 A Flexible Minimization and Partitioning Method Petr Fišer, Jan Hlavička Czech Technical University in Prague e-mail: fiserp@fel.cvut.cz, hlavicka@fel.cvut.cz

2. 2 Problem Statement The Goal: Two-level Boolean minimization performed together with the partitioning into stand-alone blocks (PLAs, GALs) Current methods (SIS): First do the minimization, then the decomposition  could be inefficient

3. 3 Problem Statement Restrictions: Number of blocks’ inputs Number of blocks’ outputs Complexity of blocks Maximum load Size of the cones...

4. 4 The Idea: 1. Not all primary inputs are needed to produce required output values 2. Some input variables can be substituted by others  Sometimes there is a great flexibility The function can be synthesized in many ways

5. 5 What is it for? Partitioning into blocks with restricted # of inputs / outputs Load balancing DFT design (synthesis) - smaller cones Synthesis for better placement & wiring Can help in multi-level synthesis ???

6. 6 Based on BOOM – the Boolean Minimizer Two main minimization phases: Implicant generation: CD-Search, Implicant Expansion, Implicant Reduction Covering problem solution  Both have to be modified

7. 7 CD-Search Top-down approach – literals are added to a term until it becomes an implicant Literals are being selected by a scoring function Originally: Most frequent literals in the on-set are preferred Partitioning forces – literals not included in the block are penalized by some factor Load balancing – literals causing branchings are penalized by some factor

8. 8 Covering Problem Solution Must be incremental - terms are added one by one  Some heuristic based on some scoring function Partitioning forces – terms constructed of literals not included in blocks are penalized Load balancing – terms constructed of literals causing branchings are penalized by some factor

9. 9 The Whole Minimization Process: 1. Assign all the output variables to blocks 2. Generate implicants (CD-Search, IE, IR) for each output separately 3. Solve the group covering problem for each block separately 4. If the result do not meet the requirements, reassign the outputs and repeat from 2

10. 10 Influence of the Partitioning Forces 50 inputs, 20 outputs, 200 care terms, 4x5-output blocks

11. 11 Illustrative Example 100 inputs, 20 outputs, 100 care terms, 5x4-output blocks Partitioning appliedLoad balancing applied

12. 12 Illustrative Example 100 inputs, 20 outputs, 100 care terms, 5x4-output blocks

13. 13 Conclusions The minimization method combined with a single-level decomposition was presented We have shown that the top-down minimization method used in the BOOM minimizer suits very well for modifications like partitioning, load balancing and DFT The result of the minimization can suit our needs Both the CD-Search and CP solution must be modified