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ECE 465 Petrick’s Algorithm for 2-level Minimization. Shantanu Dutt University of Illinois at Chicago. Acknowledgement: Transcribed to Powerpoint by Huan Ren from Prof. Shantanu Dutt’s handwritten notes. Petrick’s Algorithm for Choosing Minimal Cost cover.
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ECE 465Petrick’s Algorithm for 2-level Minimization Shantanu Dutt University of Illinois at Chicago Acknowledgement: Transcribed to Powerpoint by Huan Ren from Prof. Shantanu Dutt’s handwritten notes
Petrick’s Algorithm for Choosing Minimal Cost cover • The PIT portion of Q-M can get optimal in most cases and near-optimal cost coverings, but will not be optimal in all cases. • Can use an algebraic method called Petrick’s algorithm.
Petrick’s Algorithm • 1. Obtain all PIs using Q-M. • 2. Create a PIT and remove all EPIs and corresponding columns (MTs). • 3. Write a POS expr. representing all possible covers of remaining MTs • a) For each MT mi write an expr. C(mi) that is the sum/OR of all PIs that cover it --- the PIs are the variables in this expression. • b) Form a POS expression C that is the product/AND of all C(mi)’s --- this indicates that all MTs need to be covered. • 4. Convert the POS expr. to SOP using the distr. law. Use involution and absorption to simplify ((a*a=a, a+a=a) (a+ab=a))—note that these minimization rules come about in this problem from concepts of non-replication and lower cost than strictly from Boolean algeabra. • Each product term in the SOP expr. represents one possible cover (correct functional expression = sum of the Pis in the product term). • 5. Select the cover with the lowest cost. total # of literals + # of PIs
Minimal solution Cost=8 +3 Cost=9 +3 Absorb PI1+ PI6 PI2+ PI3 PI1+ PI2 PI4+ PI5 PI5+ PI6 PI3+ PI4 PI1PI3PI5+PI2PI4PI6+PI1PI3PI4PI6+PI2PI3PI5PI6+PI1PI2PI4PI5+ PI2PI3PI4PI6+PI1PI2PI4PI6+ PI2PI4 PI5PI6 C=(PI1+PI2PI6)(PI5+PI4PI6)(PI3+PI2PI4) =(PI1PI5+PI1PI4PI6+PI2PI5PI6+PI2PI4PI6)(PI3+PI2 PI4) Petrick’s Algorithm (Contd.) Example 1 X’s
Petrick’s Algorithm (Contd.) • Exercise: Use this algorithm to obtain the least cost cover for the example in which we used the max. MT covering heuristic to get the minimal solution.
Petrick’s Algorithm (Contd.) • Computational complexity of Petrick’s algorithm is very high If m=# of MTs n = # of vars, pi = # of PIs covering MTi, pav = avg # of PIs covering a MT Since MTi=an-1…..a0, a PI covering MTi will have X some positions j. Can choose, say, n/2 X’s with in ways, w/ each way not covering any other way, i.e., each way is a PI covering MTi The time complexity T (in terms of # of time to gen. product terms in C) = m*Pi=1mpi T = O(m*2nm = O(2n*2**n)) Or T = O(m*(pav)m ) = O(m*(pav)2**n)) worst-case # of basic operations to gen. the reqd. product terms in C 1 trillion=240 T=32 trillion trillion trillion operations (in the worst case) If each oper takes 1 ns, time taken can be 295 ~ 32k trillion trillion secs ! (in the worst case)
QM PIT (Covering Stage) Complexity If m=# of MTs n = # of vars, pi = # of PIs covering MTi, pav = avg # of PIs covering a MT, p = total # of PIs • The time taken to determine row covering is O(mp2) (look at all PI pairs—naïve method, and each pair takes O(m) time to determine if there is covering betw. them) • Similarly, O(m2p) for determining column covering relations • If after every row covering, we detect an EPI, or if we detect a col. covering, we reduce # of MTs by at least 1; so we would require m iterations, and thus O(max(m2p2, m3p) time until all MTs are covered. • Or, we do not get any EPIs or col. covering until we do lots of row coverings until finally 1 PI is left that covers all MTs (worst case). This will take p iterations, and thus a total time of O(max(mp3, m2p2)). • So overall complexity (using no appropriate data structures) is O(max(p3m, m2p2, mp3)). • Compare this to O(m*(pav)m) ~ O((p/m)m) for a large enough m, for Petrick’s, and we see that QM is much less complex (m appears as an exponent in Petrick’s as opposed to as the base of a low-order polynomial in QM).
Graphical comparison of Petrick’s and QM’s time complexities • As can be seen, Petrick’s run-time quickly becomes huge and impractical for n > 7, while QM’s is much more reasonable and thus could potentially be used in a CAD tool. • Further, as we have seen, besides the cyclic PIT scenario, QM’s non-optimality stems from the rare (and thus very low probability) case of “bad” row covering (cost(covering PI) > cost(covered PI). Thus on the average, QM produces good (near-optimal) solutions. • Thus overall, QM is a very good and practical algorithm to use for, say, n > 5.
