πDD: A New Decision Diagram for Efficient Problem Solving in Permutation Space
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Permutations and combinations are two basic concepts in elementary combinatorics. Permutations appear in various problems such as sorting, ordering, matching, coding and many other real-life situations. While conventional SAT problems are discussed in combinatorial space, “permutatorial” SAT and CSPs also constitute an interesting and practical research topic.
In this paper, we propose a new type of decision diagram named “πDD,” for compact and canonical representation of a set of permutations. Similarly to an ordinary BDD or ZDD, πDD has efficient algebraic set operations such as union, intersection, etc. In addition, πDDs hava a special Cartesian product operation which generates all possible composite permutations for two given sets of permutations. This is a beautiful and powerful property of πDDs.
We present two examples of πDD applications, namely, designing permutation networks and analysis of Rubik’s Cube. The experimental results show that a πDD-based method can explore billions of permutations within feasible time and space limits by using simple algebraic operations.
KeywordsBoolean Function Canonical Representation Reduction Rule Algebraic Operation Decision Node
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