Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Transactions on Computers C-35(8), 677–691 (1986)
Chatalic, P., Simon, L.: Zres: The old davis-putnam procedure meets ZBDDs. In: McAllester, D. (ed.) CADE 2000. LNCS(LNAI), vol. 1831, pp. 449–454. Springer, Heidelberg (2000)
GAP Forum. GAP – Groups, Algorithms, Programming – a System for Computational Discrete Algebra (2008), http://www.gap-system.org/
Knuth, D.E.: Combinatorial properties of permutations. The Art of Computer Programming, vol. 3, ch. 5.1, pp. 11–72. Addison-Wesley, Reading (1998)
Knuth, D.E.: The Art of Computer Programming: Bitwise Tricks & Techniques; Binary Decision Diagrams. fascicle 1, vol. 4. Addison-Wesley, Reading (2009)
Minato, S.: Zero-suppressed BDDs for set manipulation in combinatorial problems. In: Proc. of 30th ACM/IEEE Design Automation Conference, pp. 272–277 (1993)
Rokicki, T., Kociemba, H., Davidson, M., Dethridge, J.: God’s number is 20 (2010), http://www.cube20.org/
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Minato, Si. (2011). πDD: A New Decision Diagram for Efficient Problem Solving in Permutation Space. In: Sakallah, K.A., Simon, L. (eds) Theory and Applications of Satisfiability Testing - SAT 2011. SAT 2011. Lecture Notes in Computer Science, vol 6695. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21581-0_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-21581-0_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21580-3
Online ISBN: 978-3-642-21581-0
eBook Packages: Computer ScienceComputer Science (R0)