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From MDD to BDD and Arc consistency

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Abstract

In this paper, we present a new conversion of multivalued decision diagrams (MDD) to binary decision diagrams (BDD) which can be used to improve MDD-based fil- tering algorithms such as MDDC or MDD-4R. We also propose BDDF, an algorithm that copies modified parts of the BDD “on the fly” during the search of a solution, and yields a better incrementality than a pure MDDC-like approach. MDDC is not very efficient when used to represent poorly structured positive table constraints. Our new representation combined with BDDF retains the properties of the MDD representation and has comparable performances to the STR2 algorithm by Ullmann (2007) and Lecoutre (Constraints, 16.4, 341–371 2011).

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Notes

  1. Negative table constraints have received much less attention but are reasonably handled by generic AC algorithms such as AC-3rm [23] and efficient indexation of the table.

  2. Note that sparse sets can only be used when all operations on the set are monotonic on a branch of the search tree, i.e., items are only added along the whole branch, or removed along the whole branch.

  3. The name BDDC was already used by Cheng and Yap [10] to denote a preliminary version of MDDC restricted to binary domains.

  4. For most applications, a correct rationale for Λ is that it represents the set of all relations.

  5. Actually Θ(N) where N is the number of bits used to represent domain values.

  6. This complexity implies that indexing a vertex has a complexity linear in the number of outgoing edges (i.e., does not depend on d), which can be obtained easily by using appropriately chosen data structures, e.g., linked lists or hash tables.

  7. Note that the instances used by Demeulenaere et al. do not include any of the heavily structured relations described by e.g., Cheng and Yap [9] (definite state automatons, super-row density, slidingsum constraints…), that can only be handled by MDD or ad-hoc representations.

  8. Alternative implementations of sets can also be used provided they have expected performance similar to sparse sets.

  9. We published the idea of filtering MDD in French [38] in 2013.

  10. If X ∉ Modif, there is no need to control validity of values on Line 6 for this vertex and all its siblings. The removal of X from Modif on Line 7 can also be omitted in this case. These simple optimizations are not included in the algorithms to improve concision and preserve readability, but have some impact on the speed of processing.

  11. |V (B)| < k requires that the BDD is shallower than the arity of the constraint. This can happen only when all values of the deeper variables are supported by all instantiations of the other variables, which means that the constraint can be easily reformulated to omit these variables. Anyway, this issue did not appear in our experiments.

  12. \(\frac {|E(mdd)|-| E(bdd)|}{|E(mdd)|}\)

  13. Note that to achieve pure immutability, the timestamp technique used in this paper should be replaced by some sort of lookup table, which is likely to be much slower.

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Acknowledgments

This research was partially financed by the French Ministry of National Education, Research and Technology, The Nord/Pas-de-Calais Region, the French National Center of Scientific Research (CNRS) and the International Campus on Safety and Intermodality in Transportation.

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  1. LAMIH CNRS UMR 8201, Université de Valenciennes et du Hainaut Cambrésis, Valenciennes, France

    Julien Vion & Sylvain Piechowiak

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Correspondence to Julien Vion.

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Vion, J., Piechowiak, S. From MDD to BDD and Arc consistency. Constraints 23, 451–480 (2018). https://doi.org/10.1007/s10601-018-9286-5

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