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International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 274-285 | Cite as

Sequence Covering Arrays and Linear Extensions

  • Patrick C. Murray
  • Charles J. ColbournEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

Abstract

Covering subsequences by sets of permutations arises in numerous applications. Given a set of permutations that cover a specific set of subsequences, it is of interest not just to know how few permutations can be used, but also to find a set of size equal to or close to the minimum. These permutation construction problems have proved to be computationally challenging; few explicit constructions have been found for small sets of permutations of intermediate length, mostly arising from greedy algorithms. A different strategy is developed here. Starting with a set that covers the specific subsequences required, we determine local changes that can be made in the permutations without losing the required coverage. By selecting these local changes (using linear extensions) so as to make one or more permutations less ‘important’ for coverage, the method attempts to make a permutation redundant so that it can be removed and the set size reduced. A post-optimization method to do this is developed, and preliminary results on sequence covering arrays show that it is surprisingly effective.

Notes

Acknowledgments

Thanks to Sunil Chandran, Marty Golumbic, Rogers Mathew, and Deepak Rajendraprasad for interesting discussions about permutation coverings and geometric representations of graphs and hypergraphs.

References

  1. 1.
    Banbara, M., Tamura, N., Inoue, K.: Generating event-sequence test cases by answer set programming with the incidence matrix. In: Technical Communications of the 28th International Conference on Logic Programming (ICLP 2012), pp. 86–97 (2012)Google Scholar
  2. 2.
    Basavaraju, M., Chandran, L.S., Golumbic, M.C., Mathew, R., Rajendraprasad, D.: Boxicity and separation dimension. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 81–92. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  3. 3.
    Brain, M., Erdem, E., Inoue, K., Oetsch, J., Pührer, J., Tompits, H., Yilmaz, C.: Event-sequence testing using answer-set programming. Int. J. Adv. Softw. 5(3–4), 237–251 (2012)Google Scholar
  4. 4.
    Brightwell, G., Winkler, P.: Counting linear extensions. Order 8(3), 225–242 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Chee, Y.M., Colbourn, C.J., Horsley, D., Zhou, J.: Sequence covering arrays. SIAM J. Discrete Math. 27(4), 1844–1861 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chor, B., Sudan, M.: A geometric approach to betweenness. SIAM J. Discrete Math. 11(4), 511–523 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Colbourn, C.J., Nayeri, P.: Randomized Post-optimization for t-Restrictions. In: Aydinian, H., Cicalese, F., Deppe, C. (eds.) Ahlswede Festschrift. LNCS, vol. 7777, pp. 597–608. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  8. 8.
    Dushnik, B.: Concerning a certain set of arrangements. Proc. Amer. Math. Soc. 1, 788–796 (1950)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Erdem, E., Inoue, K., Oetsch, J., Pührer, J., Tompits, H., Yilmaz, C.: Answer-set programming as a new approach to event-sequence testing. In: Proceedings of the Second International Conference on Advances in System Testing and Validation Lifecycle, pp. 25–34. Xpert Publishing Services (2011)Google Scholar
  10. 10.
    Fishburn, P.C., Trotter, W.T.: Dimensions of hypergraphs. J. Combin. Theory Ser. B 56(2), 278–295 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Füredi, Z.: Scrambling permutations and entropy of hypergraphs. Random Struct. Alg. 8(2), 97–104 (1996)zbMATHCrossRefGoogle Scholar
  12. 12.
    Hazli, M.M.Z., Zamli, K.Z., Othman, R.R.: Sequence-based interaction testing implementation using bees algorithm. In: 2012 IEEE Symposium on Computers and Informatics, pp. 81–85. IEEE (2012)Google Scholar
  13. 13.
    Huber, M.: Fast perfect sampling from linear extensions. Discrete Math. 306(4), 420–428 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Ishigami, Y.: Containment problems in high-dimensional spaces. Graphs Combin. 11(4), 327–335 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Ishigami, Y.: An extremal problem of \(d\) permutations containing every permutation of every \(t\) elements. Discrete Math. 159(1–3), 279–283 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Karzanov, A., Khachiyan, L.: On the conductance of order Markov chains. Order 8(1), 7–15 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kuhn, D.R., Higdon, J.M., Lawrence, J.F., Kacker, R.N., Lei, Y.: Combinatorial methods for event sequence testing. CrossTalk: J. Defense Software Eng. 25(4), 15–18 (2012)Google Scholar
  18. 18.
    Kuhn, D.R., Higdon, J.M., Lawrence, J.F., Kacker, R.N., Lei, Y.: Combinatorial methods for event sequence testing. In: IEEE Fifth International Conference on Software Testing, Verification and Validation (ICST), pp. 601–609 (2012)Google Scholar
  19. 19.
    Levenshteĭn, V.I.: Perfect codes in the metric of deletions and insertions. Diskret. Mat. 3(1), 3–20 (1991)MathSciNetGoogle Scholar
  20. 20.
    Margalit, O.: Better bounds for event sequence testing. In: The 2nd International Workshop on Combinatorial Testing (IWCT 2013), pp. 281–284 (2013)Google Scholar
  21. 21.
    Mathon, R.: Tran Van Trung: Directed \(t\)-packings and directed \(t\)-Steiner systems. Des. Codes Cryptogr. 18(1–3), 187–198 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Nayeri, P., Colbourn, C.J., Konjevod, G.: Randomized postoptimization of covering arrays. Eur. J. Comb. 34, 91–103 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Opatrný, J.: Total ordering problem. SIAM J. Comput. 8(1), 111–114 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Radhakrishnan, J.: A note on scrambling permutations. Random Struct. Alg. 22(4), 435–439 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Spencer, J.: Minimal scrambling sets of simple orders. Acta Math. Acad. Sci. Hungar. 22, 349–353 (1971/72)Google Scholar
  26. 26.
    Tarui, J.: On the minimum number of completely 3-scrambling permutations. Discrete Math. 308(8), 1350–1354 (2008)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Computing, Informatics, and Decision Systems EngineeringArizona State UniversityTempeUSA

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