Ordering the Blocks of Designs

  • Megan Dewar
  • Brett Stevens
Chapter
Part of the CMS Books in Mathematics book series (CMSBM)

Abstract

In Chap. 3 we discussed the various ways to order subsets, partitions and permutations. Similarly, there are various ways to order the blocks of a design. A listing of the blocks of a design such that there is a minimal change between the content of consecutive blocks is a combinatorial Gray code. Note that the minimal change definition will depend on the design parameters. A listing of points such that the blocks of a design are induced by the consecutive subwords is a more structural way of defining minimal change. Universal cycles (Ucycles) are sequences that adhere to such a definition. Another rule for ordering blocks of designs that relates both to content and structure employs configurations. A configuration describes a pattern of intersection for a set of blocks. Unlike Gray codes and Ucycles, configuration ordering is a paradigm unique to designs and can include orderings which are neither Gray codes nor Ucycles. In this chapter we formally introduce the various ways to order the blocks of a design and illustrate them with examples. We include basic results on such orderings, including bounds and fundamental enumerations. This chapter is recommended both to the researcher introducing themselves to this subject, and to the researcher familiar with some previous literature on block design orderings who is interested in generalizations of previous orderings and new ordering definitions.

References

  1. 1.
    Bryce, R.C., Colbourn, C.J.: Prioritized interaction testing for pairwise coverage with seeding and constraints. Inform. Software Tech. 48, 960–970 (2006)CrossRefGoogle Scholar
  2. 2.
    Buhler, J., Jackson, B.: Private communication (2006)Google Scholar
  3. 3.
    Cohen, M.B., Colbourn, C.J.: Optimal and pessimal orderings of Steiner triple systems in disk arrays. Theoret. Comput. Sci. 297, 103–117 (2003)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dewar, M.: Gray codes, universal cycles and configuration orderings for block designs. Ph.D. thesis, Carleton University, Ottawa, ON (2007)Google Scholar
  5. 5.
    Horák, P., Rosa, A.: Decomposing Steiner triple systems into small configurations. Ars Combin. 26, 91–105 (1988)Google Scholar
  6. 6.
    Jackson, B.W.: Universal cycles of k-subsets and k-permutations. Discrete Math. 117, 141–150 (1993)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Jackson, B.W., Buhler, J., Mayer, R.: A recursive construction for universal cycles of 2-subspaces. Discrete Math. 309(17), 5328–5331 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    McSorley, J.P.: Single-change circular covering designs. Discrete Math. 197/198, 561–588 (1999)Google Scholar
  9. 9.
    Rothermel, G., Untch, R.H., Chu, C., Harrold, M.J.: Prioritizing test cases for regression testing. IEEE Trans. Software Eng. 27(10), 929–948 (2001)CrossRefGoogle Scholar
  10. 10.
    Wallis, W., Yucas, J.L., Zhang, G.H.: Single-change covering designs. Des. Codes Cryptogr. 3, 9–19 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wallis, W.D.: Sequential covering designs. In: Graph Theory, Combinatorics, and Algorithms, (Kalamazoo, MI, 1992), Wiley-Interscience Publication, vol. 1–2, pp. 1203–1210. Wiley, New York (1995)Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Megan Dewar
    • 1
  • Brett Stevens
    • 1
  1. 1.School of Mathematics & StatisticsCarleton UniversityOttawaCanada

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