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Ordering Block Designs pp 173–202Cite as

Applications

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Abstract

In this chapter we look at a diverse set of applications of ordering the blocks of designs. We consider erasure-correcting codes, tournament scheduling, reliability testing, group testing and design of specific statistical experiments. The different orderings encountered span the range of possible definitions of orderings. In erasure-correcting codes we see configuration orderings and their initial motivation. In tournament scheduling we see several orderings which are equivalent to configurations of blocks disjoint on points or edges. We also see induced set system orderings for home and away game balance and minimizing the carry-over effect. Reliability testing makes use of orderings which maximize objective functions and additionally, minimal change orderings. Group testing sees minimal change orderings and also induced set system orderings. Orderings in statistics vary from configuration orderings, universal cycles and induced set systems. This chapter will be of interest to readers from the applied fields discussed to learn about the connections to design theory and combinatorial orderings. It will also be of interest to mathematicians interested in understanding the original motivations for many of the ordering definitions and learning how the more abstract methods and theorems connect to applied practise.

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References

  1. Anderson, I.: Balancing carry-over effects in tournaments. In: Combinatorial Designs and their Applications (Milton Keynes, 1997), vol. 403, pp. 1–16. Chapman & Hall/CRC, Boca Raton, FL (1999)

    Google Scholar 

  2. Bedford, D., Ollis, M.A., Whitaker, R.M.: On bipartite tournaments balanced with respect to carry-over effects for both teams. Discrete Math. 231(1–3), 81–87 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brownlie, R., Prowse, J., Phadke, M.S.: Robust testing of AT&T PMX/StarMAIL using OATS. AT&T Technical Journal 71, 41–47 (1992)

    Google Scholar 

  4. Bryce, R., Colbourn, C.J.: The density algorithm for pairwise interaction testing. Software Testing, Verification, and Reliability 17, 159–182 (2007)

    Article  Google Scholar 

  5. Bryce, R.C., Colbourn, C.J.: Prioritized interaction testing for pairwise coverage with seeding and constraints. Inform. Software Tech. 48, 960–970 (2006)

    Article  Google Scholar 

  6. Burr, K., Young, W.: Combinatorial test techniques: table-based automation, test generation, and code coverage. In: Proceedings of International Conference on Software Testing Analysis and Review, San Diego, CA, pp. 503–513 (1998)

    Google Scholar 

  7. Cawse, J.N.: Experimental design for combinatorial and high throughput materials development. GE Global Research 29, 769–781 (2002)

    Google Scholar 

  8. Cohen, M.B., Colbourn, C.J.: Ordering disks for double erasure codes. In: Proceedings of Symposium Parallel Algorithms and Architectures (SPAA01), Crete Island, Greece, pp. 229–236 (2001)

    Google Scholar 

  9. Cohen, M.B., Colbourn, C.J.: Optimal and pessimal orderings of Steiner triple systems in disk arrays. Theoret. Comput. Sci. 297, 103–117 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cohen, M.B., Colbourn, C.J.: Ladder orderings of pairs and RAID performance. Disc. Appl. Math. 138(1), 35–46 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cohen, M.B., Colbourn, C.J., Fronček, D.: Cluttered orderings for the complete graph. In: Lecture Notes in Computer Science, vol. 2108, pp. 420–431 (2001)

    Article  Google Scholar 

  12. Colbourn, C.J.: Group testing for consecutive positives. Annals of Combin. 3, 37–41 (1999)

    Article  MATH  Google Scholar 

  13. Colbourn, C.J.: Combinatorial aspects of covering arrays. Matematiche (Catania) 59(1–2), 125–172 (2004)

    Google Scholar 

  14. Colbourn, C.J.: Covering Arrays, chap. VI.10, pp. 361–365. In: Colbourn and Dinitz [15] (2007)

    Google Scholar 

  15. Colbourn, C.J., Dinitz, J.H. (eds.): Handbook of Combinatorial Designs, second edn. Chapman & Hall/CRC, Boca Raton, FL (2007)

    MATH  Google Scholar 

  16. Colbourn, C.J., Hwang, F.K.: Superimposed Codes and Combinatorial Group Testing, chap. VI.56, pp. 629–633. In: Colbourn and Dinitz [15] (2007)

    Google Scholar 

  17. Dalal, S.R., Jain, A., Karunanithi, N., Leaton, J.M., Lott, C.M., Patton, G.C., Horowitz, B.M.: Model-based testing in practice. In: Proceedings of International Conference on Software Engineering (ICSE 99), Los Angeles, CA, pp. 285–294 (1999)

    Google Scholar 

  18. Davis, A.W., Hall, W.B.: Cyclic change-over designs. Biometrika 56, 283–293 (1969)

    MathSciNet  MATH  Google Scholar 

  19. Dinitz, J.H., Fronček, D., Lamken, E.R., Wallis, W.D.: Scheduling a Tournament, chap. VI.51, pp. 591–606. In: Colbourn and Dinitz [15] (2007)

    Google Scholar 

  20. Dunietz, S., Ehrlich, W.K., Szablak, B.D., Mallows, C.L., Iannino, A.: Applying design of experiments to software testing. In: Proceedings of International Conference on Software Engineering (ICSE 97), Boston, Massachusetts, pp. 205–215 (1997)

    Google Scholar 

  21. Elbaum, S., Malishevsky, A.G., Rothermel, G.: Test case prioritization: a family of empirical studies. IEEE Trans. Software Eng. 28(2), 159–182 (2002)

    Article  Google Scholar 

  22. Elbaum, S., Rothermel, G., Kanduri, S.: Selecting a cost-effective test case prioritization technique. Software Q. J. 12(2), 185–210 (2004)

    Article  Google Scholar 

  23. Folkman, J., Fulkerson, D.R.: Edge colorings in bipartite graphs. In: Combinatorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C., 1967), pp. 561–577. Univ. North Carolina Press, Chapel Hill, N.C. (1969)

    Google Scholar 

  24. Freund, J.E.: Classroom notes: round robin mathematics. Amer. Math. Monthly 63(2), 112–114 (1956)

    Article  MathSciNet  Google Scholar 

  25. Goos, P., Großmann, H.: Optimal design of paired comparison experiments in the presence of within-pair order effects. Draft Manuscript (2011)

    Google Scholar 

  26. Hartman, A.: Software and hardware testing using combinatorial covering suites. In: Graph Theory, Combinatorics and Algorithms Edited by M. Charles Golumbic and I. Ben-Arroyo Hartman, pp. 237–266. Springer, New York (2005)

    Google Scholar 

  27. Hellerstein, L., Gibson, G.A., Karp, R.M., Katz, R.H., Patterson, D.A.: Coding techniques for handling failures in large disk arrays. Algorithmica 12, 182–208 (1994)

    Article  MathSciNet  Google Scholar 

  28. Hwang, F.K.: How to design round robin schedules. In: Combinatorics, Computing and Complexity Edited by D. Zhu Du and G. Ding Hu (Tianjing and Beijing, 1988), Math. Appl. (Chinese Ser.), vol. 1, pp. 142–160. Kluwer Academic Publishers, Dordrecht (1989)

    Google Scholar 

  29. Jimbo, M.: Serially balanced designs related to balanced incomplete block designs. J. Japan Statist. Soc. 15(2), 161–175 (1985)

    MathSciNet  MATH  Google Scholar 

  30. Kuhn, D.R., Reilly, M.: An investigation of the applicability of design of experiments to software testing. In: Proceedings of 27th Annual NASA Goddard/IEEE Software Engineering Workshop, Greenbelt, Maryland, pp. 91–95 (2002)

    Google Scholar 

  31. Kuhn, D.R., Wallace, D.R., Gallo, A.M.: Software fault interactions and implications for software testing. IEEE Trans. Software Eng. 30, 418–421 (2004)

    Article  Google Scholar 

  32. Mandl, R.: Orthogonal Latin squares: an application of experiment design to compiler testing. Comm. ACM 28, 1054–1058 (1985)

    Article  Google Scholar 

  33. Momihara, K., Jimbo, M.: Some constructions for block sequences of Steiner quadruple systems with error-correcting consecutive unions. J. Combin. Des. 16(2), 152–163 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Momihara, K., Jimbo, M.: On a cyclic sequence of a packing by triples with error-correcting consecutive unions. Util. Math. 78, 93–105 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Müller, M., Adachi, T., Jimbo, M.: Cluttered orderings for the complete bipartite graph. Discrete Appl. Math. 152(1–3), 213–228 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Müller, M., Jimbo, M.: Consecutive positive detectable matrices and group testing for consecutive positives. Discrete Math. 279(1–3), 369–381 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Müller, M., Jimbo, M.: Cyclic sequences of k-subsets with distinct consecutive unions. Discrete Math. 308(2–3), 457–464 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Nair, C.R.: Sequences balanced for pairs of residual effects. J. Amer. Statist. Assoc. 62, 205–225 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  39. Qu, X., Cohen, M.B., Rothermel, G.: Configuration-aware regression testing: an empirical study of sampling and prioritization. In: International Symposium on Software Testing and Analysis (ISSTA), Seattle, WA, pp. 75–85 (2008)

    Google Scholar 

  40. Qu, X., Cohen, M.B., Woolf, K.M.: Combinatorial interaction regression testing: a study of test case generation and prioritization. In: IEEE International Conference on Software Maintenance (ICSM), Maison Internationale, Paris, France, pp. 255–264 (2007)

    Google Scholar 

  41. Rodney, P.: Balance in tournament designs. Ph.D. thesis, University of Toronto, Toronto, ON (1993)

    Google Scholar 

  42. 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)

    Article  Google Scholar 

  43. Russell, K.G.: Balancing carry-over effects in round robin tournaments. Biometrika 67(1), 127–131 (1980)

    Article  MathSciNet  Google Scholar 

  44. Sagols, F., Riccio, L.P., Colbourn, C.J.: Dominated error-correcting codes with distance two. J. Combin. Des. 10(5), 294–302 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Seroussi, G., Bshouty, N.H.: Vector sets for exhaustive testing of logic circuits. IEEE Trans. Inform. Theory 34(3), 513–522 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  46. Shasha, D.E., Kouranov, A.Y., Lejay, L.V., F., C.M., Coruzzi, G.M.: Using combinatorial design to study regulation by multiple input signals: a tool for parsimony in the post-genomics era. Plant Physiology 127, 1590–1594 (2001)

    Article  Google Scholar 

  47. Simmons, G.J., Davis, J.A.: Pair designs. Comm. Statist. 4, 255–272 (1975)

    MathSciNet  MATH  Google Scholar 

  48. base sixteen.org: binary magic trick. http://cse4k12.org/binary/magic_trick.html. Accessed 23 Sept 2011

  49. Srikanth, H., Cohen, M.B., Qu, X.: Reducing field failures in system configurable software: cost-based prioritization. In: International Symposium on Software Reliability Engineering (ISSRE), Mysuru, India, pp. 61–70 (2009)

    Google Scholar 

  50. Street, A.P., Street, D.J.: Combinatorics of Experimental Design. Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1987)

    MATH  Google Scholar 

  51. Tang, D.T., Chen, C.L.: Iterative exhaustive pattern generation for logic testing. IBM Journal Research and Development 28, 212–219 (1984)

    Article  MATH  Google Scholar 

  52. de Werra, D.: Some models of graphs for scheduling sports competitions. Discrete Appl. Math. 21(1), 47–65 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  53. Whittaker, J.A., Poore, J.H.: Markov analysis of software specifications. ACM Trans. Software Eng. Meth. 2(1), 93–106 (1993)

    Article  Google Scholar 

  54. Whittaker, J.A., Thomason, M.G.: A Markov chain model for statistical software testing. IEEE Trans. Software Eng. 20(10), 812–824 (1994)

    Article  Google Scholar 

  55. Wikipedia: Pairwise comparison — wikipedia, the free encyclopedia. http://en.wikipedia.org/w/index.php?title=Pairwise_comparison&oldid=408248092. Accessed 21 June 2011

    Google Scholar 

  56. Williams, A.W., Probert, R.L.: A measure for component interaction test coverage. In: Proceedings of ACS/IEEE International Conference on Computer Systems & Applications, Beirut, Lebanon, pp. 301–311 (2001)

    Google Scholar 

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  1. School of Mathematics & Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada

    Megan Dewar & Brett Stevens

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  1. Megan Dewar
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Dewar, M., Stevens, B. (2012). Applications. In: Ordering Block Designs. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4325-4_6

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