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Partial Covering Arrays: Algorithms and Asymptotics

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Abstract

A covering array CA(N;t, k, v) is an N × k array with entries in {1,2,…, v}, for which every N × t subarray contains each t-tuple of {1,2,…, v}t among its rows. Covering arrays find application in interaction testing, including software and hardware testing, advanced materials development, and biological systems. A central question is to determine or bound CAN(t, k, v), the minimum number N of rows of a CA(N;t, k, v). The well known bound CAN(t, k, v) = O((t − 1)v t log k) is not too far from being asymptotically optimal. Sensible relaxations of the covering requirement arise when (1) the set {1,2,…, v}t need only be contained among the rows of at least \((1-\epsilon )\binom {k}{t}\) of the N × t subarrays and (2) the rows of every N × t subarray need only contain a (large) subset of {1,2,…, v}t. In this paper, using probabilistic methods, significant improvements on the covering array upper bound are established for both relaxations, and for the conjunction of the two. In each case, a randomized algorithm constructs such arrays in expected polynomial time.

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Acknowledgments

Research of KS and CJC was supported in part by the National Science Foundation under Grant No. 1421058.

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Authors and Affiliations

  1. CIDSE, Arizona State University, Arizona, USA

    Kaushik Sarkar & Charles J. Colbourn

  2. Dipartimento di Informatica, University of Salerno, Fisciano, Italy

    Annalisa De Bonis & Ugo Vaccaro

Authors
  1. Kaushik Sarkar
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  2. Charles J. Colbourn
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  3. Annalisa De Bonis
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  4. Ugo Vaccaro
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Correspondence to Kaushik Sarkar.

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This article is part of the Topical Collection on Special Issue on Combinatorial Algorithms

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Sarkar, K., Colbourn, C.J., De Bonis, A. et al. Partial Covering Arrays: Algorithms and Asymptotics. Theory Comput Syst 62, 1470–1489 (2018). https://doi.org/10.1007/s00224-017-9782-9

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  • DOI: https://doi.org/10.1007/s00224-017-9782-9

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