Partial Covering Arrays: Algorithms and Asymptotics

Sarkar, Kaushik; Colbourn, Charles J.; Bonis, Annalisa De; Vaccaro, Ugo

doi:10.1007/978-3-319-44543-4_34

Cited by 10 publications

(10 citation statements)

References 26 publications

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“…The authors would like to express their gratitude toward Noga Alon, Charlie Colbourn, Gal Kronenberg, and Benny Sudakov for helpful discussions at various stages of this project. Special thanks go to Ugo Vaccaro for bringing to our attention, prompting the extension of our initial results on set shattering to general covering arrays. We are also grateful to the anonymous referee for several beneficial suggestions and references.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…This line of investigation was pursued by Sarkar et al. , who introduced the notion of almost‐covering arrays. Definition Given a set S of v symbols, N rows, k columns and a strength

t \in N

, let

A \in S^{N \times k}

be an

N \times k

array.…”

Section: Introductionmentioning

confidence: 99%

“…However, when ε is a small constant, Sarkar et al. showed the existence of very small almost‐covering arrays, whose size is independent of the number of columns k . More precisely, they showed

ACAN false(t, k, v, ε false) \leq v^{t} ln (\frac{v^{t - 1}}{ε}),

with an improvement to

ACAN false(t, k, v, ε false) \leq v^{t} ln false(\frac{2 v^{t - 2}}{ε} false) + v

whenever v is a prime power.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Small arrays of maximum coverage

Das

Mészáros

2018

J of Combinatorial Designs

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Given a set S of v≥2 symbols, and integers k≥t≥2 and N≥1, an N×k array A∈SN×k is said to cover a t‐set of columns if all sequences in St appear as rows in the corresponding N×t subarray of A. If A covers all t‐subsets of columns, it is called an (N;t,k,v)‐covering array. These arrays have a wide variety of applications, driving the search for small covering arrays. Here, we consider an inverse problem: rather than aiming to cover all t‐sets of columns with the smallest possible array, we fix the size N of the array to be equal to vt and try to maximize the number of covered t‐sets. With the machinery of hypergraph Lagrangians, we provide an upper bound on the number of t‐sets that can be covered. A linear algebraic construction shows this bound to be tight; exactly so in the case when v is a prime power and vt−1v−1 divides k, and asymptotically so in other cases. As an application, by combining our construction with probabilistic arguments, we match the best‐known asymptotics of the covering array number CAN(t,k,v), which is the smallest N for which an (N;t,k,v)‐covering array exists, and improve the upper bounds on the almost‐covering array number ACAN(t,k,v,ε).

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Section: Acknowledgmentsmentioning

confidence: 99%

“…This line of investigation was pursued by Sarkar et al. , who introduced the notion of almost‐covering arrays. Definition Given a set S of v symbols, N rows, k columns and a strength

t \in N

, let

A \in S^{N \times k}

be an

N \times k

array.…”

Section: Introductionmentioning

confidence: 99%

ACAN false(t, k, v, ε false) \leq v^{t} ln (\frac{v^{t - 1}}{ε}),

with an improvement to

ACAN false(t, k, v, ε false) \leq v^{t} ln false(\frac{2 v^{t - 2}}{ε} false) + v

whenever v is a prime power.…”

Section: Introductionmentioning

confidence: 99%

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Small arrays of maximum coverage

Das

Mészáros

2018

J of Combinatorial Designs

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“…In [4] and [12], partial covering arrays are related to an Erdős-Ko-Rado property. Partial covering arrays are studied exhaustively and extensively in [19]. In the statistically relevant paper [15], only consecutive sets of t columns are considered.…”

Section: Introductionmentioning

confidence: 99%

Covering arrays for some equivalence classes of words

Cassels

Godbole

2019

J of Combinatorial Designs

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Covering arrays for words of length t over a d-letter alphabet are k n × arrays with entries from the alphabet so that for each choice of t columns, each of the d t t-letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case known as partitioning hash families, words are equivalent if they induce the same partition of a t element set. In the second case, words of the same weight are equivalent. In both cases, we produce logarithmic upper bounds on the minimum size k k n = ( ) of a covering array. Definitive results for t = 2, 3, 4, as well as general results, are provided. K E Y W O R D S covering arrays, lovász local lemma, partitioning hash families, set partitions, word weight How to cite this article: Cassels J, Godbole A. Covering arrays for some equivalence classes of words. J Combin Des. 2019;27:506-521. https://doi.

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“…In the general case the CAN(t, k, v) is unknown. Upper bounds of CANs have been studied in [8][9][10]. Research in the construction of covering arrays is mainly devoted to improve the current upper bounds.…”

Section: Introductionmentioning

confidence: 99%

Improved pairwise test suites for non‐prime‐power orders

Ávila-George

Torres-Jiménez²,

Izquierdo-Marquez³

2018

IET softw.

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Software testing has become a critical component of the modern software development process. Therefore, a lot of research has been done in this area in recent years, and as a result new algorithms, methodologies, and tools have been created. One of the most used testing strategies is pairwise testing; this technique ensures that all possible combinations of values between any two input parameters are covered by at least one test. In this work, a new algorithm called add factor and stochastic optimisation (AFSO) is used to build small pairwise test suites for non-prime-power orders. Starting from an orthogonal array of order v ∈ {10, 12, 14, 15, 18, 20, 21, 22, 24}, AFSO iteratively adds a factor and then reduces to zero the number of uncovered combinations by means of a simulated annealing algorithm. The results of the AFSO algorithm improved the size of 92 pairwise test suites with non-prime-power orders. One of these improved test suites is used in a real-word application to show the usefulness of the new results.

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Partial Covering Arrays: Algorithms and Asymptotics

Cited by 10 publications

References 26 publications

Small arrays of maximum coverage

Small arrays of maximum coverage

Covering arrays for some equivalence classes of words

Improved pairwise test suites for non‐prime‐power orders

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