Asymptotic Size of Covering Arrays: An Application of Entropy Compression

Francetić, Nevena; Stevens, Brett

doi:10.1002/jcd.21553

“…Developing A over the cyclic group we obtain a PCA(N ; t, k, v, m) with Figure 1 compares (12) and (6). In Figure 1a we plot the size of the partial m-covering array as obtained by (12) and (6) for v t − 6v + 1 ≤ m ≤ v t and (12). Similarly, Figure 1b shows that for m = v t − v = 4092, (6) consistently outperforms (12) for all values of k when t = 6, v = 4.…”

Section: Almost Partial Covering Arraysmentioning

confidence: 92%

Partial Covering Arrays: Algorithms and Asymptotics

Sarkar

¹

,

Colbourn

²

,

Bonis

³

et al. 2016

Lecture Notes in Computer Science

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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 ttuple 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 − ) 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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“…In this paper, we first sought a small, structured array of

v^{t}

rows that covered columns as efficiently as possible, and then took random copies of this array to obtain a covering array. In the earlier work of Francetić and Stevens and Sarkar and Colbourn , the method employed was to first build a random array to ensure each t ‐set of columns hit the orbits of some appropriate group action, and then use the group to algebraically extend this random array into a covering array. In , the authors obtained the best‐known bounds by replacing the algebra with linear algebra; they first randomly constructed a covering perfect hash family —an array of vectors in

F_{v}^{t}

, such that for every t ‐set of columns, there is a row where the corresponding vectors are linearly independent.…”

Section: Covering and Almost‐covering Arraysmentioning

confidence: 99%

Small arrays of maximum coverage

Das

¹

,

Mészáros

²

2018

J of Combinatorial Designs

3

0

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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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“…A slight improvement on (2) has recently been proved [12,28]. An (essentially) equivalent but more convenient form of (2) is:…”

Section: Background and Motivationmentioning

confidence: 99%

“…In Figure 1a we plot the size of the partial m-covering array as obtained by (12) and (6) (12) and (6). (6) outperforms (12) for all values of k.…”

Section: [K]mentioning

confidence: 99%

Partial Covering Arrays: Algorithms and Asymptotics

Sarkar

¹

,

Colbourn

²

,

Bonis

³

et al. 2017

View full text Add to dashboard Cite

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 ttuple 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 − ) 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.

show abstract

Asymptotic Size of Covering Arrays: An Application of Entropy Compression

Cited by 18 publications

References 21 publications

Partial Covering Arrays: Algorithms and Asymptotics

Partial Covering Arrays: Algorithms and Asymptotics

Small arrays of maximum coverage

Partial Covering Arrays: Algorithms and Asymptotics

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