scholarly journals Binary Covering Arrays on Tournaments

10.37236/6149 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Elizabeth Maltais ◽  
Lucia Moura ◽  
Mike Newman
Keyword(s):  
Covering Array ◽  
Covering Arrays ◽  
Column Graph

We introduce graph-dependent covering arrays which generalize covering arrays on graphs, introduced by Meagher and Stevens (2005), and graph-dependent partition systems, studied by Gargano, Körner, and Vaccaro (1994). A covering array $\hbox{CA}(n; 2, G, H)$ (of strength 2) on column graph $G$ and alphabet graph $H$ is an $n\times |V(G)|$ array with symbols $V(H)$ such that for every arc $ij \in E(G)$ and for every arc $ab\in E(H)$, there exists a  row $\vec{r} = (r_{1},\dots, r_{|V(G)|})$   such that $(r_{i}, r_{j}) = (a,b)$.  We prove bounds on $n$ when $G$ is a tournament graph and $E(H)$ consists of the edge $(0,1)$, which corresponds to a directed version of Sperner's 1928 theorem. For two infinite families of column graphs, transitive and so-called circular tournaments, we give constructions of covering arrays which are optimal infinitely often.

New optimal covering arrays using an orderly algorithm

2018 ◽  
Vol 10 (01) ◽  
pp. 1850011 ◽  
Author(s):  
Idelfonso Izquierdo-Marquez ◽  
Jose Torres-Jimenez
Keyword(s):  
Small Subset ◽  
Computational Results ◽  
Covering Array ◽  
Covering Arrays ◽  
Computational Search ◽  
Minimum Number ◽  
Orderly Algorithm

A covering array [Formula: see text] is an [Formula: see text] array such that every [Formula: see text] subarray covers at least once each [Formula: see text]-tuple from [Formula: see text] symbols. For given [Formula: see text], [Formula: see text], and [Formula: see text], the minimum number of rows for which exists a CA is denoted by [Formula: see text] (CAN stands for Covering Array Number) and the corresponding CA is optimal. Optimal covering arrays have been determined algebraically for a small subset of cases; but another alternative to find CANs is the use of computational search. The present work introduces a new orderly algorithm to construct non-isomorphic covering arrays; this algorithm is an improvement of a previously reported algorithm for the same purpose. The construction of non-isomorphic covering arrays is used to prove the nonexistence of certain covering arrays whose nonexistence implies the optimality of other covering arrays. From the computational results obtained, the following CANs were established: [Formula: see text] for [Formula: see text], [Formula: see text], and [Formula: see text]. In addition, the new result [Formula: see text], and the already known existence of [Formula: see text], imply [Formula: see text].

Research Notes: Binary Test-Suites Using Covering Arrays

2018 ◽  
Vol 28 (09) ◽  
pp. 1321-1337 ◽  
Author(s):  
Jose Torres-Jimenez ◽  
Himer Avila-George ◽  
Ezra Federico Parra-González
Keyword(s):  
Software Testing ◽  
Test Generation ◽  
Software Systems ◽  
Combinatorial Testing ◽  
Covering Array ◽  
Covering Arrays ◽  
Almost All ◽  
Generation Problem ◽  
Test Suites

Software testing is an essential activity to ensure the quality of software systems. Combinatorial testing is a method that facilitates the software testing process; it is based on an empirical evidence where almost all faults in a software component are due to the interaction of very few parameters. The test generation problem for combinatorial testing can be represented as the construction of a matrix that has certain properties; typically this matrix is a covering array. Covering arrays have a small number of tests, in comparison with an exhaustive approach, and provide a level of interaction coverage among the parameters involved. This paper presents a repository that contains binary covering arrays involving many levels of interaction. Also, it discusses the importance of covering array repositories in the construction of better covering arrays. In most of the cases, the size of the covering arrays included in the repository reported here are the best upper bounds known, moreover, the files containing the matrices of these covering arrays are available to be downloaded. The final purpose of our Binary Covering Arrays Repository (BCAR) is to provide software testing practitioners the best-known binary test-suites.

Mixed covering arrays on graphs of small treewidth

2021 ◽  
pp. 2150085
Author(s):  
Soumen Maity ◽  
Charles J. Colbourn
Keyword(s):  
Weighted Graph ◽  
Minimum Size ◽  
Index Formula ◽  
Covering Array ◽  
Covering Arrays ◽  
Combinatorial Objects ◽  
The Family ◽  
Positive Integers ◽  
Basic Graph ◽  
Test Suites

Covering arrays are combinatorial objects that have been successfully applied in design of test suites for testing systems such as software, hardware, and networks where failures can be caused by the interaction between their parameters. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text]. Two vectors [Formula: see text] and [Formula: see text] are qualitatively independent if for any ordered pair [Formula: see text], there exists an index [Formula: see text] such that [Formula: see text]. Let [Formula: see text] be a graph with [Formula: see text] vertices [Formula: see text] with respective vertex weights [Formula: see text]. A mixed covering array on[Formula: see text] , denoted by [Formula: see text], is a [Formula: see text] array such that row [Formula: see text] corresponds to vertex [Formula: see text], entries in row [Formula: see text] are from [Formula: see text]; and if [Formula: see text] is an edge in [Formula: see text], then the rows [Formula: see text] are qualitatively independent. The parameter [Formula: see text] is the size of the array. Given a weighted graph [Formula: see text], a mixed covering array on [Formula: see text] with minimum size is optimal. In this paper, we introduce some basic graph operations to provide constructions for optimal mixed covering arrays on the family of graphs with treewidth at most three.

t-Covering Arrays: Upper Bounds and Poisson Approximations

1996 ◽  
Vol 5 (2) ◽  
pp. 105-117 ◽  
Author(s):  
Anant P. Godbole ◽  
Daphne E. Skipper ◽  
Rachel A. Sunley
Keyword(s):  
Upper Bound ◽  
Poisson Approximation ◽  
Upper Bounds ◽  
Covering Array ◽  
Covering Arrays ◽  
Lovász Local Lemma ◽  
Letter Alphabet ◽  
Local Lemma ◽  
Minimum Number ◽  
Poisson Approximations

A k×n array with entries from the q-letter alphabet {0, 1, …, q − 1} is said to be t-covering if each k × t submatrix has (at least one set of) qt distinct rows. We use the Lovász local lemma to obtain a general upper bound on the minimal number K = K(n, t, q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let Kλ = Kλ(n, t, q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the qt possible ‘words’ of length t at least λ times. The Lovász lemma yields an upper bound on Kλ that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k × n array, the Stein–Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e. missing at least one word.

Construction of non-isomorphic covering arrays

2016 ◽  
Vol 08 (02) ◽  
pp. 1650033 ◽  
Author(s):  
Jose Torres-Jimenez ◽  
Idelfonso Izquierdo-Marquez
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Hard Problem ◽  
Covering Array ◽  
Covering Arrays ◽  
Algebraic Analysis ◽  
Column Permutation ◽  
Computational Search ◽  
Strength Formula

A covering array CA([Formula: see text]) of strength [Formula: see text] and order [Formula: see text] is an [Formula: see text] array over [Formula: see text] with the property that every [Formula: see text] subarray covers all members of [Formula: see text] at least once. When the value of [Formula: see text] is the minimum possible it is named as the covering array number (CAN) i.e. [Formula: see text]. Two CAs are isomorphic if one of them can be derived from the other by a combination of a row permutation, a column permutation, and a symbol permutation in a subset of columns. Isomorphic CAs have equivalent coverage properties, and can be considered as the same CA; the truly distinct CAs are those which are non-isomorphic among them. An interesting and hard problem is to construct all the non-isomorphic CAs that exist for a particular combination of the parameters [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We constructed the non-isomorphic CAs for 70 combinations of values of the parameters [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the results allow us to determine CAN(3,13,2) =16, CAN(3,14,2) =16, CAN(3,15,2) =17, CAN(3,16,2) =17, and CAN(2,10,3) =14. The exact lower bound for these covering arrays numbers had not been determined before either by computational search or by algebraic analysis.

scholarly journals DEO: A Dynamic Event Order Strategy for t-way Sequence Covering Array Test Data Generation

2020 ◽  
Vol 17 (2) ◽  
pp. 0575
Author(s):  
Mohammed Issam Younis
Keyword(s):  
Research Area ◽  
Test Case Generation ◽  
Test Case ◽  
Generation Process ◽  
Data Generation ◽  
Covering Array ◽  
Covering Arrays ◽  
Dynamic Event ◽  
Active Research ◽  
Event Order

Sequence covering array (SCA) generation is an active research area in recent years. Unlike the sequence-less covering arrays (CA), the order of sequence varies in the test case generation process. This paper reviews the state-of-the-art of the SCA strategies, earlier works reported that finding a minimal size of a test suite is considered as an NP-Hard problem. In addition, most of the existing strategies for SCA generation have a high order of complexity due to the generation of all combinatorial interactions by adopting one-test-at-a-time fashion. Reducing the complexity by adopting one-parameter- at-a-time for SCA generation is a challenging process. In addition, this reduction facilitates the supporting for a higher strength of coverage. Motivated by such challenge, this paper proposes a novel SCA strategy called Dynamic Event Order (DEO), in which the test case generation is done using one-parameter-at-a-time fashion. The details of the DEO are presented with a step-by-step example to demonstrate the behavior and show the correctness of the proposed strategy. In addition, this paper makes a comparison with existing computational strategies. The practical results demonstrate that the proposed DEO strategy outperforms the existing strategies in term of minimal test size in most cases. Moreover, the significance of the DEO increases as the number of sequences increases and/ or the strength of coverage increases. Furthermore, the proposed DEO strategy succeeds to generate SCAs up to t=7. Finally, the DEO strategy succeeds to find new upper bounds for SCA. In fact, the proposed strategy can act as a research vehicle for variants future implementation.

scholarly journals New Bounds for Ternary Covering Arrays Using a Parallel Simulated Annealing

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Himer Avila-George ◽  
Jose Torres-Jimenez ◽  
Vicente Hernández
Keyword(s):  
Simulated Annealing ◽  
Direct Methods ◽  
Combinatorial Structure ◽  
Covering Array ◽  
Covering Arrays ◽  
Construction Problem ◽  
Cooperative Approach ◽  
Benchmark Instances ◽  
Best Execution ◽  
Greedy Methods

A covering array (CA) is a combinatorial structure specified as a matrix ofNrows andkcolumns over an alphabet onvsymbols such that for each set oftcolumns everyt-tuple of symbols is covered at least once. Given the values oft,k, andv, the optimal covering array construction problem (CAC) consists in constructing a CA (N;t,k,v) with the minimum possible value ofN. There are several reported methods to attend the CAC problem, among them are direct methods, recursive methods, greedy methods, and metaheuristics methods. In this paper, There are three parallel approaches for simulated annealing: the independent, semi-independent, and cooperative searches are applied to the CAC problem. The empirical evidence supported by statistical analysis indicates that cooperative approach offers the best execution times and the same bounds as the independent and semi-independent approaches. Extensive experimentation was carried out, using 182 well-known benchmark instances of ternary covering arrays, for assessing its performance with respect to the best-known bounds reported previously. The results show that cooperative approach attains 134 new bounds and equals the solutions for other 29 instances.

scholarly journals A Survey of Binary Covering Arrays

10.37236/571 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jim Lawrence ◽  
Raghu N. Kacker ◽  
Yu Lei ◽  
D. Richard Kuhn ◽  
Michael Forbes
Keyword(s):  
Software Testing ◽  
Covering Array ◽  
Covering Arrays ◽  
Search Techniques ◽  
Explicit Constructions ◽  
The Matrix

Binary covering arrays of strength $t$ are 0–1 matrices having the property that for each $t$ columns and each of the possible $2^t$ sequences of $t$ 0's and 1's, there exists a row having that sequence in that set of $t$ columns. Covering arrays are an important tool in certain applications, for example, in software testing. In these applications, the number of columns of the matrix is dictated by the application, and it is desirable to have a covering array with a small number of rows. Here we survey some of what is known about the existence of binary covering arrays and methods of producing them, including both explicit constructions and search techniques.

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