Constructing strength three covering arrays with augmented annealing

A covering arrayCA(N;t,k,v) is an N×k array such that every N×t sub-array contains all t-tuples from v symbols at least once, where t is the strength of the array. One application of these objects is to generate software test suites to cover all t-sets of component interactions. Methods for construction of covering arrays for software testing have focused on two main areas. The first is finding new algebraic and combinatorial constructions that produce smaller covering arrays. The second is refining computational search algorithms to find smaller covering arrays more quickly. In this paper, we examine some new cut-and-paste techniques for strength three covering arrays that combine recursive combinatorial constructions with computational search; when simulated annealing is the base method, this is augmented annealing. This method leverages the computational efficiency and optimality of size obtained through combinatorial constructions while benefiting from the generality of a heuristic search. We present a few examples of specific constructions and provide new bounds for some strength three covering arrays.     

Randomized post-optimization of covering arrays

The construction of covering arrays with the fewest rows remains a challenging problem. Most computational and recursive constructions result in extensive repetition of coverage. While some is necessary, some is not. By reducing the repeated coverage, metaheuristic search techniques typically outperform simpler computational methods, but they have been applied in a limited set of cases. Time constraints often prevent them from finding an array of competitive size. We examine a different approach. Having used a simple computation or construction to find a covering array, we employ a post-optimization technique that repeatedly adjusts the array in an attempt to reduce its number of rows. At every stage the array retains full coverage. We demonstrate its value on a collection of previously best known arrays by eliminating, in some cases, 10% of their rows. In the well-studied case of strength two with twenty factors having ten values each, post-optimization produces a covering array with only 162 rows, improving on a wide variety of computational and combinatorial methods. We identify certain important features of covering arrays for which post-optimization is successful.     

Covering and radius-covering arrays: Constructions and classification

The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays, explicit constructions from covering codes are developed. Lower bounds are improved upon using connections to orthogonal arrays, partition matrices, and covering codes, and in specific cases by computation. Consequently for some parameter sets the minimum size of a covering array is determined precisely. For some of these, a complete classification of all inequivalent covering arrays is determined, again using computational techniques. Existence tables for up to 10 columns, up to 8 symbols, and all possible strengths are presented to report the best current lower and upper bounds, and classifications of inequivalent arrays.     

A two-stage algorithm for combinatorial testing

Covering arrays are combinatorial structures which have applications in fields like software testing and hardware Trojan detection. In this paper we proposed a two-stage simulated annealing algorithm to construct covering arrays. The proposed algorithm is instanced in this paper through the construction of ternary covering arrays of strength three. We were able to get 579 new upper bounds. In order to show the generality of our proposal, we defined a new benchmark composed of 25 instances of MCAs taken from the literature, all instances were improved.     

On a Combinatorial Framework for Fault Characterization

Covering arrays have been widely used to detect the presence of faults in large software and hardware systems. Indeed, finding failures that result from faulty interactions requires that all interactions that may cause faults be covered by a test case. However, finding the actual faults requires more, because the failures resulting from two potential sets of faults must not be the same. The combinatorial requirements on test suites to enable a tester to locate the faults are developed, and set in the context of similar combinatorial search questions. Test suites known as locating and detecting arrays to locate faults both in principle and in practice generalize covering arrays, thereby addressing combinatorial fault characterization. In common with covering arrays, these locating and detecting arrays scale logarithmically in size with the number of factors, but unlike covering arrays they support complete characterization of the interactions that underlie faults.     

Covering arrays of higher strength from permutation vectors

A covering array CA(N;t,k,v) is an N × k array such that every N × t sub‐array contains all t‐tuples from v symbols at least once, where t is the strength of the array. Covering arrays are used to generate software test suites to cover all t‐sets of component interactions. We introduce a combinatorial technique for their construction, focussing on covering arrays of strength 3 and 4. With a computer search, covering arrays with improved parameters have been found. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 202–213, 2006     

Optimal and near-optimal mixed covering arrays by column expansion

This paper describes constructions for strength-2 mixed covering arrays developed from index-1 orthogonal arrays, ordered designs and covering arrays. The constructed arrays have optimal or near-optimal sizes. Conditions for achieving optimal size are described. An optimization among the different ingredient arrays to maximize the number of factors of each alphabet size is also presented.     

Generalized orthogonal arrays: Constructions and related graphs

Generalized orthogonal arrays were first defined to provide a combinatorial characterization of (t, m, s)-nets. In this article we describe three new constructions for generalized orthogonal arrays. Two of these constructions are straightforward generalizations of constructions for orthogonal arrays and one employs new techniques. We construct families of generalized orthogonal arrays using orthogonal arrays and provide net parameters obtained from our constructions. In addition, we define a set of graphs associated with a generalized orthogonal array which provide information about the structure of the array. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 31–39, 1999     

On the construction of authentication and secrecy codes

We present several recursive constructions for authentication and secrecy codes using t-designs. These constructions are based on combinatorial structures called authentication perpendicular arrays, introduced by Stinson. As a by-product we obtain a method for constructing sets of permutations which are uniform and t-homogeneous for arbitrarily large t. A table of parameters for codes whose existence is known is included.     

Frequency permutation arrays

Motivated by recent interest in permutation arrays, we introduce and investigate the more general concept of frequency permutation arrays (FPAs). An FPA of length n = mλ and distance d is a set T of multipermutations on a multiset of m symbols, each repeated with frequency λ, such that the Hamming distance between any distinct x,y ∈ T is at least d. Such arrays have potential applications in powerline communication. In this article, we establish basic properties of FPAs, and provide direct constructions for FPAs using a range of combinatorial objects, including polynomials over finite fields, combinatorial designs, and codes. We also provide recursive constructions, and give bounds for the maximum size of such arrays. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 463–478, 2006     

On linear programming bounds for nets

It is well known that there are close connections between low-discrepancy point sets and sequences on the one hand, and certain combinatorial and algebraic structures on the other hand. E. g., Niederreiter [1] showed the equivalence between (t, t + 2, s)-nets and orthogonal arrays of strength 2. Some years later this was generalized and made precise in the work of Lawrence [2] as well as Mullen and Schmid [3] by introducing ordered orthogonal arrays. This large class of combinatorial structures yields both new constructions and bounds for the existence of nets and sequences. The linear programming bound for ordered orthogonal arrays was first derived by Martin and Stinson [4]. As in the case of error-correcting codes and orthogonal arrays, it yields a very strong bound for ordered orthogonal arrays, and consequently for nets and sequences. Solving linear programming problems in exact arithmetics is very time-consuming. Using different approaches to reduce the computing time, we have calculated the linear programming bound for a wide parameter range. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New constructions for perfect hash families and related structures using combinatorial designs and codes

In this paper, we consider explicit constructions of perfect hash families using combinatorial methods. We provide several direct constructions from combinatorial structures related to orthogonal arrays. We also simplify and generalize a recursive construction due to Atici, Magliversas, Stinson and Wei [3]. Using similar methods, we also obtain efficient constructions for separating hash families which result in improved existence results for structures such as separating systems, key distribution patterns, group testing algorithms, cover‐free families and secure frameproof codes. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:189–200, 2000     

Covering Arrays of Strength Three

A covering array of size N, degree k, order v and strength t is a k × N array with entries from a set of v symbols such that in any t × N subarray every t × 1 column occurs at least once. Covering arrays have been studied for their applications to drug screening and software testing. We present explicit constructions and give constructive upper bounds for the size of a covering array of strength three.     

Asymptotic and constructive methods for covering perfect hash families and covering arrays

Covering perfect hash families represent certain covering arrays compactly. Applying two probabilistic methods to covering perfect hash families improves upon the asymptotic upper bound for the minimum number of rows in a covering array with v symbols, k columns, and strength t. One bound can be realized by a randomized polynomial time construction algorithm using column resampling, while the other can be met by a deterministic polynomial time conditional expectation algorithm. Computational results are developed for both techniques. Further, a random extension algorithm further improves on the best known sizes for covering arrays in practice. An extensive set of computations with column resampling and random extension yields explicit constructions when \(k \le 75\) for strength seven, \(k \le 200\) for strength six, \(k \le 600\) for strength five, and \(k \le 2500\) for strength four. When \(v > 3\), almost all known explicit constructions are improved upon. For strength \(t=3\), restrictions on the covering perfect hash family ensure the presence of redundant rows in the covering array, which can be removed. Using restrictions and random extension, computations for \(t=3\) and \(k \le 10{,}000\) again improve upon known explicit constructions in the majority of cases. Computations for strengths three and four demonstrate that a conditional expectation algorithm can produce further improvements at the expense of a larger time and storage investment.     

Cover-Free Families and Topology-Transparent Scheduling for MANETs

We examine the combinatorial requirements of topology-transparent transmission schedules in a mobile ad hoc network (MANET). Specifically, if each of the N nodes has at most D active neighbors, we require the schedule to guarantee a collision-free transmission to each neighbor. This requirement is met by a cover-free family. We show that existing constructions for topology-transparent schedules correspond to an orthogonal array. Moreover, we show that Steiner systems support the largest number of nodes for a given schedule length. Both of these combinatorial objects are special cases of cover-free families. Analytically and numerically, we examine slot guarantees, expected throughput, and normalized expected throughput for systems of small strength, exploring the sensitivity of the response to D. Expected throughput provides a better performance metric than the minimum throughput results obtained earlier. The impact of a more realistic model of acknowledgments is also examined. The extension of the schedule to multiple frames returns us to the orthogonal arrays. The very density of Steiner systems that afforded an improvement over orthogonal arrays in one frame impedes the best extension to more frames.     

Codismantlability and projective dimension of the Stanley–Reisner ring of special hypergraphs

In this paper, we generalize the concept of codismantlable graphs to hypergraphs and show that some special vertex decomposable hypergraphs are codismantlable. Then we generalize the concept of bouquet in graphs to hypergraphs to extend some combinatorial invariants of graphs about disjointness of a set of bouquets. We use these invariants to characterize the projective dimension of Stanley–Reisner ring of special hypergraphs in some sense.     

Quasi-random hypergraphs

We introduce an equivalence class of varied properties for hypergraphs. Any hypergraph possessing any one of these properties must of necessity possess them all. Since almost all random hypergraphs share these properties, we term these properties quasi-random. With these results, it becomes quite easy to show that many natural explicit constructions result in hypergraphs which imitate random hypergraphs in a variety of ways.     

Covering arrays from m-sequences and character sums

A covering array of strength t on v symbols is an array with the property that, for every t-set of column vectors, every one of the \(v^t\) possible t-tuples of symbols appears as a row at least once in the sub-array defined by these column vectors. Arrays constructed using m-sequences over a finite field possess many combinatorial properties and have been used to construct various combinatorial objects; see the recent survey Moura et al. (Des Codes Cryptogr 78(1):197–219, 2016). In this paper we construct covering arrays whose elements are the remainder of the division by some integer of the discrete logarithm applied to selected m-sequence elements. Inspired by the work of Colbourn (Des Codes Cryptogr 55(2–3):201–219, 2010), we prove our results by connecting the covering array property to a character sum, and we evaluate this sum by taking advantage of the balanced way in which the m-sequence elements are distributed. Our results include new infinite families of covering arrays of arbitrary strength.     

A Survey on Hypergraph Products

A surprising diversity of different products of hypergraphs have been discussed in the literature. Most of the hypergraph products can be viewed as generalizations of one of the four standard graph products. The most widely studied variant, the so-called square product, does not have this property, however. Here we survey the literature on hypergraph products with an emphasis on comparing the alternative generalizations of graph products and the relationships among them. In this context the so-called 2-sections and L2-sections are considered. These constructions are closely linked to related colored graph structures that seem to be a useful tool for the prime factor decompositions w.r.t. specific hypergraph products. We summarize the current knowledge on the propagation of hypergraph invariants under the different hypergraph multiplications. While the overwhelming majority of the material concerns finite (undirected) hypergraphs, the survey also covers a summary of the few results on products of infinite and directed hypergraphs.     

Theory of perpendicular arrays

The cryptographical theory of unconditional secrecy and authentication is based on design-like structures called perpendicular arrays in the combinatorial literature. In order to meet the cryptographical requirements some additional and rather natural homogeneity conditions have to be satisfied. We develop a theory of these structures. Topics include bounds on the size, recursive and direct constructions using designs and permutation groups, as well as a link to Room cubes. © 1994 John Wiley & Sons, Inc.