Variable strength covering arrays

Abstract

Recently, covering arrays have been the subject of considerable research attention as they hold both theoretical interest and practical importance due to their applications to testing. In this thesis, we perform the first comprehensive study of a generalization of covering arrays called variable strength covering arrays, where we dictate the interactions to be covered in the array by modeling them as facets of an abstract simplicial complex.We outline the necessary background in the theory of hypergraphs, combinatorial testing, and design theory that is relevant to the study of variable strength covering arrays. We then approach questions that arise in variable strength covering arrays in a number of ways. We demonstrate their connections to hypergraph homomorphisms, and explore the properties of a particular family of abstract simplicial complexes, the qualitative independence hypergraphs. These hypergraphs are tightly linked to variable strength covering arrays, and we determine and identify several of their important properties and subhypergraphs.We give a detailed study of constructions for variable strength covering arrays, and provide several operations and divide-and-conquer techniques that can be used in building them. In addition, we give a construction using linear feedback shift registers from primitive polynomials of degree 3 over arbitrary finite fields to find variable strength covering arrays, which we extend to strength-3 covering arrays whose sizes are smaller than many of the best known sizes of covering arrays.We then give an algorithm for creating variable strength covering arrays over arbitrary abstract simplicial complexes, which builds the arrays one row at a time, using a density concept to guarantee that the size of the resultant array is asymptotic in the logarithm of the number of facets in the abstact simplicial complex. This algorithm is of immediate practical importance, as it can be used to create test suites for combinatorial testing.Finally, we use the Lovász Local Lemma to nonconstructively determine upper bounds on the sizes of arrays for a number of different families of hypergraphs. We lay out a framework that can be used for many hypergraphs, and then discuss possible strategies that can be taken in asymmetric problems.ii Acknowledgements This thesis certainly could not have been completed without the help of many people.Firstly and most importantly, I would like to extend my immeasurable thanks to both of my supervisors, Lucia Moura and Brett Stevens. As a lowly undergraduate student many years ago, I had the extreme fortune of being chosen to serve as a teaching assistant to work with Lucia Moura. I could never have predicted that that was to be the first step on a long and rewarding journey into academia. Through all the opportunities she gave me, I caught her infectious enthusiasm for research and had my eyes opened to the wonderful fields of design theory and covering arrays. Without her endless patience and compassion with my quirkiness, this thesis would never...

Upper bounds on the sizes of variable strength covering arrays using the Lovász local lemma

“…, which does grow with k. Thus we have an example of a family of hypergraphs where N prob is substantially better than N dens . The homomorphism construction (see [19,Chapter 3]…”

“…, where k ≤ 2t−1 and the ln(2t−1) term in Equation 3.1 in place of the usual ln k term suggests that N prob "recognizes" this homomorphism while N dens does not. However, experiments show that running VarDens algorithm for H k,t c does seem to generate arrays where the array size is independent of k [19].…”

“…The bound N dens also grows with k and, in contrast to the previous class of hypergraphs, experiments running VarDens on random triangulation hypergraphs suggest that the size of the arrays produced indeed increases as some function of k, see Table 2. For more details about generating the random triangulation hypergraphs see [19].…”

“…We begin defining VCAs and indicate how covering arrays are a special case. For more information on VCAs, see [19,22]. Definition 1.1.…”

Covering arrays for some equivalence classes of words

“…In the statistically relevant paper [15], only consecutive sets of t columns are considered. The paper [17] is just one of many in which variable strength covering arrays (where the interactions to be covered in the array modeled as facets of an abstract simplicial complex), covering arrays on graphs, and mixed covering arrays (different alphabet sets in different columns) are studied. See also the contributed talks in the sessions on Generalizations of Covering Arrays at https://canadam.math.ca/2011/program/schedule_con-tributed_mini.…”