Tower of covering arrays

Abstract

a b s t r a c tCovering arrays are combinatorial objects that have several practical applications, specially in the design of experiments for software and hardware testing. A covering array of strength t and order v is an N ×k array over Z v with the property that every N ×t subarray covers all members of Z t v at least once. In this work we explore the construction of a Tower of Covering Arrays (TCA) as a way to produce covering arrays that improve or match some current upper bounds. A TCA of height h is a succession of h + 1 covering arrays C 0 , C 1 , . . . , C h in which for i = 1, 2, . . . , h the covering array C i is one unit greater in the number of factors and the strength of the covering array C i−1 ; this way, if the covering array C 0 is of strength t and has k factors then the covering arrays C 1 , . . . , C h are of strength t + 1, . . . , t + h and have k + 1, . . . , k + h factors respectively. We note that the ratio between the number of rows of the last covering array C h in a TCA of height h and the number of rows of the best known covering array for the same values of t, k, and v as for C h is reduced as h grows. Therefore, we search for TCAs with the greatest height possible. The relevant results are the improvement of nineteen current upper bounds for v = 2 and t ∈ {7, 8, 9, 10, 11}, and the construction of twenty-one covering arrays that matched current upper bounds.

Methods to Construct Uniform Covering Arrays

“…Recursive algorithms are also fast algorithms that generally do not perform a computational search to construct the final CA; some exceptions are augmented annealing [43] that uses simulated annealing to construct some parts of the final CA, and tower of covering arrays [47] where the next CA in the tower is constructed by exploring a number of arrangements of the columns of the input CA. On the other hand, the classical recursive techniques of product, Roux-type constructions, and powering only takes the input ingredients and construct the output CA by following a fixed procedure to combine the inputs; in this methods the size of the final CA is known in advance, and so the quality of the output CA depends on the quality of the input ingredients.…”

“…These constructions are known as Roux-type Constructions. Other approaches take smaller CAs as inputs as in [46], [47]; or employ other combinatorial designs such as ordered designs [43] and difference matrices [48]. Another class of recursive methods construct a base array whose elements will be replaced by columns of another array [21].…”

Wrapper for Building Classification Models Using Covering Arrays

“…4, where in the first two rows contain the CA(2; 1, 10, 2), additionally the first seven rows contain the CA(7; 2, 10, 2), furthermore the first thirteen rows contain the CA(13; 3, 10, 2), and lastly, all its twenty four rows contain the CA(24; 4, 10, 2). An ICA differs from a classical CA, in that it is built on the CAs of lowest strength, whereas the classical CA is constructed independently of the previous ones [14]. This characteristic of the ICA is useful when performing the experiments to analyze the influence of the strength on the results of feature selection, specifically in cases where it is desired to increase the strength, wherein to test at a greater strength (coverage), implies adding some rows (or tests) extra, ultimately resulting in a highly efficient process, unlike classic CAs, wherein increasing the strength, involves evaluating an array with many new rows and therefore a greater number of tests for the wrapper.…”

“…Incremental Covering Arrays (ICA) are a variation of CAs. They work in the following manner: Within a matrix of strength t are sub-matrices of CAs with strengths less than t, and if a new matrix is desired with strength t + 1 or greater, all that is required is to add a certain number of rows, without having to completely regenerate the array [14].…”