Sequence Covering Arrays

Abstract

Sequential processes can encounter faults as a result of improper ordering of subsets of the events. In order to reveal faults caused by the relative ordering of t or fewer of v events, for some fixed t, a test suite must provide tests so that every ordering of every set of t or fewer events is exercised. Such a test suite is equivalent to a sequence covering array, a set of permutations on v events for which every subsequence of t or fewer events arises in at least one of the permutations. Equivalently it is a (different) set of permutations, a completely t-scrambling set of permutations, in which the images of every set of t chosen events include each of the t! possible "patterns." In event sequence testing, minimizing the number of permutations used is the principal objective. By developing a connection with covering arrays, lower bounds on this minimum in terms of the minimum number of rows in covering arrays are obtained. An existing bound on the largest v for which the minimum can equal t! is improved. A conditional expectation algorithm is developed to generate sequence covering arrays whose number of permutations never exceeds a specified logarithmic function of v when t is fixed, and this method is shown to operate in polynomial time. A recursive product construction is established when t = 3 to construct sequence covering arrays on vw events from ones on v and w events. Finally computational results are given for t ∈ {3, 4, 5} to demonstrate the utility of the conditional expectation algorithm and the product construction.

Perfect sequence covering arrays

“…We see that (2) provides logarithmic upper and lower bounds for f (n, k), so the order of magnitude of f (n, k) for fixed k, is known. A natural design-theoretic question that arises when studying sequence covering arrays is that of perfectness.…”

“…Sequence covering arrays have been extensively studied, see [2,8] and the references therein which also provide some important applications of sequence covering arrays to the area of event sequence testing. Observe first that f (n, 2) = 2 as can be seen by taking any permutation and its reverse.…”

“…The conjecture is known to hold for k = 3, 5, 6 and it is also known that g(7, 4) > 1 [10]. For general k ≥ 3, a result of Chee et al [2] shows that g(2k, k) > 1.…”

Covering arrays for some equivalence classes of words

“…For example, we have the ucycle abcbccccddcdeec of the set partitions of [4], where, as in this paper, the substring dcde encodes the partition | | 13 2 4. Note that the alphabet used was, in this case, of size 5, though an alphabet of (minimum) size 5 is shown to suffice to encode the partitions of [5] as…”

“…There have been several variations on the basic definition of covering arrays. In [5], and [9], the authors considered the notion of covering arrays of permutations. In [4] and [12], partial covering arrays are related to an Erdős-Ko-Rado property.…”