# Partial Covering Arrays: Algorithms and Asymptotics

Partial Covering Arrays: Algorithms and Asymptotics A covering array CA(N;t, k, v) is an N × k array with entries in {1,2,…, v}, for which every N × t subarray contains each t-tuple of {1,2,…, v} t among its rows. Covering arrays find application in interaction testing, including software and hardware testing, advanced materials development, and biological systems. A central question is to determine or bound CAN(t, k, v), the minimum number N of rows of a CA(N;t, k, v). The well known bound CAN(t, k, v) = O((t − 1)v t log k) is not too far from being asymptotically optimal. Sensible relaxations of the covering requirement arise when (1) the set {1,2,…, v} t need only be contained among the rows of at least ( 1 − 𝜖 ) k t $(1-\epsilon )\binom {k}{t}$ of the N × t subarrays and (2) the rows of every N × t subarray need only contain a (large) subset of {1,2,…, v} t . In this paper, using probabilistic methods, significant improvements on the covering array upper bound are established for both relaxations, and for the conjunction of the two. In each case, a randomized algorithm constructs such arrays in expected polynomial time. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Theory of Computing Systems Springer Journals

# Partial Covering Arrays: Algorithms and Asymptotics

, Volume 62 (6) – May 27, 2017
20 pages

/lp/springer_journal/partial-covering-arrays-algorithms-and-asymptotics-Il0WmDpX6q
Publisher
Springer Journals
Subject
Computer Science; Theory of Computation
ISSN
1432-4350
eISSN
1433-0490
D.O.I.
10.1007/s00224-017-9782-9
Publisher site
See Article on Publisher Site

### Abstract

A covering array CA(N;t, k, v) is an N × k array with entries in {1,2,…, v}, for which every N × t subarray contains each t-tuple of {1,2,…, v} t among its rows. Covering arrays find application in interaction testing, including software and hardware testing, advanced materials development, and biological systems. A central question is to determine or bound CAN(t, k, v), the minimum number N of rows of a CA(N;t, k, v). The well known bound CAN(t, k, v) = O((t − 1)v t log k) is not too far from being asymptotically optimal. Sensible relaxations of the covering requirement arise when (1) the set {1,2,…, v} t need only be contained among the rows of at least ( 1 − 𝜖 ) k t $(1-\epsilon )\binom {k}{t}$ of the N × t subarrays and (2) the rows of every N × t subarray need only contain a (large) subset of {1,2,…, v} t . In this paper, using probabilistic methods, significant improvements on the covering array upper bound are established for both relaxations, and for the conjunction of the two. In each case, a randomized algorithm constructs such arrays in expected polynomial time.

### Journal

Theory of Computing SystemsSpringer Journals

Published: May 27, 2017

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