t-Covering arrays: Upper bounds and poisson approximations

Anant P. Godbole, Daphne E. Skipper, Rachel A. Sunley

Research output: Contribution to journalArticle

52 Citations (Scopus)

Abstract

A k × n array with entries from the g-letter alphabet {0,1,..., q-1} is said to be t-covering if each k × t submatrix has (at least one set of) qt distinct rows. We use the Lovász local lemma to obtain a general upper bound on the minimal number K = K(n,t,q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let Kλ = Kλ(n,t,q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the qt possible 'words' of length t at least λ times. The Lovász lemma yields an upper bound on Kλ that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k × n array, the Stein-Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e. missing at least one word.

Original languageEnglish (US)
Pages (from-to)105-117
Number of pages13
JournalCombinatorics Probability and Computing
Volume5
Issue number2
DOIs
StatePublished - Jan 1 1996

Fingerprint

Covering Array
Poisson Approximation
Upper bound
Lemma
Covering
Stein-Chen Method
Denote
Distinct

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Computational Theory and Mathematics
  • Applied Mathematics

Cite this

t-Covering arrays : Upper bounds and poisson approximations. / Godbole, Anant P.; Skipper, Daphne E.; Sunley, Rachel A.

In: Combinatorics Probability and Computing, Vol. 5, No. 2, 01.01.1996, p. 105-117.

Research output: Contribution to journalArticle

Godbole, Anant P. ; Skipper, Daphne E. ; Sunley, Rachel A. / t-Covering arrays : Upper bounds and poisson approximations. In: Combinatorics Probability and Computing. 1996 ; Vol. 5, No. 2. pp. 105-117.
@article{a96be5f2c82d4143aa07c681ba9ebf1e,
title = "t-Covering arrays: Upper bounds and poisson approximations",
abstract = "A k × n array with entries from the g-letter alphabet {0,1,..., q-1} is said to be t-covering if each k × t submatrix has (at least one set of) qt distinct rows. We use the Lov{\'a}sz local lemma to obtain a general upper bound on the minimal number K = K(n,t,q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let Kλ = Kλ(n,t,q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the qt possible 'words' of length t at least λ times. The Lov{\'a}sz lemma yields an upper bound on Kλ that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k × n array, the Stein-Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e. missing at least one word.",
author = "Godbole, {Anant P.} and Skipper, {Daphne E.} and Sunley, {Rachel A.}",
year = "1996",
month = "1",
day = "1",
doi = "10.1017/S0963548300001905",
language = "English (US)",
volume = "5",
pages = "105--117",
journal = "Combinatorics Probability and Computing",
issn = "0963-5483",
publisher = "Cambridge University Press",
number = "2",

}

TY - JOUR

T1 - t-Covering arrays

T2 - Upper bounds and poisson approximations

AU - Godbole, Anant P.

AU - Skipper, Daphne E.

AU - Sunley, Rachel A.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - A k × n array with entries from the g-letter alphabet {0,1,..., q-1} is said to be t-covering if each k × t submatrix has (at least one set of) qt distinct rows. We use the Lovász local lemma to obtain a general upper bound on the minimal number K = K(n,t,q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let Kλ = Kλ(n,t,q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the qt possible 'words' of length t at least λ times. The Lovász lemma yields an upper bound on Kλ that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k × n array, the Stein-Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e. missing at least one word.

AB - A k × n array with entries from the g-letter alphabet {0,1,..., q-1} is said to be t-covering if each k × t submatrix has (at least one set of) qt distinct rows. We use the Lovász local lemma to obtain a general upper bound on the minimal number K = K(n,t,q) of rows for which a t-covering array exists; for t = 3 and q = 2, we are able to match the best-known such bound. Let Kλ = Kλ(n,t,q), (λ ≥ 2), denote the minimum number of rows that guarantees the existence of an array for which each set of t columns contains, amongst its rows, each of the qt possible 'words' of length t at least λ times. The Lovász lemma yields an upper bound on Kλ that reveals how substantially fewer rows are needed to accomplish subsequent t-coverings (beyond the first). Finally, given a random k × n array, the Stein-Chen method is employed to obtain a Poisson approximation for the number of sets of t columns that are deficient, i.e. missing at least one word.

UR - http://www.scopus.com/inward/record.url?scp=0030538878&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030538878&partnerID=8YFLogxK

U2 - 10.1017/S0963548300001905

DO - 10.1017/S0963548300001905

M3 - Article

AN - SCOPUS:0030538878

VL - 5

SP - 105

EP - 117

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 2

ER -