Asymptotic and constructive methods for covering perfect hash families and covering arrays

Charles Colbourn, Erin Lanus, Kaushik Sarkar

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

Covering perfect hash families represent certain covering arrays compactly. Applying two probabilistic methods to covering perfect hash families improves upon the asymptotic upper bound for the minimum number of rows in a covering array with v symbols, k columns, and strength t. One bound can be realized by a randomized polynomial time construction algorithm using column resampling, while the other can be met by a deterministic polynomial time conditional expectation algorithm. Computational results are developed for both techniques. Further, a random extension algorithm further improves on the best known sizes for covering arrays in practice. An extensive set of computations with column resampling and random extension yields explicit constructions when (Formula presented.) for strength seven, (Formula presented.) for strength six, (Formula presented.) for strength five, and (Formula presented.) for strength four. When (Formula presented.), almost all known explicit constructions are improved upon. For strength (Formula presented.), restrictions on the covering perfect hash family ensure the presence of redundant rows in the covering array, which can be removed. Using restrictions and random extension, computations for (Formula presented.) and (Formula presented.) again improve upon known explicit constructions in the majority of cases. Computations for strengths three and four demonstrate that a conditional expectation algorithm can produce further improvements at the expense of a larger time and storage investment.

Original languageEnglish (US)
Pages (from-to)1-31
Number of pages31
JournalDesigns, Codes, and Cryptography
DOIs
StateAccepted/In press - May 26 2017

Fingerprint

Perfect Hash Family
Covering Array
Covering
Polynomials
Conditional Expectation
Resampling
Polynomial time
Restriction
Probabilistic Methods
Computational Results
Upper bound

Keywords

  • Asymptotic bound
  • Conditional expectation algorithm
  • Covering array
  • Covering perfect hash family

ASJC Scopus subject areas

  • Computer Science Applications
  • Applied Mathematics

Cite this

Asymptotic and constructive methods for covering perfect hash families and covering arrays. / Colbourn, Charles; Lanus, Erin; Sarkar, Kaushik.

In: Designs, Codes, and Cryptography, 26.05.2017, p. 1-31.

Research output: Contribution to journalArticle

@article{a57b119c1fd842ce81c6b09a9d3c590f,
title = "Asymptotic and constructive methods for covering perfect hash families and covering arrays",
abstract = "Covering perfect hash families represent certain covering arrays compactly. Applying two probabilistic methods to covering perfect hash families improves upon the asymptotic upper bound for the minimum number of rows in a covering array with v symbols, k columns, and strength t. One bound can be realized by a randomized polynomial time construction algorithm using column resampling, while the other can be met by a deterministic polynomial time conditional expectation algorithm. Computational results are developed for both techniques. Further, a random extension algorithm further improves on the best known sizes for covering arrays in practice. An extensive set of computations with column resampling and random extension yields explicit constructions when (Formula presented.) for strength seven, (Formula presented.) for strength six, (Formula presented.) for strength five, and (Formula presented.) for strength four. When (Formula presented.), almost all known explicit constructions are improved upon. For strength (Formula presented.), restrictions on the covering perfect hash family ensure the presence of redundant rows in the covering array, which can be removed. Using restrictions and random extension, computations for (Formula presented.) and (Formula presented.) again improve upon known explicit constructions in the majority of cases. Computations for strengths three and four demonstrate that a conditional expectation algorithm can produce further improvements at the expense of a larger time and storage investment.",
keywords = "Asymptotic bound, Conditional expectation algorithm, Covering array, Covering perfect hash family",
author = "Charles Colbourn and Erin Lanus and Kaushik Sarkar",
year = "2017",
month = "5",
day = "26",
doi = "10.1007/s10623-017-0369-x",
language = "English (US)",
pages = "1--31",
journal = "Designs, Codes, and Cryptography",
issn = "0925-1022",
publisher = "Springer Netherlands",

}

TY - JOUR

T1 - Asymptotic and constructive methods for covering perfect hash families and covering arrays

AU - Colbourn, Charles

AU - Lanus, Erin

AU - Sarkar, Kaushik

PY - 2017/5/26

Y1 - 2017/5/26

N2 - Covering perfect hash families represent certain covering arrays compactly. Applying two probabilistic methods to covering perfect hash families improves upon the asymptotic upper bound for the minimum number of rows in a covering array with v symbols, k columns, and strength t. One bound can be realized by a randomized polynomial time construction algorithm using column resampling, while the other can be met by a deterministic polynomial time conditional expectation algorithm. Computational results are developed for both techniques. Further, a random extension algorithm further improves on the best known sizes for covering arrays in practice. An extensive set of computations with column resampling and random extension yields explicit constructions when (Formula presented.) for strength seven, (Formula presented.) for strength six, (Formula presented.) for strength five, and (Formula presented.) for strength four. When (Formula presented.), almost all known explicit constructions are improved upon. For strength (Formula presented.), restrictions on the covering perfect hash family ensure the presence of redundant rows in the covering array, which can be removed. Using restrictions and random extension, computations for (Formula presented.) and (Formula presented.) again improve upon known explicit constructions in the majority of cases. Computations for strengths three and four demonstrate that a conditional expectation algorithm can produce further improvements at the expense of a larger time and storage investment.

AB - Covering perfect hash families represent certain covering arrays compactly. Applying two probabilistic methods to covering perfect hash families improves upon the asymptotic upper bound for the minimum number of rows in a covering array with v symbols, k columns, and strength t. One bound can be realized by a randomized polynomial time construction algorithm using column resampling, while the other can be met by a deterministic polynomial time conditional expectation algorithm. Computational results are developed for both techniques. Further, a random extension algorithm further improves on the best known sizes for covering arrays in practice. An extensive set of computations with column resampling and random extension yields explicit constructions when (Formula presented.) for strength seven, (Formula presented.) for strength six, (Formula presented.) for strength five, and (Formula presented.) for strength four. When (Formula presented.), almost all known explicit constructions are improved upon. For strength (Formula presented.), restrictions on the covering perfect hash family ensure the presence of redundant rows in the covering array, which can be removed. Using restrictions and random extension, computations for (Formula presented.) and (Formula presented.) again improve upon known explicit constructions in the majority of cases. Computations for strengths three and four demonstrate that a conditional expectation algorithm can produce further improvements at the expense of a larger time and storage investment.

KW - Asymptotic bound

KW - Conditional expectation algorithm

KW - Covering array

KW - Covering perfect hash family

UR - http://www.scopus.com/inward/record.url?scp=85019738541&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019738541&partnerID=8YFLogxK

U2 - 10.1007/s10623-017-0369-x

DO - 10.1007/s10623-017-0369-x

M3 - Article

SP - 1

EP - 31

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

SN - 0925-1022

ER -