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 language | English (US) |
|---|---|
| Pages (from-to) | 1-31 |
| Number of pages | 31 |
| Journal | Designs, Codes, and Cryptography |
| DOIs | |
| State | Accepted/In press - May 26 2017 |
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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 journal › Article
}
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
AN - SCOPUS:85019738541
SP - 1
EP - 31
JO - Designs, Codes, and Cryptography
JF - Designs, Codes, and Cryptography
SN - 0925-1022
ER -