Given their several applications, covering arrays have become a topic of significance over the last twenty years in both the mathematical and computer science fields. A covering array is a N × k array with strength t, k rows of length N, entries from the set {0, 1, ..., v − 1}, and all vt possible combinations occur between any t columns, where N,k,t, and v are positive integers. The focus of our research was to explore the different constructions of strength two and strength three covering arrays, to find better covering arrays (i.e. more cost and time efficient covering arrays), and to see if covering arrays can detect a fault in a system. Through analyzing the covering arrays that we constructed, we were able to successfully prove that in general, a covering array of strength k + 1 can detect a single fault between any k or fewer variables in a system. Some areas of future research would include finding the location of a fault in a system or detecting two or more faults in a system

Anderson, Emily

LeFevre, Brooke

English

ValpoScholar

Valparaiso University
ValpoScholar
Symposium on Undergraduate Research and
Creative Expression (SOURCE) Office of Sponsored and Undergraduate Research
Summer 7-28-2016
Covering Arrays and Fault Detection
Brooke LeFevre
Valparaiso University, brooke.lefevre@valpo.edu
Emily Anderson
Valparaiso University, emily.anderson@valpo.edu
Follow this and additional works at: https://scholar.valpo.edu/cus
This Poster Presentation is brought to you for free and open access by the Office of Sponsored and Undergraduate Research at ValpoScholar. It has been
accepted for inclusion in Symposium on Undergraduate Research and Creative Expression (SOURCE) by an authorized administrator of ValpoScholar.
For more information, please contact a ValpoScholar staff member at scholar@valpo.edu.
Recommended Citation
LeFevre, Brooke and Anderson, Emily, "Covering Arrays and Fault Detection" (2016). Symposium on Undergraduate Research and
Creative Expression (SOURCE). 588.
https://scholar.valpo.edu/cus/588
Covering Arrays and Fault Detection
Emily Anderson and Brooke LeFevre
Advisor: Professor Jon Beagley
Abstract
Given their several applications, covering arrays have become a topic of sig-
nificance over the last twenty years in both the mathematical and computer
science fields. A covering array is a N × k array with strength t, k rows of
length N, entries from the set {0, 1, ..., v−1}, and all vt possible combinations
occur between any t columns, where N, k, t, and v are positive integers. The
focus of our research was to explore the different constructions of strength
two and strength three covering arrays, to find better covering arrays (i.e.
more cost and time efficient covering arrays), and to see if covering arrays can
detect a fault in a system. Through analyzing the covering arrays that we
constructed, we were able to successfully prove that in general, a covering ar-
ray of strength k+1 can detect a single fault between any k or fewer variables
in a system. Some areas of future research would include finding the location
of a fault in a system or detecting two or more faults in a system.
Definitions
•A covering array is a N × k array with strength t, k rows of length
N , entries from the set {0, 1, . . . , v − 1}, and all vt possible combinations
occur between any t columns, where N, k, t, and v are positive integers. It
is often denoted as CA(N ; k, t, v). See Figure 1.
•A binary covering array is a N × k covering array with v = 2, where
v can be 0 or 1 and N, k, t, and v are positive integers. See Figure 1.
•CA(k, t, v) = N denotes the minimum number of rows or tests that a cov-
ering array must have in order to be valid. See Figure 2.
CA(4; 3, 2, 2)
0 0 0
1 0 1
0 1 1
1 1 0
Figure 1: This is a 4× 3 binary covering array of strength 2 with 4 tests and 3 variables.
CA(8; 3, 3, 2)
0 0 0
0 1 0
0 0 1
1 1 0
1 0 0
1 1 1
1 0 1
0 1 1
Figure 2: This is a 8× 3 binary covering array of strength 3 with 8 tests and 3 variables.
robots do not have a common sense of right and left on the line. Each robot can move on
the line until it meets another. One wishes to determine the minimum over all possible
strategies of the maximum over all possible starting permutations of the robots, of the
time by which they can arrive at the same point. Hartman shows that this value is
⌈k/2⌉+ ⌈log2 k⌉+ 1, by making use of a bound described in 3.2.1, below.
For given small values of k = s + t and t, Table 1 records the smallest number of
rows (n, denoted by CAN(k, t)) in a binary covering array having those parameters,
when this is known. When a range is given, the numbers represent lower and upper
bounds on the smallest number of rows. This table is included to aid in the exposition.
Much more thoroughgoing tables may be found at the website of Colbourn [28], from
which many of the upper bounds in the table were derived; and we note that the upper
bounds for the intervals in the lower right corner of this table represent recent results
and will probably change again soon!
The tables of [28] do not include actual covering arrays (and we have not inde-
pendently verified in all cases that covering arrays of the indicated sizes exist). For an
extensive collection of actual covering arrays with relatively few rows, visit the webpage
[87], and follow the link to the covering array library. Also, covering arrays are available
at the website of Nurmela [67], and covering arrays can be downloaded, by request, from
the website maintained by Torres-Jimenez [81]. The website of Torres-Jimenez includes,
in particular, covering arrays yielding all of the upper bounds in Table 1.
s\t 1 2 3 4 5 6
0 2 4 8 16 32 64
1 2 4 8 16 32 64
2 2 5 10 21 42 85
3 2 6 12 24 48–52 96–108
4 2 6 12 24 48–54 96–116
5 2 6 12 24 48–56 96–118
6 2 6 12 24 48–64 96–128
7 2 6 12 24 48–64 96–128
8 2 6 12 24 48–64 96–128
9 2 7 15 30–32 60–64 120–128
10 2 7 15–16 30–35 60–79 120–179
Table 1. Values of CAN(s+ t, t).
In Section 2 we describe several ways of “looking at” covering arrays, including
alternative terminology and definitions that have been used. Section 3 gives the values
of CAN that are known exactly, including descriptions of the arguments and covering
arrays that have been used to establish the lower and (equal) upper bounds. Sec-
tion 4 describes the lower and upper bounds on values of CAN. Section 5 describes
methods used in construction of covering arrays. Section 6 briefly addresses issues of
the electronic journal of combinatorics 18 (2011), #P84 4
Figure 3: This table represents the minimum number of tests needed for known binary
covering arrays, where s+ t represents the number of columns and t represents the strength.
Constructions
Theorem 1 (Generalized Direct Product, Theorem 4.1,[2]).When a CA(N ; k, 2, v) and a CA(M ; l, 2, v)
both exist, a CA(N + M,kl, 2, v) also exists.
A=CA(4;3,2,2) B=CA(5;4,2,2) C=CA(9;12,2,2)
0 0 0
1 0 1
0 1 1
1 1 0
0 0 0 0
1 0 1 1
0 1 1 1
1 1 0 1
1 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 0 1 1 0 1 1 0 1
0 1 1 0 1 1 0 1 1 0 1 1
1 1 0 1 1 0 1 1 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 1 0 1 1 1 0 1 1
0 1 1 1 0 1 1 1 0 1 1 1
1 1 0 1 1 1 0 1 1 1 0 1
1 1 1 0 1 1 1 0 1 1 1 0
Figure 4: In this example, matrix A is written l = 4 times. Below that, matrix B is written k = 3 times. This construction
then produces matrix C.
Theorem 2 (Roux Construction, Theorem 1, [1]).When a CA(N ; k, 3, v), CA(N ; k, 3, v), and
CA(M ; k, 2, v) exist, a CA(N + M ; 2k, 3, v) also exists.
A=CA(8;3,3,2) B=CA(8;3,3,2) C=CA(4;3,2,2) D=CA(12;6,3,2)
0 0 0
0 1 0
0 0 1
1 1 0
1 0 0
1 1 1
1 0 1
0 1 1
0 0 0
0 1 0
0 0 1
1 1 0
1 0 0
1 1 1
1 0 1
0 1 1
0 0 0
1 0 1
0 1 1
1 1 0
0 0 0 0 0 0
0 1 0 0 1 0
0 0 1 0 0 1
1 1 0 1 1 0
1 0 0 1 0 0
1 1 1 1 1 1
1 0 1 1 0 1
0 1 1 0 1 1
0 0 0 1 1 1
1 0 1 0 1 0
0 1 1 1 0 0
1 1 0 0 0 1
Figure 5: The matrix D is produced as a result of a quad formation. It places A in the upper left corner, B in the upper
right corner, C in the bottom left corner, and C complement in the bottom right corner.
Application
CA(5;4,2,2)
0 0 0 0
1 0 1 1
0 1 1 1
1 1 0 1
1 1 1 0
Google Chat
Wifi/Android (0,0)
Wifi/iOS (0,1)
Cellular/Android (1,1)
Cellular/iOS (1,0)
Figure 6: Here is an application of how covering arrays are used in the real world. Suppose Google wants to offer its consumers
a way to communicate with others that has both visual and audio features. They have decided to call it Google Chat. Before
releasing Google Chat, they want to make sure that it works between both Android and iOS phones as well as on both cellular
network and wifi. The covering array to the left of the table codes the different combinations that are possible for those trying
to communicate via Google Chat. Google can be pretty certain that their new application will work for these 5 combinations or
“tests”, but they cannot be 100% certain unless they test all 16 combinations of the 4 variables.
Fault Detection
Goal: To identify and characterize failures caused by specific combinations
of option settings
Theorem 3. If C is a covering array of strength k+1, then we can detect
a single fault between any k variables in the system. Note that if C is a
strength k+ 1 covering array, it is also a strength k, k− 1, . . . , 1 covering
array.
Proof.We show the result by contrapositive. Let s be a subset of size k of
the columns of C, p(s) be a set of values on these columns, and r be a subset
of rows of C. Define f : s × p(s) → r to be the set of rows that contain the
values p(s) on the subset s.
Now, take s1, s2 to be distinct subsets of size k of the columns of C with
values p(s1), p(s2), respectively. We know that s1 ∪ s2 must contain at least
k + 1 elements. So, suppose f (s1, p(s1)) = f (s2, p(s2)). Let r ∈ f (s1, p(s1)),
which means that column s1 contains p(s1) and column s2 contains p(s2). Let
these contain distinct columns c1, c2, . . . , ck+1. In r, these values are fixed by
our choice of p(s1), p(s2). This means that p(c1), p(c2), . . . , p(ck+1)
{, where
p(ck+1)
{ represents a different value from the alphabet, does not appear in
any row of C. Therefore, C is not a covering array of strength k + 1, which
contradicts our original assumption. So, f is injective meaning that it has a
distinct set of rows for the subset of size k of the columns of C and the set of
values on these columns. Thus, we can conclude that if C is a covering array
of strength k + 1, then we can detect a single fault between any k variables
in the system.
Future Work
•Detecting two faults in a system as opposed to just one fault
•Continuing to find “better” small covering arrays
• Improving the construction of both strength two and strength three covering
arrays
Acknowledgments
•Valparaiso University Mathematics and Statistics Department
•MSEED Program (NSF Grant No. 1068346)
•Professor Jon Beagley
References
[1] Cohen, Myra B., Charles J. Colbourn, and Alan C. H. Ling. Constructing strength three
covering arrays with augmented annealing. Discrete Mathematics, vol. 308, Elsevier B.V,
2008..doi:10.1016/j.disc.2006.06.036.
[2] Colbourn, Charles, and Jose Torres-Jimenez. Profiles of Covering Arrays of Strength
Two. Journal of Algorithms and Computation 44.1 (2013): 31-59. Web. 29 June 2016.
[3] Lawrence, J., et al. A Survey of Binary Covering Arrays. Electronic Journal of Combina-
torics 18.1 (2011) Web.
[4] Nie, Changhai, and Hareton Leung. A survey of combinatorial testing. ACM Computing
Surveys (CSUR), vol. 43, ACM, Baltimore, 2011..doi:10.1145/1883612.1883618.
[5] Yilmaz, C., M. B. Cohen, and A. A. Porter. Covering Arrays for Efficient Fault Char-
acterization in Complex Configuration Spaces. IEEE Transactions on Software Engineering
32.1 (2006): 20-34. Web.


Covering Arrays and Fault Detection

Given their several applications, covering arrays have become a topic of significance over the last twenty years in both the mathematical and computer science fields. A covering array is a N × k array with strength t, k rows of length N, entries from the set {0, 1, ..., v − 1}, and all v^t possible combinations occur between any t columns, where N,k,t, and v are positive integers. The focus of our research was to explore the different constructions of strength two and strength three covering arrays, to find better covering arrays (i.e. more cost and time efficient covering arrays), and to see if covering arrays can detect a fault in a system. Through analyzing the covering arrays that we constructed, we were able to successfully prove that in general, a covering array of strength k + 1 can detect a single fault between any k or fewer variables in a system. Some areas of future research would include finding the location of a fault in a system or detecting two or more faults in a system

Valparaiso University
ValpoScholar
Fall Interdisciplinary Research Symposium Office of Sponsored and Undergraduate Research
Fall 10-28-2016
Covering Arrays and Fault Detection
Brooke LeFevre
Valparaiso University, brooke.lefevre@valpo.edu
Emily Anderson
Valparaiso University, emily.anderson@valpo.edu
Follow this and additional works at: http://scholar.valpo.edu/fires
This Poster Presentation is brought to you for free and open access by the Office of Sponsored and Undergraduate Research at ValpoScholar. It has been
accepted for inclusion in Fall Interdisciplinary Research Symposium by an authorized administrator of ValpoScholar. For more information, please
contact a ValpoScholar staff member at scholar@valpo.edu.
Recommended Citation
LeFevre, Brooke and Anderson, Emily, "Covering Arrays and Fault Detection" (2016). Fall Interdisciplinary Research Symposium. Paper
29.
http://scholar.valpo.edu/fires/29
Covering Arrays and Fault Detection
Emily Anderson and Brooke LeFevre
Advisor: Professor Jon Beagley
Abstract
Given their several applications, covering arrays have become a topic of sig-
nificance over the last twenty years in both the mathematical and computer
science fields. A covering array is a N × k array with strength t, k rows of
length N, entries from the set {0, 1, ..., v−1}, and all vt possible combinations
occur between any t columns, where N, k, t, and v are positive integers. The
focus of our research was to explore the different constructions of strength
two and strength three covering arrays, to find better covering arrays (i.e.
more cost and time efficient covering arrays), and to see if covering arrays can
detect a fault in a system. Through analyzing the covering arrays that we
constructed, we were able to successfully prove that in general, a covering ar-
ray of strength k+1 can detect a single fault between any k or fewer variables
in a system. Some areas of future research would include finding the location
of a fault in a system or detecting two or more faults in a system.
Definitions
•A covering array is a N × k array with strength t, k rows of length
N , entries from the set {0, 1, . . . , v − 1}, and all vt possible combinations
occur between any t columns, where N, k, t, and v are positive integers. It
is often denoted as CA(N ; k, t, v). See Figure 1.
•A binary covering array is a N × k covering array with v = 2, where
v can be 0 or 1 and N, k, t, and v are positive integers. See Figure 1.
•CA(k, t, v) = N denotes the minimum number of rows or tests that a cov-
ering array must have in order to be valid. See Figure 2.
CA(4; 3, 2, 2)
0 0 0
1 0 1
0 1 1
1 1 0
Figure 1: This is a 4× 3 binary covering array of strength 2 with 4 tests and 3 variables.
CA(8; 3, 3, 2)
0 0 0
0 1 0
0 0 1
1 1 0
1 0 0
1 1 1
1 0 1
0 1 1
Figure 2: This is a 8× 3 binary covering array of strength 3 with 8 tests and 3 variables.
robots do not have a common sense of right and left on the line. Each robot can move on
the line until it meets another. One wishes to determine the minimum over all possible
strategies of the maximum over all possible starting permutations of the robots, of the
time by which they can arrive at the same point. Hartman shows that this value is
⌈k/2⌉+ ⌈log2 k⌉+ 1, by making use of a bound described in 3.2.1, below.
For given small values of k = s + t and t, Table 1 records the smallest number of
rows (n, denoted by CAN(k, t)) in a binary covering array having those parameters,
when this is known. When a range is given, the numbers represent lower and upper
bounds on the smallest number of rows. This table is included to aid in the exposition.
Much more thoroughgoing tables may be found at the website of Colbourn [28], from
which many of the upper bounds in the table were derived; and we note that the upper
bounds for the intervals in the lower right corner of this table represent recent results
and will probably change again soon!
The tables of [28] do not include actual covering arrays (and we have not inde-
pendently verified in all cases that covering arrays of the indicated sizes exist). For an
extensive collection of actual covering arrays with relatively few rows, visit the webpage
[87], and follow the link to the covering array library. Also, covering arrays are available
at the website of Nurmela [67], and covering arrays can be downloaded, by request, from
the website maintained by Torres-Jimenez [81]. The website of Torres-Jimenez includes,
in particular, covering arrays yielding all of the upper bounds in Table 1.
s\t 1 2 3 4 5 6
0 2 4 8 16 32 64
1 2 4 8 16 32 64
2 2 5 10 21 42 85
3 2 6 12 24 48–52 96–108
4 2 6 12 24 48–54 96–116
5 2 6 12 24 48–56 96–118
6 2 6 12 24 48–64 96–128
7 2 6 12 24 48–64 96–128
8 2 6 12 24 48–64 96–128
9 2 7 15 30–32 60–64 120–128
10 2 7 15–16 30–35 60–79 120–179
Table 1. Values of CAN(s+ t, t).
In Section 2 we describe several ways of “looking at” covering arrays, including
alternative terminology and definitions that have been used. Section 3 gives the values
of CAN that are known exactly, including descriptions of the arguments and covering
arrays that have been used to establish the lower and (equal) upper bounds. Sec-
tion 4 describes the lower and upper bounds on values of CAN. Section 5 describes
methods used in construction of covering arrays. Section 6 briefly addresses issues of
the electronic journal of combinatorics 18 (2011), #P84 4
Figure 3: This table represents the minimum number of tests needed for known binary
covering arrays, where s+ t represents the number of columns and t represents the strength.
Constructions
Theorem 1 (Generalized Direct Product, Theorem 4.1,[2]).When a CA(N ; k, 2, v) and a CA(M ; l, 2, v)
both exist, a CA(N + M,kl, 2, v) also exists.
A=CA(4;3,2,2) B=CA(5;4,2,2) C=CA(9;12,2,2)
0 0 0
1 0 1
0 1 1
1 1 0
0 0 0 0
1 0 1 1
0 1 1 1
1 1 0 1
1 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 0 1 1 0 1 1 0 1
0 1 1 0 1 1 0 1 1 0 1 1
1 1 0 1 1 0 1 1 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 1 0 1 1 1 0 1 1
0 1 1 1 0 1 1 1 0 1 1 1
1 1 0 1 1 1 0 1 1 1 0 1
1 1 1 0 1 1 1 0 1 1 1 0
Figure 4: In this example, matrix A is written l = 4 times. Below that, matrix B is written k = 3 times. This construction
then produces matrix C.
Theorem 2 (Roux Construction, Theorem 1, [1]).When a CA(N ; k, 3, v), CA(N ; k, 3, v), and
CA(M ; k, 2, v) exist, a CA(N + M ; 2k, 3, v) also exists.
A=CA(8;3,3,2) B=CA(8;3,3,2) C=CA(4;3,2,2) D=CA(12;6,3,2)
0 0 0
0 1 0
0 0 1
1 1 0
1 0 0
1 1 1
1 0 1
0 1 1
0 0 0
0 1 0
0 0 1
1 1 0
1 0 0
1 1 1
1 0 1
0 1 1
0 0 0
1 0 1
0 1 1
1 1 0
0 0 0 0 0 0
0 1 0 0 1 0
0 0 1 0 0 1
1 1 0 1 1 0
1 0 0 1 0 0
1 1 1 1 1 1
1 0 1 1 0 1
0 1 1 0 1 1
0 0 0 1 1 1
1 0 1 0 1 0
0 1 1 1 0 0
1 1 0 0 0 1
Figure 5: The matrix D is produced as a result of a quad formation. It places A in the upper left corner, B in the upper
right corner, C in the bottom left corner, and C complement in the bottom right corner.
Application
CA(5;4,2,2)
0 0 0 0
1 0 1 1
0 1 1 1
1 1 0 1
1 1 1 0
Google Chat
Wifi/Android (0,0)
Wifi/iOS (0,1)
Cellular/Android (1,1)
Cellular/iOS (1,0)
Figure 6: Here is an application of how covering arrays are used in the real world. Suppose Google wants to offer its consumers
a way to communicate with others that has both visual and audio features. They have decided to call it Google Chat. Before
releasing Google Chat, they want to make sure that it works between both Android and iOS phones as well as on both cellular
network and wifi. The covering array to the left of the table codes the different combinations that are possible for those trying
to communicate via Google Chat. Google can be pretty certain that their new application will work for these 5 combinations or
“tests”, but they cannot be 100% certain unless they test all 16 combinations of the 4 variables.
Fault Detection
Goal: To identify and characterize failures caused by specific combinations
of option settings
Theorem 3. If C is a covering array of strength k+1, then we can detect
a single fault between any k variables in the system. Note that if C is a
strength k+ 1 covering array, it is also a strength k, k− 1, . . . , 1 covering
array.
Proof.We show the result by contrapositive. Let s be a subset of size k of
the columns of C, p(s) be a set of values on these columns, and r be a subset
of rows of C. Define f : s × p(s) → r to be the set of rows that contain the
values p(s) on the subset s.
Now, take s1, s2 to be distinct subsets of size k of the columns of C with
values p(s1), p(s2), respectively. We know that s1 ∪ s2 must contain at least
k + 1 elements. So, suppose f (s1, p(s1)) = f (s2, p(s2)). Let r ∈ f (s1, p(s1)),
which means that column s1 contains p(s1) and column s2 contains p(s2). Let
these contain distinct columns c1, c2, . . . , ck+1. In r, these values are fixed by
our choice of p(s1), p(s2). This means that p(c1), p(c2), . . . , p(ck+1)
{, where
p(ck+1)
{ represents a different value from the alphabet, does not appear in
any row of C. Therefore, C is not a covering array of strength k + 1, which
contradicts our original assumption. So, f is injective meaning that it has a
distinct set of rows for the subset of size k of the columns of C and the set of
values on these columns. Thus, we can conclude that if C is a covering array
of strength k + 1, then we can detect a single fault between any k variables
in the system.
Future Work
•Detecting two faults in a system as opposed to just one fault
•Continuing to find “better” small covering arrays
• Improving the construction of both strength two and strength three covering
arrays
Acknowledgments
•Valparaiso University Mathematics and Statistics Department
•MSEED Program (NSF Grant No. 1068346)
•Professor Jon Beagley
References
[1] Cohen, Myra B., Charles J. Colbourn, and Alan C. H. Ling. Constructing strength three
covering arrays with augmented annealing. Discrete Mathematics, vol. 308, Elsevier B.V,
2008..doi:10.1016/j.disc.2006.06.036.
[2] Colbourn, Charles, and Jose Torres-Jimenez. Profiles of Covering Arrays of Strength
Two. Journal of Algorithms and Computation 44.1 (2013): 31-59. Web. 29 June 2016.
[3] Lawrence, J., et al. A Survey of Binary Covering Arrays. Electronic Journal of Combina-
torics 18.1 (2011) Web.
[4] Nie, Changhai, and Hareton Leung. A survey of combinatorial testing. ACM Computing
Surveys (CSUR), vol. 43, ACM, Baltimore, 2011..doi:10.1145/1883612.1883618.
[5] Yilmaz, C., M. B. Cohen, and A. A. Porter. Covering Arrays for Efficient Fault Char-
acterization in Complex Configuration Spaces. IEEE Transactions on Software Engineering
32.1 (2006): 20-34. Web.


Valparaiso University

Valparaiso UniversityValpoScholarSymposium on Undergraduate Research andCreative Expression (SOURCE) Office of Sponsored and Undergraduate ResearchSummer 7-28-2016Covering Arrays and Fault DetectionBrooke LeFevreValparaiso University, brooke.lefevre@valpo.eduEmily AndersonValparaiso University, emily.anderson@valpo.eduFollow this and additional works at: https://scholar.valpo.edu/cusThis Poster Presentation is brought to you for free and open access by the Office of Sponsored and Undergraduate Research at ValpoScholar. It has beenaccepted for inclusion in Symposium on Undergraduate Research and Creative Expression (SOURCE) by an authorized administrator of ValpoScholar.For more information, please contact a ValpoScholar staff member at scholar@valpo.edu.Recommended CitationLeFevre, Brooke and Anderson, Emily, "Covering Arrays and Fault Detection" (2016). Symposium on Undergraduate Research andCreative Expression (SOURCE). 588.https://scholar.valpo.edu/cus/588Covering Arrays and Fault DetectionEmily Anderson and Brooke LeFevreAdvisor: Professor Jon BeagleyAbstractGiven their several applications, covering arrays have become a topic of sig-nificance over the last twenty years in both the mathematical and computerscience fields. A covering array is a N × k array with strength t, k rows oflength N, entries from the set {0, 1, ..., v−1}, and all vt possible combinationsoccur between any t columns, where N, k, t, and v are positive integers. Thefocus of our research was to explore the different constructions of strengthtwo and strength three covering arrays, to find better covering arrays (i.e.more cost and time efficient covering arrays), and to see if covering arrays candetect a fault in a system. Through analyzing the covering arrays that weconstructed, we were able to successfully prove that in general, a covering ar-ray of strength k+1 can detect a single fault between any k or fewer variablesin a system. Some areas of future research would include finding the locationof a fault in a system or detecting two or more faults in a system.Definitions•A covering array is a N × k array with strength t, k rows of lengthN , entries from the set {0, 1, . . . , v − 1}, and all vt possible combinationsoccur between any t columns, where N, k, t, and v are positive integers. Itis often denoted as CA(N ; k, t, v). See Figure 1.•A binary covering array is a N × k covering array with v = 2, wherev can be 0 or 1 and N, k, t, and v are positive integers. See Figure 1.•CA(k, t, v) = N denotes the minimum number of rows or tests that a cov-ering array must have in order to be valid. See Figure 2.CA(4; 3, 2, 2)0 0 01 0 10 1 11 1 0Figure 1: This is a 4× 3 binary covering array of strength 2 with 4 tests and 3 variables.CA(8; 3, 3, 2)0 0 00 1 00 0 11 1 01 0 01 1 11 0 10 1 1Figure 2: This is a 8× 3 binary covering array of strength 3 with 8 tests and 3 variables.robots do not have a common sense of right and left on the line. Each robot can move onthe line until it meets another. One wishes to determine the minimum over all possiblestrategies of the maximum over all possible starting permutations of the robots, of thetime by which they can arrive at the same point. Hartman shows that this value is⌈k/2⌉+ ⌈log2 k⌉+ 1, by making use of a bound described in 3.2.1, below.For given small values of k = s + t and t, Table 1 records the smallest number ofrows (n, denoted by CAN(k, t)) in a binary covering array having those parameters,when this is known. When a range is given, the numbers represent lower and upperbounds on the smallest number of rows. This table is included to aid in the exposition.Much more thoroughgoing tables may be found at the website of Colbourn [28], fromwhich many of the upper bounds in the table were derived; and we note that the upperbounds for the intervals in the lower right corner of this table represent recent resultsand will probably change again soon!The tables of [28] do not include actual covering arrays (and we have not inde-pendently verified in all cases that covering arrays of the indicated sizes exist). For anextensive collection of actual covering arrays with relatively few rows, visit the webpage[87], and follow the link to the covering array library. Also, covering arrays are availableat the website of Nurmela [67], and covering arrays can be downloaded, by request, fromthe website maintained by Torres-Jimenez [81]. The website of Torres-Jimenez includes,in particular, covering arrays yielding all of the upper bounds in Table 1.s\t 1 2 3 4 5 60 2 4 8 16 32 641 2 4 8 16 32 642 2 5 10 21 42 853 2 6 12 24 48–52 96–1084 2 6 12 24 48–54 96–1165 2 6 12 24 48–56 96–1186 2 6 12 24 48–64 96–1287 2 6 12 24 48–64 96–1288 2 6 12 24 48–64 96–1289 2 7 15 30–32 60–64 120–12810 2 7 15–16 30–35 60–79 120–179Table 1. Values of CAN(s+ t, t).In Section 2 we describe several ways of “looking at” covering arrays, includingalternative terminology and definitions that have been used. Section 3 gives the valuesof CAN that are known exactly, including descriptions of the arguments and coveringarrays that have been used to establish the lower and (equal) upper bounds. Sec-tion 4 describes the lower and upper bounds on values of CAN. Section 5 describesmethods used in construction of covering arrays. Section 6 briefly addresses issues ofthe electronic journal of combinatorics 18 (2011), #P84 4Figure 3: This table represents the minimum number of tests needed for known binarycovering arrays, where s+ t represents the number of columns and t represents the strength.ConstructionsTheorem 1 (Generalized Direct Product, Theorem 4.1,[2]).When a CA(N ; k, 2, v) and a CA(M ; l, 2, v)both exist, a CA(N + M,kl, 2, v) also exists.A=CA(4;3,2,2) B=CA(5;4,2,2) C=CA(9;12,2,2)0 0 01 0 10 1 11 1 00 0 0 01 0 1 10 1 1 11 1 0 11 1 1 00 0 0 0 0 0 0 0 0 0 0 01 0 1 1 0 1 1 0 1 1 0 10 1 1 0 1 1 0 1 1 0 1 11 1 0 1 1 0 1 1 0 1 1 00 0 0 0 0 0 0 0 0 0 0 01 0 1 1 1 0 1 1 1 0 1 10 1 1 1 0 1 1 1 0 1 1 11 1 0 1 1 1 0 1 1 1 0 11 1 1 0 1 1 1 0 1 1 1 0Figure 4: In this example, matrix A is written l = 4 times. Below that, matrix B is written k = 3 times. This constructionthen produces matrix C.Theorem 2 (Roux Construction, Theorem 1, [1]).When a CA(N ; k, 3, v), CA(N ; k, 3, v), andCA(M ; k, 2, v) exist, a CA(N + M ; 2k, 3, v) also exists.A=CA(8;3,3,2) B=CA(8;3,3,2) C=CA(4;3,2,2) D=CA(12;6,3,2)0 0 00 1 00 0 11 1 01 0 01 1 11 0 10 1 10 0 00 1 00 0 11 1 01 0 01 1 11 0 10 1 10 0 01 0 10 1 11 1 00 0 0 0 0 00 1 0 0 1 00 0 1 0 0 11 1 0 1 1 01 0 0 1 0 01 1 1 1 1 11 0 1 1 0 10 1 1 0 1 10 0 0 1 1 11 0 1 0 1 00 1 1 1 0 01 1 0 0 0 1Figure 5: The matrix D is produced as a result of a quad formation. It places A in the upper left corner, B in the upperright corner, C in the bottom left corner, and C complement in the bottom right corner.ApplicationCA(5;4,2,2)0 0 0 01 0 1 10 1 1 11 1 0 11 1 1 0Google ChatWifi/Android (0,0)Wifi/iOS (0,1)Cellular/Android (1,1)Cellular/iOS (1,0)Figure 6: Here is an application of how covering arrays are used in the real world. Suppose Google wants to offer its consumersa way to communicate with others that has both visual and audio features. They have decided to call it Google Chat. Beforereleasing Google Chat, they want to make sure that it works between both Android and iOS phones as well as on both cellularnetwork and wifi. The covering array to the left of the table codes the different combinations that are possible for those tryingto communicate via Google Chat. Google can be pretty certain that their new application will work for these 5 combinations or“tests”, but they cannot be 100% certain unless they test all 16 combinations of the 4 variables.Fault DetectionGoal: To identify and characterize failures caused by specific combinationsof option settingsTheorem 3. If C is a covering array of strength k+1, then we can detecta single fault between any k variables in the system. Note that if C is astrength k+ 1 covering array, it is also a strength k, k− 1, . . . , 1 coveringarray.Proof.We show the result by contrapositive. Let s be a subset of size k ofthe columns of C, p(s) be a set of values on these columns, and r be a subsetof rows of C. Define f : s × p(s) → r to be the set of rows that contain thevalues p(s) on the subset s.Now, take s1, s2 to be distinct subsets of size k of the columns of C withvalues p(s1), p(s2), respectively. We know that s1 ∪ s2 must contain at leastk + 1 elements. So, suppose f (s1, p(s1)) = f (s2, p(s2)). Let r ∈ f (s1, p(s1)),which means that column s1 contains p(s1) and column s2 contains p(s2). Letthese contain distinct columns c1, c2, . . . , ck+1. In r, these values are fixed byour choice of p(s1), p(s2). This means that p(c1), p(c2), . . . , p(ck+1){, wherep(ck+1){ represents a different value from the alphabet, does not appear inany row of C. Therefore, C is not a covering array of strength k + 1, whichcontradicts our original assumption. So, f is injective meaning that it has adistinct set of rows for the subset of size k of the columns of C and the set ofvalues on these columns. Thus, we can conclude that if C is a covering arrayof strength k + 1, then we can detect a single fault between any k variablesin the system.Future Work•Detecting two faults in a system as opposed to just one fault•Continuing to find “better” small covering arrays• Improving the construction of both strength two and strength three coveringarraysAcknowledgments•Valparaiso University Mathematics and Statistics Department•MSEED Program (NSF Grant No. 1068346)•Professor Jon BeagleyReferences[1] Cohen, Myra B., Charles J. Colbourn, and Alan C. H. Ling. Constructing strength threecovering arrays with augmented annealing. Discrete Mathematics, vol. 308, Elsevier B.V,2008..doi:10.1016/j.disc.2006.06.036.[2] Colbourn, Charles, and Jose Torres-Jimenez. Profiles of Covering Arrays of StrengthTwo. Journal of Algorithms and Computation 44.1 (2013): 31-59. Web. 29 June 2016.[3] Lawrence, J., et al. A Survey of Binary Covering Arrays. Electronic Journal of Combina-torics 18.1 (2011) Web.[4] Nie, Changhai, and Hareton Leung. A survey of combinatorial testing. ACM ComputingSurveys (CSUR), vol. 43, ACM, Baltimore, 2011..doi:10.1145/1883612.1883618.[5] Yilmaz, C., M. B. Cohen, and A. A. Porter. Covering Arrays for Efficient Fault Char-acterization in Complex Configuration Spaces. IEEE Transactions on Software Engineering32.1 (2006): 20-34. Web.

Valparaiso UniversityValpoScholarFall Interdisciplinary Research Symposium Office of Sponsored and Undergraduate ResearchFall 10-28-2016Covering Arrays and Fault DetectionBrooke LeFevreValparaiso University, brooke.lefevre@valpo.eduEmily AndersonValparaiso University, emily.anderson@valpo.eduFollow this and additional works at: http://scholar.valpo.edu/firesThis Poster Presentation is brought to you for free and open access by the Office of Sponsored and Undergraduate Research at ValpoScholar. It has beenaccepted for inclusion in Fall Interdisciplinary Research Symposium by an authorized administrator of ValpoScholar. For more information, pleasecontact a ValpoScholar staff member at scholar@valpo.edu.Recommended CitationLeFevre, Brooke and Anderson, Emily, "Covering Arrays and Fault Detection" (2016). Fall Interdisciplinary Research Symposium. Paper29.http://scholar.valpo.edu/fires/29Covering Arrays and Fault DetectionEmily Anderson and Brooke LeFevreAdvisor: Professor Jon BeagleyAbstractGiven their several applications, covering arrays have become a topic of sig-nificance over the last twenty years in both the mathematical and computerscience fields. A covering array is a N × k array with strength t, k rows oflength N, entries from the set {0, 1, ..., v−1}, and all vt possible combinationsoccur between any t columns, where N, k, t, and v are positive integers. Thefocus of our research was to explore the different constructions of strengthtwo and strength three covering arrays, to find better covering arrays (i.e.more cost and time efficient covering arrays), and to see if covering arrays candetect a fault in a system. Through analyzing the covering arrays that weconstructed, we were able to successfully prove that in general, a covering ar-ray of strength k+1 can detect a single fault between any k or fewer variablesin a system. Some areas of future research would include finding the locationof a fault in a system or detecting two or more faults in a system.Definitions•A covering array is a N × k array with strength t, k rows of lengthN , entries from the set {0, 1, . . . , v − 1}, and all vt possible combinationsoccur between any t columns, where N, k, t, and v are positive integers. Itis often denoted as CA(N ; k, t, v). See Figure 1.•A binary covering array is a N × k covering array with v = 2, wherev can be 0 or 1 and N, k, t, and v are positive integers. See Figure 1.•CA(k, t, v) = N denotes the minimum number of rows or tests that a cov-ering array must have in order to be valid. See Figure 2.CA(4; 3, 2, 2)0 0 01 0 10 1 11 1 0Figure 1: This is a 4× 3 binary covering array of strength 2 with 4 tests and 3 variables.CA(8; 3, 3, 2)0 0 00 1 00 0 11 1 01 0 01 1 11 0 10 1 1Figure 2: This is a 8× 3 binary covering array of strength 3 with 8 tests and 3 variables.robots do not have a common sense of right and left on the line. Each robot can move onthe line until it meets another. One wishes to determine the minimum over all possiblestrategies of the maximum over all possible starting permutations of the robots, of thetime by which they can arrive at the same point. Hartman shows that this value is⌈k/2⌉+ ⌈log2 k⌉+ 1, by making use of a bound described in 3.2.1, below.For given small values of k = s + t and t, Table 1 records the smallest number ofrows (n, denoted by CAN(k, t)) in a binary covering array having those parameters,when this is known. When a range is given, the numbers represent lower and upperbounds on the smallest number of rows. This table is included to aid in the exposition.Much more thoroughgoing tables may be found at the website of Colbourn [28], fromwhich many of the upper bounds in the table were derived; and we note that the upperbounds for the intervals in the lower right corner of this table represent recent resultsand will probably change again soon!The tables of [28] do not include actual covering arrays (and we have not inde-pendently verified in all cases that covering arrays of the indicated sizes exist). For anextensive collection of actual covering arrays with relatively few rows, visit the webpage[87], and follow the link to the covering array library. Also, covering arrays are availableat the website of Nurmela [67], and covering arrays can be downloaded, by request, fromthe website maintained by Torres-Jimenez [81]. The website of Torres-Jimenez includes,in particular, covering arrays yielding all of the upper bounds in Table 1.s\t 1 2 3 4 5 60 2 4 8 16 32 641 2 4 8 16 32 642 2 5 10 21 42 853 2 6 12 24 48–52 96–1084 2 6 12 24 48–54 96–1165 2 6 12 24 48–56 96–1186 2 6 12 24 48–64 96–1287 2 6 12 24 48–64 96–1288 2 6 12 24 48–64 96–1289 2 7 15 30–32 60–64 120–12810 2 7 15–16 30–35 60–79 120–179Table 1. Values of CAN(s+ t, t).In Section 2 we describe several ways of “looking at” covering arrays, includingalternative terminology and definitions that have been used. Section 3 gives the valuesof CAN that are known exactly, including descriptions of the arguments and coveringarrays that have been used to establish the lower and (equal) upper bounds. Sec-tion 4 describes the lower and upper bounds on values of CAN. Section 5 describesmethods used in construction of covering arrays. Section 6 briefly addresses issues ofthe electronic journal of combinatorics 18 (2011), #P84 4Figure 3: This table represents the minimum number of tests needed for known binarycovering arrays, where s+ t represents the number of columns and t represents the strength.ConstructionsTheorem 1 (Generalized Direct Product, Theorem 4.1,[2]).When a CA(N ; k, 2, v) and a CA(M ; l, 2, v)both exist, a CA(N + M,kl, 2, v) also exists.A=CA(4;3,2,2) B=CA(5;4,2,2) C=CA(9;12,2,2)0 0 01 0 10 1 11 1 00 0 0 01 0 1 10 1 1 11 1 0 11 1 1 00 0 0 0 0 0 0 0 0 0 0 01 0 1 1 0 1 1 0 1 1 0 10 1 1 0 1 1 0 1 1 0 1 11 1 0 1 1 0 1 1 0 1 1 00 0 0 0 0 0 0 0 0 0 0 01 0 1 1 1 0 1 1 1 0 1 10 1 1 1 0 1 1 1 0 1 1 11 1 0 1 1 1 0 1 1 1 0 11 1 1 0 1 1 1 0 1 1 1 0Figure 4: In this example, matrix A is written l = 4 times. Below that, matrix B is written k = 3 times. This constructionthen produces matrix C.Theorem 2 (Roux Construction, Theorem 1, [1]).When a CA(N ; k, 3, v), CA(N ; k, 3, v), andCA(M ; k, 2, v) exist, a CA(N + M ; 2k, 3, v) also exists.A=CA(8;3,3,2) B=CA(8;3,3,2) C=CA(4;3,2,2) D=CA(12;6,3,2)0 0 00 1 00 0 11 1 01 0 01 1 11 0 10 1 10 0 00 1 00 0 11 1 01 0 01 1 11 0 10 1 10 0 01 0 10 1 11 1 00 0 0 0 0 00 1 0 0 1 00 0 1 0 0 11 1 0 1 1 01 0 0 1 0 01 1 1 1 1 11 0 1 1 0 10 1 1 0 1 10 0 0 1 1 11 0 1 0 1 00 1 1 1 0 01 1 0 0 0 1Figure 5: The matrix D is produced as a result of a quad formation. It places A in the upper left corner, B in the upperright corner, C in the bottom left corner, and C complement in the bottom right corner.ApplicationCA(5;4,2,2)0 0 0 01 0 1 10 1 1 11 1 0 11 1 1 0Google ChatWifi/Android (0,0)Wifi/iOS (0,1)Cellular/Android (1,1)Cellular/iOS (1,0)Figure 6: Here is an application of how covering arrays are used in the real world. Suppose Google wants to offer its consumersa way to communicate with others that has both visual and audio features. They have decided to call it Google Chat. Beforereleasing Google Chat, they want to make sure that it works between both Android and iOS phones as well as on both cellularnetwork and wifi. The covering array to the left of the table codes the different combinations that are possible for those tryingto communicate via Google Chat. Google can be pretty certain that their new application will work for these 5 combinations or“tests”, but they cannot be 100% certain unless they test all 16 combinations of the 4 variables.Fault DetectionGoal: To identify and characterize failures caused by specific combinationsof option settingsTheorem 3. If C is a covering array of strength k+1, then we can detecta single fault between any k variables in the system. Note that if C is astrength k+ 1 covering array, it is also a strength k, k− 1, . . . , 1 coveringarray.Proof.We show the result by contrapositive. Let s be a subset of size k ofthe columns of C, p(s) be a set of values on these columns, and r be a subsetof rows of C. Define f : s × p(s) → r to be the set of rows that contain thevalues p(s) on the subset s.Now, take s1, s2 to be distinct subsets of size k of the columns of C withvalues p(s1), p(s2), respectively. We know that s1 ∪ s2 must contain at leastk + 1 elements. So, suppose f (s1, p(s1)) = f (s2, p(s2)). Let r ∈ f (s1, p(s1)),which means that column s1 contains p(s1) and column s2 contains p(s2). Letthese contain distinct columns c1, c2, . . . , ck+1. In r, these values are fixed byour choice of p(s1), p(s2). This means that p(c1), p(c2), . . . , p(ck+1){, wherep(ck+1){ represents a different value from the alphabet, does not appear inany row of C. Therefore, C is not a covering array of strength k + 1, whichcontradicts our original assumption. So, f is injective meaning that it has adistinct set of rows for the subset of size k of the columns of C and the set ofvalues on these columns. Thus, we can conclude that if C is a covering arrayof strength k + 1, then we can detect a single fault between any k variablesin the system.Future Work•Detecting two faults in a system as opposed to just one fault•Continuing to find “better” small covering arrays• Improving the construction of both strength two and strength three coveringarraysAcknowledgments•Valparaiso University Mathematics and Statistics Department•MSEED Program (NSF Grant No. 1068346)•Professor Jon BeagleyReferences[1] Cohen, Myra B., Charles J. Colbourn, and Alan C. H. Ling. Constructing strength threecovering arrays with augmented annealing. Discrete Mathematics, vol. 308, Elsevier B.V,2008..doi:10.1016/j.disc.2006.06.036.[2] Colbourn, Charles, and Jose Torres-Jimenez. Profiles of Covering Arrays of StrengthTwo. Journal of Algorithms and Computation 44.1 (2013): 31-59. Web. 29 June 2016.[3] Lawrence, J., et al. A Survey of Binary Covering Arrays. Electronic Journal of Combina-torics 18.1 (2011) Web.[4] Nie, Changhai, and Hareton Leung. A survey of combinatorial testing. ACM ComputingSurveys (CSUR), vol. 43, ACM, Baltimore, 2011..doi:10.1145/1883612.1883618.[5] Yilmaz, C., M. B. Cohen, and A. A. Porter. Covering Arrays for Efficient Fault Char-acterization in Complex Configuration Spaces. IEEE Transactions on Software Engineering32.1 (2006): 20-34. Web.

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