Abstract
Covering arrays (CA) of strength t, mixed level or fixed level, have been applied to software testing to aim for a minimum coverage of all t-way interactions among components. The size of CA increases with the increase of strength interaction t, which increase the cost of software testing. However, it is quite often that some certain components have strong interactions, while others may have fewer or none. Hence, a better way to test software system is to identify the subsets of components which are involved in stronger interactions and apply high strength interaction testing only on these subsets. For this, in 2003, the notion of variable strength covering arrays was proposed by Cohen et al. to satisfy the need to vary the size of t in an individual test suite. In this paper, an effective deterministic construction of variable strength covering arrays is presented. Based on the construction, some series of variable strength covering arrays are then obtained, which are all optimal in the sense of their sizes. In the procedure, two classes of new difference matrices of strength 3 are also mentioned.
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References
Beth, T., Jungnickel, D., Lenz, H. Design theory. Cambridge University Press, Cambridge, 1999
Brouwer, A.E., Cohen, A.M., Nguyen, M.V.M. Orthogonal arrays of strength 3 and small run sizes. J. Statist. Plann. Inference, 136: 3268–3280 (2006)
Bush, K.A. A generalization of the theorem due to MacNeish. Ann. Math. Stat., 23: 293–295 (1952)
Bush, K.A. Orthogonal arrays of index unity. Ann. Math. Stat., 23: 426–434 (1952)
Cohen, M.B., Gibbons, P.B., Mugridge, W.B., Colbourn, C.J., Collofello, J.S. Variable strength interaction testing of components. Proceedings of the 27th Annual International Computer Software and Applications Conference (COMPSAC2003), Dallas, TX, USA, 413–418 (2003)
Colbourn, C.J., Dinitz, J.H. The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton, FL, 2007
Colbourn, C.J., Martirosyan, S.S., Mullen, G.L., Shasha, D.E., Sherwood, G.B., Yucas, J.L. Products of mixed covering arrays of strength two. J. Combin. Des., 14: 124–138 (2006)
Colbourn, C.J., McClary, D.W. Locating and detecting arrays for interaction faults. Journal of Combinatorial Optimization, 15: 17–48 (2008)
Colbourn, C.J., Shi, C., Wang, C.M., Yan, J. Mixed covering arrays of strength three with few factors. J. Statist. Plann. Inference, 141: 3640–3647 (2011)
Hedayat, A.S., Sloane, N.J.A., Stufken, J. Orthogonal arrays. Springer, New York, 1999
Hedayat, A.S., Stufken, J., Su, G. On difference schemes and orthogonal arrays of strengtht. J. Statist. Plann. Inference, 56: 307–324 (1996)
Ji, L., Yin, J. Constructions of new orthogonal arrays and covering arrays of strength three. J. Combin. Theory A, 117(3): 236–247 (2010)
Jiang, L., Yin, J. An approach of constructing mixed-level orthogonal arrays of strength ≥ 3. Science China Mathematics, 56(6): 1109–1115 (2013)
Johnson, K. A., Entringer, R. Largest induced subgraphs of the n-cube that contain no 4-cycles. J. Combin. Theory B, 46: 346–355 (1989)
Kuhn, D.R., Reilly, M.J. An investigation of the applicability of design of experiments to software testing. In: Proceedings of the Annual NASA/IEEE Software Engineering Workshop (SEW), IEEE Press, Los Alamitos, 91–95 (2002)
Moura, L., Raaphorst, S., Stevens, B. The Lovász Local Lemma and Variable Strength Covering Arrays. Electronic Notes in Discrete Mathematics, 65: 43–49 (2018)
Nguyen, M.V.M. Some new constructions of strength 3 mixed orthogonal arrays. J. Statist. Plann. Inference, 138: 220–233 (2008)
Raaphorst, S., Moura, L., Stevens, B. Variable Strength Covering Arrays. J. Combin. Des., 26: 417–438 (2018)
Schoen, E.D., Eendebak, P.T., Nguyen, M.V.M. Complete enumeration of pure-level and mixed-level orthogonal arrays. J. Comb. Des., 18: 123–140 (2010)
Shi, C., Tang, Y., Yin, J. Optimum mixed level detecting arrays. The Annals of Statistics, 42(4): 1546–1563 (2014)
Wang, C.M., Yan, J., Yin, J. Resolvable generalized difference matrices: Existence and Applications. Finite fields and Theire Applications, 42: 39–56 (2016)
Wang, Z., Xu, B., Nie, C. Survey of combinatorial test generation. Journal of Frontiers of Computer Science and Technology, 2(6): 571–588 (2008) (in Chinese)
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The authors would like to thank the referees for their helpful suggestions and comments.
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This paper is supported by the National Natural Science Foundation of China (Nos. 11301342, 61972241), the Natural Science Foundation of Shanghai (No. 17ZR1419900) and President Foundation of Shanghai Ocean University (NO. A2-2006-20-200212).
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Jiang, L., Shi, C. A Construction of Variable Strength Covering Arrays. Acta Math. Appl. Sin. Engl. Ser. 37, 240–250 (2021). https://doi.org/10.1007/s10255-021-1006-z
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DOI: https://doi.org/10.1007/s10255-021-1006-z
Keywords
- software testing
- variable strength covering arrays
- resolvable orthogonal arrays
- divisible orthogonal arrays
- existence