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Covering Arrays with Row Limit: Bounds and Constructions Nevena Franceti c Supervised by Prof. P. Danziger and Prof. E. Mendelsohn Discrete Maths Research Group Monash University September 22, 2014. Covering arrays N. Franceti c

  1. Covering Arrays with Row Limit: Bounds and Constructions Nevena Franceti´ c Supervised by Prof. P. Danziger and Prof. E. Mendelsohn Discrete Maths Research Group Monash University September 22, 2014.

  2. Covering arrays N. Franceti´ c (Monash) CARL s September 22, 2014.

  3. Covering arrays ... is a test suite ... N. Franceti´ c (Monash) CARL s September 22, 2014.

  4. Covering arrays ... is a test suite ... ... for verification of interactions between components. N. Franceti´ c (Monash) CARL s September 22, 2014.

  5. Example N. Franceti´ c (Monash) CARL s September 22, 2014.

  6. Example N. Franceti´ c (Monash) CARL s September 22, 2014.

  7. Example N. Franceti´ c (Monash) CARL s September 22, 2014.

  8. Example Components: outline, shadow, blinking, hidden N. Franceti´ c (Monash) CARL s September 22, 2014.

  9. Example Components: outline, shadow, blinking, hidden Levels for each component: present or absent (1 or 0) N. Franceti´ c (Monash) CARL s September 22, 2014.

  10. Example Components: outline, shadow, blinking, hidden Levels for each component: present or absent (1 or 0) What interactions? Pairwise interactions. N. Franceti´ c (Monash) CARL s September 22, 2014.

  11. Example Components: outline, shadow, blinking, hidden Levels for each component: present or absent (1 or 0) What interactions? Pairwise interactions. � 4 � 2 2 = 24 tests. Testing two at time: it would take 2 N. Franceti´ c (Monash) CARL s September 22, 2014.

  12. Example Outline Shadow Blinking Hidden 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

  13. Example Outline Shadow Blinking Hidden 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

  14. Example Outline Shadow Blinking Hidden 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 Testing can be done in only 5 iterations. N. Franceti´ c (Monash) CARL s September 22, 2014.

  15. Example Outline Shadow Blinking Hidden 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

  16. Parameters of a Covering Array A covering array is characterized by: k : the number of components (columns) v : the number of levels for each component (alphabet size) t : strength ⇒ testing interactions between t columns N. Franceti´ c (Monash) CARL s September 22, 2014.

  17. Parameters of a Covering Array A covering array is characterized by: k : the number of components (columns) v : the number of levels for each component (alphabet size) t : strength ⇒ testing interactions between t columns Goal: find the smallest number of rows, called size N of a covering array. Size of an optimal CA is usually denoted by CAN ( t , k , v ). N. Franceti´ c (Monash) CARL s September 22, 2014.

  18. Some facts about covering arrays N. Franceti´ c (Monash) CARL s September 22, 2014.

  19. Some facts about covering arrays N = O (log k )) (Gargano et al., 1993) N. Franceti´ c (Monash) CARL s September 22, 2014.

  20. Some facts about covering arrays N = O (log k )) (Gargano et al., 1993) The exact size of a covering array is ONLY known for one family of arrays (Kleitman and Spencer, 1973; Katona, 1973): � N − 1 � CA ( N ; 2 , k , 2) exists for all k ≤ � N � 2 − 1 N. Franceti´ c (Monash) CARL s September 22, 2014.

  21. Some facts about covering arrays N = O (log k )) (Gargano et al., 1993) The exact size of a covering array is ONLY known for one family of arrays (Kleitman and Spencer, 1973; Katona, 1973): � N − 1 � CA ( N ; 2 , k , 2) exists for all k ≤ � N � 2 − 1 Finding an optimal covering array is NP-complete when extra constraints are imposed even when v = 2 (Maltais and Moura, 2011). N. Franceti´ c (Monash) CARL s September 22, 2014.

  22. More on covering arrays http://www.pairwise.org/tools.asp contains: 39 software tools for constructing CA s both commercial and open source N. Franceti´ c (Monash) CARL s September 22, 2014.

  23. Covering array with row limit ( CARL ) N. Franceti´ c (Monash) CARL s September 22, 2014.

  24. Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 0 0 − 1 − 0 0 − 1 1 1 − 0 1 0 − − 1 0 − 0 0 0 − 1 0 1 0 − − 1 − 0 − 1 0 1 − 1 − 0 1 1 1 − 1 − 1 − 0 0 1 0 − − 1 1 − 1 0 − 0 − 0 1 1 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

  25. Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 1 2 3 4 5 6 1 0 0 − 1 − 0 k = the number of columns 2 0 − 1 1 1 − (components) 3 0 1 0 − − 1 4 0 − 0 0 0 − 5 1 0 1 0 − − 6 1 − 0 − 1 0 7 1 − 1 − 0 1 8 1 1 − 1 − 1 9 − 0 0 1 0 − 10 − 1 1 − 1 0 11 − 0 − 0 1 1 12 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

  26. Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 1 2 3 4 5 6 1 0 0 − 1 − 0 k = the number of columns 2 0 − 1 1 1 − (components) 3 0 1 0 − − 1 4 0 − 0 0 0 − v = the alphabet size; the 5 1 0 1 0 − − number of different values assigned to a column 6 1 − 0 − 1 0 7 1 − 1 − 0 1 8 1 1 − 1 − 1 9 − 0 0 1 0 − 10 − 1 1 − 1 0 11 − 0 − 0 1 1 12 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

  27. Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 1 2 3 4 5 6 1 0 0 − 1 − 0 k = the number of columns 2 0 − 1 1 1 − (components) 3 0 1 0 − − 1 4 0 − 0 0 0 − v = the alphabet size; the 5 1 0 1 0 − − number of different values assigned to a column 6 1 − 0 − 1 0 7 1 − 1 − 0 1 w = row limit; the number of 8 1 1 − 1 − 1 non-empty cells in a row 9 − 0 0 1 0 − 10 − 1 1 − 1 0 11 − 0 − 0 1 1 12 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

  28. Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 1 2 3 4 5 6 k = the number of columns 1 0 0 − 1 − 0 (components) 2 0 − 1 1 1 − v = the alphabet size; the 3 0 1 0 − − 1 number of different values 4 0 − 0 0 0 − assigned to a column 5 1 0 1 0 − − w = row limit; the number of 6 1 − 0 − 1 0 7 1 − 1 − 0 1 non-empty cells in a row 8 1 1 − 1 − 1 t = strength 9 − 0 0 1 0 − 10 − 1 1 − 1 0 11 − 0 − 0 1 1 12 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

  29. Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) 1 2 3 4 5 6 k = the number of columns 1 0 0 − 1 − 0 (components) 2 0 − 1 1 1 − v = the alphabet size; the 3 0 1 0 − − 1 number of different values 4 0 − 0 0 0 − assigned to a column 5 1 0 1 0 − − w = row limit; the number of 6 1 − 0 − 1 0 7 1 − 1 − 0 1 non-empty cells in a row 8 1 1 − 1 − 1 t = strength 9 − 0 0 1 0 − N = size; goal find minimum N 10 − 1 1 − 1 0 11 − 0 − 0 1 1 12 − 1 − 0 0 0 N. Franceti´ c (Monash) CARL s September 22, 2014.

  30. Covering arrays vs CARL s CAN ( t , k , v ) ≤ CARLN ( t , k , v : w ( k )) N. Franceti´ c (Monash) CARL s September 22, 2014.

  31. Covering arrays vs CARL s CAN ( t , k , v ) ≤ CARLN ( t , k , v : w ( k )) Theorem ( Gargano et al.(1993)) lim sup CAN (2 , k , v ) = Θ (log k ) . k →∞ N. Franceti´ c (Monash) CARL s September 22, 2014.

  32. Covering arrays vs CARL s CAN ( t , k , v ) ≤ CARLN ( t , k , v : w ( k )) Theorem ( Gargano et al.(1993)) lim sup CAN (2 , k , v ) = Θ (log k ) . k →∞ Ω (log k ) = CAN (2 , k , v ) ≤ CAN ( t , k , v ) ≤ CARLN ( t , k , v : w ( k )) N. Franceti´ c (Monash) CARL s September 22, 2014.

  33. Sch¨ onheim lower bound Theorem CARLN λ ( t , k , v : w ) ≥ SB ( t , k , v : w ) � vk � v ( k − 1) � v ( k − t + 1) � �� SB ( t , k , v : w ) = · · · . . . . w w − 1 w − t + 1 CARL (12; 2 , 6 , 2: 4) N. Franceti´ c (Monash) CARL s September 22, 2014.

  34. Sch¨ onheim lower bound w ( k ) = c , c ∈ N : � k � t � v t (1 + o (1)) CARLN ( t , k , v : w ) = � w t N. Franceti´ c (Monash) CARL s September 22, 2014.

  35. Sch¨ onheim lower bound w ( k ) = c , c ∈ N : � k � t � v t (1 + o (1)) CARLN ( t , k , v : w ) = � w t CARLN ( t , k , v : w ) = Θ ( k t ) N. Franceti´ c (Monash) CARL s September 22, 2014.

  36. Sch¨ onheim lower bound w ( k ) = c , c ∈ N : � k � t � v t (1 + o (1)) CARLN ( t , k , v : w ) = � w t CARLN ( t , k , v : w ) = Θ ( k t ) k t lim k →∞ w ( k ) t log k = 0 SB ( t , k , v : w ( k )) lim = 0 log k k →∞ N. Franceti´ c (Monash) CARL s September 22, 2014.

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