Covering Arrays with Row Limit Ottawa-Carleton Discrete Mathematics - PowerPoint PPT Presentation

Covering Arrays with Row Limit Ottawa-Carleton Discrete Mathematics Days University of Ottawa Nevena Franceti´ c Supervised by Prof. E. Mendelsohn and Prof. P. Danziger Department of Mathematics University of Toronto May 13-14, 2011.

Overview Introduction 1 Definition Some equivalent objects Goal w = o ( k ) 2 Bounds Constructions w ∼ ck , 0 < c ≤ 1 3 Bounds N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 2 / 19

Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) : components 1 2 3 4 5 6 1 0 0 ∅ 1 ∅ 0 2 0 ∅ 1 1 1 ∅ t e 3 0 1 0 ∅ ∅ 1 ∅ ∅ s 4 0 0 0 0 5 1 0 1 0 ∅ ∅ t ∅ ∅ 6 1 0 1 0 7 1 ∅ 1 ∅ 0 1 r u 8 1 1 ∅ 1 ∅ 1 9 ∅ 0 0 1 0 ∅ n s 10 ∅ 1 1 ∅ 1 0 11 ∅ 0 ∅ 0 1 1 12 ∅ 1 ∅ 0 0 0 N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 3 / 19

Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) : components 1 2 3 4 5 6 1 0 0 ∅ 1 ∅ 0 2 0 ∅ 1 1 1 ∅ t e 3 0 1 0 ∅ ∅ 1 ∅ ∅ s 4 0 0 0 0 5 1 0 1 0 ∅ ∅ t ∅ ∅ 6 1 0 1 0 7 1 ∅ 1 ∅ 0 1 r u 8 1 1 ∅ 1 ∅ 1 9 ∅ 0 0 1 0 ∅ n s 10 ∅ 1 1 ∅ 1 0 11 ∅ 0 ∅ 0 1 1 12 ∅ 1 ∅ 0 0 0 N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 4 / 19

Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) : components In general, 1 2 3 4 5 6 CARL λ ( N ; t , k , { v 1 , v 2 , . . . , v k } : w ). 1 0 0 ∅ 1 ∅ 0 2 0 ∅ 1 1 1 ∅ t e 3 0 1 0 ∅ ∅ 1 ∅ ∅ s 4 0 0 0 0 5 1 0 1 0 ∅ ∅ t ∅ ∅ 6 1 0 1 0 7 1 ∅ 1 ∅ 0 1 r u 8 1 1 ∅ 1 ∅ 1 9 ∅ 0 0 1 0 ∅ n s 10 ∅ 1 1 ∅ 1 0 11 ∅ 0 ∅ 0 1 1 12 ∅ 1 ∅ 0 0 0 N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 4 / 19

Covering array with row limit ( CARL ) CARL ( N = 12; t = 2 , k = 6 , v = 2: w = 4) : components In general, 1 2 3 4 5 6 CARL λ ( N ; t , k , { v 1 , v 2 , . . . , v k } : w ). 1 0 0 ∅ 1 ∅ 0 2 0 ∅ 1 1 1 ∅ t Applicable for testing in studies in: e 3 0 1 0 ∅ ∅ 1 pharmacology, ∅ ∅ s 4 0 0 0 0 medicine, 5 1 0 1 0 ∅ ∅ t ∅ ∅ 6 1 0 1 0 agriculture, 7 1 ∅ 1 ∅ 0 1 r chemistry, u 8 1 1 ∅ 1 ∅ 1 etc. 9 ∅ 0 0 1 0 ∅ n s 10 ∅ 1 1 ∅ 1 0 11 ∅ 0 ∅ 0 1 1 12 ∅ 1 ∅ 0 0 0 N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 4 / 19

Ck v C1 C2 C3 C4 ... Equivalent objects w = k : CA ( N ; t , k , v ). v = 1: t − ( k , w , λ ) covering ( V , B ): N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 5 / 19

Equivalent objects w = k : CA ( N ; t , k , v ). v = 1: t − ( k , w , λ ) covering ( V , B ): Ck v C1 C2 C3 C4 ... N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 5 / 19

Equivalent objects w = k : CA ( N ; t , k , v ). v = 1: t − ( k , w , λ ) covering ( V , B ): Ck v C1 C2 C3 C4 ... N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 6 / 19

What is the optimal size of a CARL ? Given t ≤ w ≤ k and v , we need to determine the smallest possible size of a CARL ( t , k , v : w ), denoted by CARLN ( t , k , v : w ). Also, we need to construct these objects. N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 7 / 19

A lower bound Theorem (Sch¨ onheim bound, (Franceti´ c et al.)) � vk � v ( k − 1) � λ v ( k − t + 1) � �� CARLN λ ( t , k , v : w ) ≥ · · · . . . . w w − 1 w − t + 1 CARL (12; 2 , 6 , 2: 4) N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 8 / 19

Sch¨ onheim bound for fixed w and t = 2 Corollary � vk � v ( k − 1) �� CARLN (2 , k , v : w ) ≥ . w w − 1 Alon, Caro and Yuster (1998): bound is optimal for large k and w fixed. Heinrich and Yin (1999): w = 3. N.F., P. Danziger, E. Mendelsohn: w = 4 with regular excess graph. N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 9 / 19

Sch¨ onheim bound for fixed w and t = 2 Corollary � vk � v ( k − 1) �� CARLN (2 , k , v : w ) ≥ . w w − 1 Alon, Caro and Yuster (1998): bound is optimal for large k and w fixed. Heinrich and Yin (1999): w = 3. N.F., P. Danziger, E. Mendelsohn: w = 4 with regular excess graph. Any t ≥ 3: R¨ odl(1985): w is fixed, v = 1, the bound is optimal . N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 9 / 19

CARL s with w = o ( k 1 / ( t +1) ) N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 10 / 19

CARL s with w = o ( k 1 / ( t +1) ) Theorem (Sch¨ onheim bound) � vk � v ( k − 1) � v ( k − t + 1) � �� CARLN ( t , k , v : w ) ≥ · · · . . . . w − 1 w − t + 1 w N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 10 / 19

CARL s with w = o ( k 1 / ( t +1) ) Theorem (Sch¨ onheim bound) � vk � v ( k − 1) � v ( k − t + 1) � �� CARLN ( t , k , v : w ) ≥ · · · . . . . w − 1 w − t + 1 w Extending the proof of R¨ odl(1985) using the interpretation of given by Alon and Spencer(2000), we get the following: Theorem (N.F.) If lim k →∞ w t +1 = 0 , then k � k � t � v t (1 + o (1)) CARLN ( t , k , v : w ) = � w t k ( k − 1) · · · ( k − t + 1) w ( w − 1) · · · ( w − t + 1) v t (1 + o (1)) . = N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 10 / 19

PARL s with w = o ( k 1 / ( t +1) ) Also, we can extend this result to the Packing Arrays with Row Limit ( PARL ): Theorem (N.F.) If lim k →∞ w t +1 = 0 , then k � k v t � t � (1 + o (1)) . PARLN ( t , k , v : w ) = � w t N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 11 / 19

CARL s with w = o ( k ) Theorem (N.F.) If lim k →∞ w k = 0 , then � k � � � w �� t � v t CARLN ( t , k , v : w ) ≤ 1 + ln . � w t t N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 12 / 19

CARL s with w = o ( k ) Theorem (N.F.) If lim k →∞ w k = 0 , then � k � � � w �� t � v t CARLN ( t , k , v : w ) ≤ 1 + ln . � w t t Conjecture (N.F.) Let w be a non-decreasing function of k. When lim k →∞ w k = 0 , there exists an optimal CARL ( t , k , v : w ) meeting the Sch¨ onheim bound for large enough k. N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 12 / 19

Wilson construction Example: If k ≡ 3 (mod 6), k ≥ 33, there exists a 4 − GDD of type k − 9 6 9 1 , ( V , G , B ). 6 ...... ...... ...... ......... ... N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 13 / 19

Wilson construction Example: If k ≡ 3 (mod 6), k ≥ 33, there exists a 4 − GDD of type k − 9 6 9 1 , ( V , G , B ). 6 Let { b 1 , b 2 , b 3 , b 4 } ∈ B . Let v = 3. N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 14 / 19

Wilson construction Example: If k ≡ 3 (mod 6), k ≥ 33, there exists a 4 − GDD of type k − 9 6 9 1 , ( V , G , B ). 6 Let { b 1 , b 2 , b 3 , b 4 } ∈ B . Let v = 3. · · · · · · b 1 b 2 b 3 b 4 · · · · · · ∅ ∅ 1 1 1 1 ∅ ∅ ∅ ∅ ∅ ∅ 1 2 2 2 ∅ ∅ 1 3 3 3 ∅ ∅ ∅ ∅ ∅ ∅ 2 1 3 2 ∅ ∅ 2 2 1 3 ∅ ∅ ∅ ∅ 2 3 2 2 ∅ ∅ ∅ ∅ 3 1 2 3 ∅ ∅ ∅ ∅ 3 2 3 1 ∅ ∅ ∅ ∅ 3 3 1 2 ∅ ∅ N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 14 / 19

Wilson construction Example: If k ≡ 3 (mod 6), k ≥ 33, there exists a 4 − GDD of type k − 9 6 9 1 , ( V , G , B ). 6 ...... ...... ...... ......... ... CARL(2,6,v:4) CARL(2,6,v:4) CARL(2,6,v:4) CARL(2,9,v:4) N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 15 / 19

CARL s with w = o ( k ) and t = 2 We can apply the Wilson construction. From affine planes we get the following: Theorem There exists an optimal CARL ( q 3 ( q + 1); 2 , q 2 , q : q ) for all prime powers q. N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 16 / 19

CARL s with w = o ( k ) and t = 2 We can apply the Wilson construction. From affine planes we get the following: Theorem There exists an optimal CARL ( q 3 ( q + 1); 2 , q 2 , q : q ) for all prime powers q. Lemma (N.F) Let q be a prime power. If there exists an optimal CARL (( q 2 − q )( q 2 − 1); 2 , q 2 − q , q : q ) , then there exists an optimal CARL (( q 2 − q )( q + 1)( q 3 + q 2 − 1); 2 , q 3 − q , q : q ) . N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 16 / 19

CARL s with w = o ( k ) and t = 3 Theorem Let q be a prime power and k ≡ 2 , 4 (mod 6) . There exists an optimal CARL ( N ; 3 , k , q : 4) with N = k ( k − 1)( k − 2) q 3 . 24 N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 17 / 19

CARL s with w = o ( k ) and t = 3 Theorem Let q be a prime power and k ≡ 2 , 4 (mod 6) . There exists an optimal CARL ( N ; 3 , k , q : 4) with N = k ( k − 1)( k − 2) q 3 . 24 Theorem Let q be a prime power. There exists an optimal CARL q ( N ; 3 , q 2 + 1 , q ; q + 1) of size N = q 5 ( q 2 + 1) . N. Franceti´ c (University of Toronto) CARL s May 13-14, 2011. 17 / 19