Combinatorial Analysis of Diagonal, Box, and Greater-Than Polynomials As Packing Functions
Appl. Math. Inf. Sci. 9, No. 6, 2757-2766 (2015) 2757 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/090601 Combinatorial Analysis of Diagonal, Box, and Greater-Than Polynomials as Packing Functions 1,2, 3 2 4 Jose Torres-Jimenez ∗, Nelson Rangel-Valdez , Raghu N. Kacker and James F. Lawrence 1 CINVESTAV-Tamaulipas, Information Technology Laboratory, Km. 5.5 Carretera Victoria-Soto La Marina, 87130 Victoria Tamps., Mexico 2 National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 3 Universidad Polit´ecnica de Victoria, Av. Nuevas Tecnolog´ıas 5902, Km. 5.5 Carretera Cd. Victoria-Soto la Marina, 87130, Cd. Victoria Tamps., M´exico 4 George Mason University, Fairfax, VA 22030, USA Received: 2 Feb. 2015, Revised: 2 Apr. 2015, Accepted: 3 Apr. 2015 Published online: 1 Nov. 2015 Abstract: A packing function is a bijection between a subset V Nm and N, where N denotes the set of non negative integers N. Packing functions have several applications, e.g. in partitioning schemes⊆ and in text compression. Two categories of packing functions are Diagonal Polynomials and Box Polynomials. The bijections for diagonal ad box polynomials have mostly been studied for small values of m. In addition to presenting bijections for box and diagonal polynomials for any value of m, we present a bijection using what we call Greater-Than Polynomial between restricted m dimensional vectors over Nm and N. We give details of two interesting applications of packing functions: (a) the application of greater-than− polynomials for the manipulation of Covering Arrays that are used in combinatorial interaction testing; and (b) the relationship between grater-than and diagonal polynomials with a special case of Diophantine equations.
[Show full text]