Time-optimal Algorithms for Generalized Dominance Computation and Related Problems on Mesh Connected Computers and Meshes with Multiple Broadcasting

The generalized dominance computation (GDC) problem is stated as follows: Let A = fa 1 ; a 2 ; : : :; a n g be a set of triplets, i.e. a i = (x i ; y i ; f i), \<" be a linear order relation deened on x-components, \<" be a linear order relation deened on y-components and \" an abelian operator deened on f-components. It is required to compute for every a i 2 A, the expression D(a i) = f j 1 f j 2 : : : f j k , where fj 1 ; j 2 ;: : :; j k g is the set of all indices j such that a j 2 A and x j < x i , y j < y i. First, this paper presents a time-optimal algorithm to solve the GDC problem in O(p n) on a mesh connected computer of size p n p n. To prove the generality of our approach, we show how a number of computational geometry problems such as ECDF (empirical cumulative distribution function) searching and two-set dominance counting, can be derived from GDC problem. Second, we deene a natural extension of the GDC, called multiple-query generalized dominance computation (MQGDC), on meshes with multiple broadcasting. By using multiple querying (MQ) paradigm of Bokka et al. 3, 4, 6] we devise a time-optimal algorithm that solves a MQGDC problem involving a set A of n items and a set Q of m queries in O(n 1 6 m 1 3) on a mesh with multiple broadcasting of size p n p n.