Colored intersection searching via sparse rectangular matrix multiplication

In a Batched Colored Intersection Searching Problem (CI), one is given a set of n geometric objects (of a certain class). Each object is colored by one of c colors, and the goal is to report all pairs of colors (c1, c2) such that there are two objects, one colored c1 and one colored c2, that intersect each other. We also consider the bipartite version of the problem, where we are interested in intersections between objects of one class with objects of another class (e.g., points and halfspaces). In a Sparse Rectangular Matrix Multiplication Problem (SRMM), one is given an n1 × n2 matrix A and an n2 × n3 matrix B, each containing at most m non-zero entries, and the goal is to compute their product AB. In this paper we present a technique for solving CI problems over a wide range of classes of geometric objects. The basic idea is first to use some decomposition method, such as geometric cuttings, to represent the intersection graph of the objects as a union of bi-cliques. Then, in ea...

• Type: Conference
• Year: 2006
• Where: COMPGEOM
• Authors: Haim Kaplan, Micha Sharir, Elad Verbin
• Keywords: Batched Colored Intersection, COMPGEOM 2006, Geometric Objects, Matrix Multiplication