SOME RESULTS IN COMPUTATIONAL AND COMBINATORIAL GEOMETRY

Abstract

OF THE DISSERTATION Some Results in Computational and Combinatorial Geometry by Mudassir Shabbir Dissertation Director: William Steiger In this thesis we present some new results in the field of discrete and computational geometry. The techniques and tools developed to achieve these results add to our understanding of important geometric objects like line arrangements, and geometric measures of depth. Small Hitting Sets Given a set S of n points, a weak -net X is a set of points (not necessarily in S) such that any convex set, called a range, that contains more than an fraction of S must meet X for a fixed > 0 [30]. Aronov et al. gave the first bounds on when the cardinality of X is a fixed small number in the plane. Later Mustafa and Ray proved that |X| = 2 can be chosen so that we hit all convex ranges that contain 4n/7 points of S [46]. We describe an O(n log n) time algorithm to find points z1 6= z2, at least one of which must meet any convex set of “size” greater than 4n/7; z1 and z2 comprise a hitting set of size two for such convex ranges. This is the first algorithm for computing the hitting sets of fixed size.