On Geometric Path Query Problems

Abstract

In this dissertation, we study several geometric path query problems. Our focus is primarily on the so-called "two-point" query problem: Given a scene of disjoint polygonal obstacles with totally n vertices in the plane, we construct efficient data structures that enable fast reporting of an "optimal" obstacle-avoiding path (or its length, cost, directions, etc.) between two arbitrary query points s and t that are given in an on-line fashion. We consider geometric paths under several optimality criteria: $\rm L\sb{p}$ length, number of edges (called links), monotonicity with respect to a certain direction, and some combinations of length and links. Our methods are centered around the notion of gateways, a small number of easily identified points in the plane that control the paths we seek. We present solutions for the general cases based upon the computation of the minimum size visibility polygon for query points. We also give better solutions for several special cases based upon new geometric observations. Very few algorithms were previously known for two-point query problems and our results represent a significant addition to the field. In addition to our theoretical results, we also perform experimental studies on issues involved with the implementation of geometric algorithms. These studies are a necessary first step in the implementation of full path-planning systems.