Combinatorial and algorithmic analysis of space decomposition problems

Candidate: Aronov,Boris

Abstract

The first part of the thesis studies geodesic Voronoi diagrams. The closest-site (respectively, furthest-site) Voronoi diagram of a finite set of sites in Euclidean space is a classical geometric structure, which partitions the space into a set of Voronoi cells, each associated with a site, so that any point in the cell of site s is closer to s (resp. further from s) than to any other site. The structure of such diagrams for point sites in the plane has been completely characterized and well-known efficient algorithms exist for computing them. Voronoi diagrams have been generalized by replacing the Euclidean distance by a more general metric and/or relaxing the assumption that sites be single points. We consider the closest- and the furthest-site Voronoi diagrams for a set of k point sites in a simple n-gon, defined by the internal geodesic distance inside the polygon. We demonstrate that the planar map defined by either diagram is comprised of O(n + k) features of bounded complexity each and describe nearly optimal algorithms for constructing the two Voronoi diagrams. Namely, the closest-site geodesic Voronoi diagram can be computed in time O((n + k)log(n + k)log n), while O((n + k)log(n + k)) time is sufficient for the furthest-site diagram. The second part of the thesis analyzes the structure of an arrangement of flat triangles in 3-space. The combined combinatorial complexity of all non-convex cells (i.e., non-convex components of the complement of the union of the triangles), maximized over all arrangements of n triangles is shown to be roughly O($n\sp{7\over 3}$), improving the best previously known upper bound of O($n\sp{3-{1\over 49}}$) for a smaller quantity--the maximum combinatorial complexity of a single cell. Our result has applications to algorithmic motion planning, stemming from the well-known technique that transforms a polyhedral body translating in a polyhedral environment into a collection of convex polygonal plates in three-dimensional space; the set of placements of the body reachable from a starting configuration along a collision-free path corresponds to a cell in the arrangement of these plates. Thus analyzing the maximum combinatorial complexity of a single cell and obtaining a comparably efficient algorithm for its calculation constitutes a satisfactory solution to the translational motion planning just mentioned. To this end, we also consider the problem of computing a single cell or a subset of cells in a three-dimensional arrangement of triangles, providing a nearly worst-case optimal randomized algorithm for solving the former problem and a less efficient procedure for the latter. In addition, we examine a few special classes of arrangements for which better estimates on the maximum single-cell complexity can be deduced and where computing a cell or any collection of cells appears easier.