Title of Dissertation : Approximate Range Searching In The Absolute Error Model Guilherme

Title of Dissertation: Approximate Range Searching In The Absolute Error Model Guilherme Dias da Fonseca, Doctor of Philosophy, 2007 Dissertation directed by: Professor David M. Mount Department of Computer Science Range searching is a well known problem in computational geometry. We consider this problem in the context of approximation, where an approximation parameter ε > 0 is provided. Most prior work on this problem has focused on the relative error model, where each range shape R is bounded, and points within distance ε · diam(R) of the range’s boundary may or may not be included. We introduce a different approximation model, called the absolute error model, in which points within distance ε of the range’s boundary may or may not be included, regardless of the diameter of the range. We consider sets of ranges consisting of general convex bodies, axis-aligned rectangles, halfspaces, Euclidean balls, and simplices. We examine a variety of problem formulations, including range searching under general commutative semigroups, idempotent semigroups, groups, range emptiness, and range reporting. We apply our data structures to several related problems, including range sketching, approximate nearest neighbor searching, exact idempotent range searching, approximate range searching in the data stream model, and approximate range searching in the relative model. Approximate Range Searching In The Absolute Error Model by Guilherme Dias da Fonseca Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2007