General Analysis Tool Box for Controlled Perturbation Algorithms and Complexity and Computation of Θ-Guarded Regions

Abstract

This thesis belongs to the field of computational geometry and addresses the following two issues. 1. The implementation of reliable and efficient geometric algorithms is a challenging task. Controlled perturbation combines the speed of floating-point arithmetic with a mechanism that guarantees reliability. We present a general tool box for the analysis of controlled perturbation algorithms. This tool box is separated into independent components. We present three alternative approaches for the derivation of the most important bounds. Furthermore, we have included polynomial-based predicates, rational function-based predicates, and object-preserving perturbations into the theory. Moreover, the tool box is designed such that it reflects the actual behavior of the algorithm at hand without simplifying assumptions. 2. Illumination and guarding problems are a wide field in computational and combinatorial geometry to which we contribute the complexity and computation of Θguarded regions. They are a generalization of the convex hull and are related to α-hulls and Θ-maxima. The difficulty in the study of Θ-guarded regions is the dependency of its shape and complexity on Θ. For all angles Θ, we prove fundamental properties of the region, derive lower and upper bounds on its worst-case complexity, and present an algorithm to compute the region.