General Analysis Tool Box for Controlled Perturbation

TL;DR

This paper introduces a comprehensive analysis tool box for controlled perturbation algorithms, enabling reliable geometric computations by balancing efficiency and accuracy through theoretical performance analysis and new approaches.

Contribution

It presents a modular analysis framework for controlled perturbation algorithms, including new methods for bounds derivation and extended predicate types, reflecting real algorithm behavior.

Findings

The tool box supports multiple approaches for bounds derivation.

It incorporates rational function predicates into the theory.

Object-preserving perturbations are introduced for enhanced reliability.

Abstract

The implementation of reliable and efficient geometric algorithms is a challenging task. The reason is the following conflict: On the one hand, computing with rounded arithmetic may question the reliability of programs while, on the other hand, computing with exact arithmetic may be too expensive and hence inefficient. One solution is the implementation of controlled perturbation algorithms which combine the speed of floating-point arithmetic with a protection mechanism that guarantees reliability, nonetheless. This paper is concerned with the performance analysis of controlled perturbation algorithms in theory. We answer this question with the presentation of a general analysis tool box. This tool box is separated into independent components which are presented individually with their interfaces. This way, the tool box supports alternative approaches for the derivation of the most crucial bounds. We present three approaches for this task. Furthermore, we have thoroughly reworked the concept of controlled perturbation in order to include rational function based predicates into the theory; polynomial based predicates are included anyway. Even more we introduce object-preserving perturbations. Moreover, the tool box is designed such that it reflects the actual behavior of the controlled perturbation algorithm at hand without any simplifying assumptions.