A general approach to the analysis of controlled perturbation algorithms

Abstract

Controlled Perturbation (CP, for short) is an approach to obtaining efficient and robust implementations of a large class of geometric algorithms using the computational speed of multiple precision floating point arithmetic (compared to exact arithmetic), while bypassing the precision problems by perturbation. It also allows algorithms to be written without consideration of degenerate cases. CP replaces the input objects by a set of randomly perturbed (moved, scaled, stretched, etc.) objects and protects the evaluation of geometric predicates by guards. The execution is aborted if a guard indicates that the evaluation of a predicate with floating point arithmetic may return an incorrect result. If the execution is aborted, the algorithm is rerun on a new perturbation and maybe with a higher precision of the floating point arithmetic. If the algorithm runs to completion, it returns the correct output for the perturbed input. The analysis of CP algorithms relates various parameters: the perturbation amount, the arithmetic precision, the range of input values, and the number of input objects. We present a general methodology for analyzing CP algorithms. It is powerful enough to analyze all geometric predicates that are formulated as signs of polynomials.