Efficient Perturbations for Handling Geometric Degeneracies

Abstract.

This article defines input perturbations so that an algorithm designed under certain restrictions on the input can execute on arbitrary instances. A syntactic definition of perturbations is proposed and certain properties are specified under which an algorithm executed on perturbed input produces an output from which the exact answer can be recovered. A general framework is adopted for linear perturbations, which are efficient from the point of view of worst-case complexity. The deterministic scheme of Emiris and Canny [1] was the first efficient scheme and is extended in a consistent manner, most notably to the InSphere primitive. We introduce a variant scheme, applicable to a restricted class of algorithms, which is almost optimal in terms of algebraic as well as bit complexity. Neither scheme requires any symbolic computation and both are simple to use as illustrated by our implementation of a convex hull algorithm in arbitrary dimension. Empirical results and a concrete application in robotics are presented.