A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets

TL;DR

This paper introduces a probabilistic algorithm for computing roadmaps in smooth, bounded real algebraic sets, achieving near-optimal efficiency with polynomial output size and subquadratic running time, advancing algebraic geometry and motion planning.

Contribution

It presents the first roadmap algorithm with output size and running time polynomial in (nD)^{n d} for smooth, bounded real algebraic sets, improving efficiency in algebraic geometry.

Findings

Algorithm computes roadmaps with polynomial size.

Running time is essentially subquadratic in output size.

First such algorithm with these efficiency guarantees.

Abstract

A roadmap for a semi-algebraic set $S$ is a curve which has a non-empty and connected intersection with all connected components of $S$. Hence, this kind of object, introduced by Canny, can be used to answer connectivity queries (with applications, for instance, to motion planning) but has also become of central importance in effective real algebraic geometry, since it is used in higher-level algorithms. In this paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets. Its output size and running time are polynomial in $(nD)^{n\log(d)}$, where $D$ is the maximum of the degrees of the input polynomials, $d$ is the dimension of the set under consideration and $n$ is the number of variables. More precisely, the running time of the algorithm is essentially subquadratic in the output size. Even under our assumptions, it is the first roadmap algorithm with output size and running time polynomial in $(nD)^{n\log(d)}$.