A baby steps/giant steps Monte Carlo algorithm for computing roadmaps in smooth compact real hypersurfaces

TL;DR

This paper introduces a new Monte Carlo algorithm with improved complexity for constructing roadmaps of smooth, compact real hypersurfaces, significantly reducing computational costs compared to previous methods.

Contribution

The paper presents a novel Monte Carlo algorithm with complexity $(nD)^{O(n^{1.5})}$ for computing roadmaps of compact hypersurfaces, improving over the prior $D^{O(n^2)}$ cost.

Findings

Achieves a complexity of $(nD)^{O(n^{1.5})}$ for hypersurfaces

Applicable to smooth, compact hypersurfaces with finite singular points

Significantly reduces computational complexity compared to previous algorithms

Abstract

We consider the problem of constructing roadmaps of real algebraic sets. The problem was introduced by Canny to answer connectivity questions and solve motion planning problems. Given $s$ polynomial equations with rational coefficients, of degree $D$ in $n$ variables, Canny's algorithm has a Monte Carlo cost of $s^n\log(s) D^{O(n^2)}$ operations in $\mathbb{Q}$; a deterministic version runs in time $s^n \log(s) D^{O(n^4)}$. The next improvement was due to Basu, Pollack and Roy, with an algorithm of deterministic cost $s^{d+1} D^{O(n^2)}$ for the more general problem of computing roadmaps of semi-algebraic sets ($d \le n$ is the dimension of an associated object). We give a Monte Carlo algorithm of complexity $(nD)^{O(n^{1.5})}$ for the problem of computing a roadmap of a compact hypersurface $V$ of degree $D$ in $n$ variables; we also have to assume that $V$ has a finite number of singular points. Even under these extra assumptions, no previous algorithm featured a cost better than $D^{O(n^2)}$.