Divide and Conquer Roadmap for Algebraic Sets

TL;DR

This paper introduces a new algorithm for constructing roadmaps of algebraic sets with improved complexity bounds, enabling efficient connectivity analysis of real algebraic varieties.

Contribution

It presents a divide-and-conquer algorithm for roadmap computation with significantly better complexity bounds than previous methods.

Findings

Algorithm computes roadmaps with complexity depending on degrees and number of points

Provides upper bounds on path length and complexity in real algebraic sets

Improves upon previous algorithms by reducing dependence on the number of points

Abstract

Let $\mathrm{R}$ be a real closed field, and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We describe an algorithm that given as input a polynomial $P \in \mathrm{D} [ X_{1},\ldots,X_{k} ]$, and a finite set, $\mathcal{A}= \{ p_{1}, \ldots,p_{m} \}$, of points contained in $V= \mathrm{Zer}( P, \mathrm{R}^{k})$ described by real univariate representations, computes a roadmap of $V$ containing $\mathcal{A}$. The complexity of the algorithm, measured by the number of arithmetic operations in $\mathrm{D} $ is bounded by $\left( \sum_{i=1}^{m} D^{O ( \log^{2} ( k ) )}_{i} +1 \right) ( k^{\log ( k )} d )^{O ( k\log^{2} ( k ))}$, where $d= \mathrm{deg} ( P )$, and $D_{i}$ is the degree of the real univariate representation describing the point $p_{i}$. The best previous algorithm for this problem had complexity $\mathrm{card} ( \mathcal{A} )^{O ( 1 )} d^{O ( k^{3/2} )}$ due to Basu, Roy, Safey-El-Din, and Schost (2012), where it is assumed that the degrees of the polynomials appearing in the representations of the points in $\mathcal{A}$ are bounded by $d^{O ( k )}$. As an application of our result we prove that for any real algebraic subset $V$ of $\mathbb{R}^{k}$ defined by a polynomial of degree $d$, any connected component $C$ of $V$ contained in the unit ball, and any two points of $C$, there exist a semi-algebraic path connecting them in $C$, of length at most $( k ^{\log (k )} d )^{O ( k\log ( k ) )}$, consisting of at most $( k ^{\log (k )} d )^{O ( k\log ( k ) )}$ curve segments of degrees bounded by $( k ^{\log ( k )} d )^{O ( k \log ( k) )}$. While it was known previously, by a result of D'Acunto and Kurdyka, that there always exists a path of length $( O ( d ) )^{k-1}$ connecting two such points, there was no upper bound on the complexity of such a path.