Algorithm for connectivity queries on real algebraic curves

TL;DR

This paper presents an efficient algorithm for answering connectivity queries on real algebraic curves, avoiding full topology computation and matching the best known complexity bounds.

Contribution

It introduces a novel algorithm that counts connected components and determines query point connectivity without computing the entire curve topology.

Findings

Algorithm runs in log-linear time in input size.

Matches the best known complexity bounds for topology computation.

Avoids full topology computation of the curve.

Abstract

We consider the problem of answering connectivity queries on a real algebraic curve. The curve is given as the real trace of an algebraic curve, assumed to be in generic position, and being defined by some rational parametrizations. The query points are given by a zero-dimensional parametrization. We design an algorithm which counts the number of connected components of the real curve under study, and decides which query point lie in which connected component, in time log-linear in $N^6$, where $N$ is the maximum of the degrees and coefficient bit-sizes of the polynomials given as input. This matches the currently best-known bound for computing the topology of real plane curves. The main novelty of this algorithm is the avoidance of the computation of the complete topology of the curve.