A Worst-case Bound for Topology Computation of Algebraic Curves

TL;DR

This paper establishes a new worst-case complexity bound for computing the topology of algebraic plane curves, significantly improving previous bounds through advanced root isolation and analysis techniques.

Contribution

The authors present a deterministic and a randomized algorithm with improved complexity bounds for topology computation of algebraic curves, enhancing efficiency over prior methods.

Findings

Deterministic complexity: O(n^8(n+ ho^2)) bit operations.

Randomized complexity: O(n^8 ho^2) bit operations.

Improved bounds by a factor of n^2 over previous work.

Abstract

Computing the topology of an algebraic plane curve $\mathcal{C}$ means to compute a combinatorial graph that is isotopic to $\mathcal{C}$ and thus represents its topology in $\mathbb{R}^2$. We prove that, for a polynomial of degree $n$ with coefficients bounded by $2^\rho$, the topology of the induced curve can be computed with $\tilde{O}(n^8(n+\rho^2))$ bit operations deterministically, and with $\tilde{O}(n^8\rho^2)$ bit operations with a randomized algorithm in expectation. Our analysis improves previous best known complexity bounds by a factor of $n^2$. The improvement is based on new techniques to compute and refine isolating intervals for the real roots of polynomials, and by the consequent amortized analysis of the critical fibers of the algebraic curve.