Approximating the (Continuous) Fr\'echet Distance

TL;DR

This paper introduces a novel subquadratic time algorithm for approximating the Fréchet distance between polygonal chains with a significantly improved approximation ratio and running time, advancing computational geometry methods.

Contribution

The authors present the first strongly subquadratic algorithm with subexponential approximation for the Fréchet distance, including a method to convert decision procedures into optimization algorithms.

Findings

Achieves an $O(n)$-approximation in near-linear time.

Provides a deterministic $O(rac{n^3}{eta^2} ext{log} n)$ time algorithm for $O(eta)$-approximate Fréchet correspondence.

Transforms approximate decision procedures into approximate optimization algorithms with minimal overhead.

Abstract

We describe the first strongly subquadratic time algorithm with subexponential approximation ratio for approximately computing the Fr\'echet distance between two polygonal chains. Specifically, let $P$ and $Q$ be two polygonal chains with $n$ vertices in $d$-dimensional Euclidean space, and let $\alpha \in [\sqrt{n}, n]$. Our algorithm deterministically finds an $O(\alpha)$-approximate Fr\'echet correspondence in time $O((n^3 / \alpha^2) \log n)$. In particular, we get an $O(n)$-approximation in near-linear $O(n \log n)$ time, a vast improvement over the previously best know result, a linear time $2^{O(n)}$-approximation. As part of our algorithm, we also describe how to turn any approximate decision procedure for the Fr\'echet distance into an approximate optimization algorithm whose approximation ratio is the same up to arbitrarily small constant factors. The transformation into an approximate optimization algorithm increases the running time of the decision procedure by only an $O(\log n)$ factor.