Constant Approximation of Fréchet Distance in Near-Linear Time

Abstract

We give the first constant-factor approximation for the Fréchet distance in near-linear time. Given two polygonal curves in $\mathbb{R}^d$ with $n$ and $m$ vertices, respectively, where $m\le n$, our deterministic algorithm runs in $O(n^{1+\varepsilon})$ time for every fixed $\varepsilon>0$ and $d$, with an approximation factor depending only on $\varepsilon$ and $d$. The same result holds for the discrete Fréchet distance. Our result extends the line of work initiated by Cheng, Huang, and Zhang (STOC 2025) from strongly subquadratic to near-linear time. Our algorithm recursively balances preprocessing and query time. An index for comparing subcurves of a single curve supports faster queries between different curves. We obtain this speedup by replacing one curve with surrogates assembled from indexed subcurves of the other and shared across starting points of matchings. Faster queries then reduce the cost of building the next index. Compact boundary reachability summaries allow the answers to be reused in later recursive calls. After a number of rounds depending only on $\varepsilon$, both preprocessing and query time are near-linear.