A Strongly-Subquadratic $(3+\varepsilon)$-Approximation for the Fréchet Distance for Paths in Metric Spaces

2026-07-09

AI Summary

This work breaks through the theoretical barrier imposed by the Strong Exponential Time Hypothesis (SETH) for approximating the Fréchet distance, presenting the first deterministic algorithm that achieves a $(3+\varepsilon)$-approximation in strongly subquadratic time within general metric spaces. Relying solely on standard properties of the free space and eschewing any additional metric assumptions, the approach combines divide-and-conquer strategies, refined grid discretization, and geometric analysis of feasible paths. In $\mathbb{R}^d$, the algorithm runs in $O(nm^{2/3} \log n \cdot \log (\frac{1}{\varepsilon} \log n))$ time; in one-dimensional Euclidean space, it further yields a tight 3-approximation in $O(nm^{2/3} \log^{5/3} n)$ time, thereby matching the SETH-based lower bound at approximation ratio 3 for the first time.

Abstract

The Fréchet distance is a well-studied distance measure for paths in a metric space. It is mostly studied for paths in $d$-dimensional Euclidean space. Here, computing the Fréchet distance between two polylines takes time roughly quadratic in the number of vertices. Assuming the strong exponential time hypothesis (SETH), it cannot be approximated to within a factor less than $3$ in strongly-subquadratic time. Recently, it was shown that for any $\varepsilon>0$, there exists a randomized algorithm that can compute a $(7+\varepsilon)$-approximation in strongly-subquadratic expected time [Cheng, Huang, and Zhang; STOC'25]. For polylines with $n$ and $m$ vertices in a Euclidean space of constant dimension, where $n \geq m$, their algorithm takes $O(nm^{0.99} \log(n/\varepsilon))$ time in expectation. We present a deterministic approximation algorithm that significantly improves upon the approximation factor and running time. Specifically, our algorithm computes a $(3+\varepsilon)$-approximation in $O(nm^{2/3} \log n \cdot \log (\frac{1}{\varepsilon} \log n))$ time. Our algorithm nearly matches the conditional lower bound on the approximation factor implied by SETH. For polylines in $\mathbb{R}$, we present a $3$-approximation algorithm that runs in $O(nm^{2/3} \log^{5/3} n)$ time, and exactly matches the conditional lower bound. For our results, we introduce a general strongly-subquadratic time $3$-approximate decision algorithm. This algorithm makes no assumptions on the ambient metric space, and relies only on standard assumptions on the so-called free space of the input paths. Under some mild assumptions, our decision algorithm leads to a $(3+\varepsilon)$-approximation algorithm in general metric spaces. These assumptions hold automatically for polylines in any metric space $(\mathbb{R}^d, L_p)$ with $p \geq 1$.

Problem

• Fréchet distance
• strongly-subquadratic time
• approximation algorithm
• metric spaces
• conditional lower bound

Innovation

• Fréchet distance
• strongly-subquadratic algorithm
• approximation algorithm
• conditional lower bound
• metric spaces