Near-tight Bounds for Computing the Fréchet Distance in d-Dimensional Grid Graphs and the Implications for λ-low Dense Curves

2026-04-27

AI Summary

This work addresses the efficient approximation of the Fréchet distance between simple paths in $d$-dimensional integer grid graphs. We present a $(1+\varepsilon)$-approximation algorithm running in $\widetilde{O}((n/\varepsilon)^{2-2/d} + n)$ time and establish its near-optimality under the Orthogonal Vectors Hypothesis (OVH): unless OVH fails, no substantially faster algorithm exists. Notably, for dimensions $d \geq 3$, we provide the first matching upper and lower bounds for this setting. Furthermore, we demonstrate that the dependence on $n$ and $\varepsilon$ in Driemel et al.'s algorithm for $\lambda$-low-density curves is already optimal, thereby resolving an open question regarding the tightness of their bounds.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The Fréchet distance is a popular distance measure between trajectories or curves in space, or between walks in graphs. We study computing the Fréchet distance between walks in the $d$-dimensional grid graphs, i.e. $\mathbb{Z}^d$ where points share an edge if they differ by one in one coordinate. We give an algorithm, that for two simple paths on $n$ vertices, $(1+\varepsilon)$-approximates the Fréchet distance in time $\widetilde{O}((\frac{n}{\varepsilon})^{2-2/d} +n)$. We complement this by a near-matching fine-grained lower bound: for constant dimensions $d \geq 3$, there is no $O((\varepsilon^{2/d}(\frac{n}{\varepsilon})^{2-2/d})^{1-δ})$ algorithm for any $δ>0$ unless the Orthogonal Vector Hypothesis fails. Thus, our results are tight up to a factor $\varepsilon^{2/d}$ and $\log(n)$-factors. We extend our results to imbalanced lower and upper bounds, where the curves have $n$ and $m$ vertices respectively, and also obtain near-tight bounds. Driemel, Har-Peled and Wenk [DCG'12] studied \emph{realistic assumptions} for curves to speed up Fréchet distance computation. One of these assumptions is $λ$-low density and they can compute a $(1+\varepsilon)$-approximation between $λ$-low dense curves in time $\widetilde{O}( \varepsilon^{-2} λ^2 n^{2(1-1/d)})$. By adapting our lower bound, we show that their algorithm has a tight dependency on $n$ and a tight dependency on $\varepsilon$ as $d$ goes to infinity. A gap remains in terms of $λ$.

Problem

• Fréchet distance
• grid graphs
• fine-grained lower bounds
• λ-low density curves
• approximation algorithms

Innovation

• Fréchet distance
• grid graphs
• fine-grained complexity
• approximation algorithms
• λ-low density curves