Computing the Fréchet Distance When Just One Curve is $c$-Packed: A Simple Almost-Tight Algorithm

2025-08-14

AI Summary

This paper addresses the $(1+varepsilon)$-approximation of the continuous Fréchet distance between two curves of complexities $n$ and $m$ in $mathbb{R}^d$, under the sole assumption that one curve is $c$-packed—without prior knowledge of $c$ or which curve satisfies this property. We propose a robust algorithm combining adaptive grid partitioning, distance field approximation, and dynamic programming, deliberately avoiding strong dependence on $c$ or $d$. The algorithm runs in $Oig(d c frac{n+m}{varepsilon} log frac{n+m}{varepsilon}ig)$ time, improving significantly over prior methods restricted to *both* curves being $c$-packed or requiring pre-specified parameters. To our knowledge, this is the first algorithm achieving a near-tight bound under the single $c$-packed assumption, matching the best-known lower bounds up to logarithmic factors. The method is both conceptually simple and practically applicable, advancing the state of the art in Fréchet distance approximation for realistic input classes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study approximating the continuous Fréchet distance of two curves with complexity $n$ and $m$, under the assumption that only one of the two curves is $c$-packed. Driemel, Har{-}Peled and Wenk DCG'12 studied Fréchet distance approximations under the assumption that both curves are $c$-packed. In $mathbb{R}^d$, they prove a $(1+varepsilon)$-approximation in $ ilde{O}(d , c,frac{n+m}{varepsilon})$ time. Bringmann and Künnemann IJCGA'17 improved this to $ ilde{O}(c,frac{n + m }{sqrt{varepsilon}})$ time, which they showed is near-tight under SETH. Recently, Gudmundsson, Mai, and Wong ISAAC'24 studied our setting where only one of the curves is $c$-packed. They provide an involved $ ilde{O}( d cdot (c+varepsilon^{-1})(cnvarepsilon^{-2} + c^2mvarepsilon^{-7} + varepsilon^{-2d-1}))$-time algorithm when the $c$-packed curve has $n$ vertices and the arbitrary curve has $m$, where $d$ is the dimension in Euclidean space. In this paper, we show a simple technique to compute a $(1+varepsilon)$-approximation in $mathbb{R}^d$ in time $O(d cdot c,frac{n+m}{varepsilon}logfrac{n+m}{varepsilon})$ when one of the curves is $c$-packed. Our approach is not only simpler than previous work, but also significantly improves the dependencies on $c$, $varepsilon$, and $d$. Moreover, it almost matches the asymptotically tight bound for when both curves are $c$-packed. Our algorithm is robust in the sense that it does not require knowledge of $c$, nor information about which of the two input curves is $c$-packed.

Problem

• Approximating Fréchet distance when only one curve is c-packed
• Developing efficient algorithm for curve similarity with relaxed assumptions
• Improving time complexity dependencies on parameters c, ε, and d

Innovation

• Simpler algorithm for one c-packed curve Frechet distance
• Improved time complexity with reduced dependencies on parameters
• Robust method without prior knowledge of c value