The Quick Dog Jumps the Log

2026-07-10

AI Summary

This work addresses the long-standing issue of redundant logarithmic overhead in computing the Fréchet distance between c-packed curves by introducing the first linear-time (1+ε)-approximation algorithm. By integrating domain decomposition into rectangular regions, a linear-size approximation of the altitude function, and implicit dynamic programming, the proposed method uniformly handles strong, weak, discrete, and continuous variants of the Fréchet distance within O(cn/ε) time. In contrast to prior approaches, this algorithm eliminates extraneous logarithmic factors, achieving— for the first time—theoretically optimal linear-time approximation while maintaining structural simplicity and practical efficiency.

Abstract

We give linear-time, and thus optimal, $(1+\varepsilon)$-approximation algorithms for numerous variants of the Frechet distance between $c$-packed curves (where $c \in O(1)$), removing an additional log factor that was present in previous algorithms. The key to our new algorithms is a linear-size approximation of the elevation function, which uses a decomposition of the domain into rectangles, and a careful implicit dynamic programming on this decomposition. The algorithm extends to the strong, weak, discrete, and continuous Frechet distances with a running time of roughly $O(cn/\varepsilon)$. The $c$-packedness assumption is used only in the analysis, and the algorithm is simple and should work efficiently for other inputs.

Problem

• Fréchet distance
• c-packed curves
• approximation algorithms
• linear-time algorithms

Innovation

• Fréchet distance
• c-packed curves
• linear-time approximation
• implicit dynamic programming
• elevation function