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arXiv · 2607.09917

The Quick Dog Jumps the Log

Lotte Blank·Anne Driemel·Sariel Har-Peled·Marena Richter
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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.

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Lotte Blank, Anne Driemel, Sariel Har-Peled, Marena Richter. 2026-07-10. The Quick Dog Jumps the Log. https://arxiv.org/abs/2607.09917

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