Computing the Fréchet Distance When Just One Curve is $c$-Packed: A Simple Almost-Tight Algorithm
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912778369892352 |
|---|---|
| author | Conradi, Jacobus van der Hoog, Ivor van der Horst, Thijs Ophelders, Tim |
| author_facet | Conradi, Jacobus van der Hoog, Ivor van der Horst, Thijs Ophelders, Tim |
| contents | 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 $\tilde{O}(d \, c\,\frac{n+m}{\varepsilon})$ time. Bringmann and Künnemann IJCGA'17 improved this to $\tilde{O}(c\,\frac{n + m }{\sqrt{\varepsilon}})$ 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 $\tilde{O}( d \cdot (c+\varepsilon^{-1})(cn\varepsilon^{-2} + c^2m\varepsilon^{-7} + \varepsilon^{-2d-1}))$-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 $O(d \cdot c\,\frac{n+m}{\varepsilon}\log\frac{n+m}{\varepsilon})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10537 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computing the Fréchet Distance When Just One Curve is $c$-Packed: A Simple Almost-Tight Algorithm Conradi, Jacobus van der Hoog, Ivor van der Horst, Thijs Ophelders, Tim Computational Geometry 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 $\tilde{O}(d \, c\,\frac{n+m}{\varepsilon})$ time. Bringmann and Künnemann IJCGA'17 improved this to $\tilde{O}(c\,\frac{n + m }{\sqrt{\varepsilon}})$ 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 $\tilde{O}( d \cdot (c+\varepsilon^{-1})(cn\varepsilon^{-2} + c^2m\varepsilon^{-7} + \varepsilon^{-2d-1}))$-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 $O(d \cdot c\,\frac{n+m}{\varepsilon}\log\frac{n+m}{\varepsilon})$ 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. |
| title | Computing the Fréchet Distance When Just One Curve is $c$-Packed: A Simple Almost-Tight Algorithm |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2508.10537 |