Near-tight Bounds for Computing the Fréchet Distance in d-Dimensional Grid Graphs and the Implications for λ-low Dense Curves
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866910223203041280 |
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| author | Conradi, Jacobus van der Hoog, Ivor Uldahl, Frederikke Rotenberg, Eva |
| author_facet | Conradi, Jacobus van der Hoog, Ivor Uldahl, Frederikke Rotenberg, Eva |
| contents | 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 $λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24135 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Near-tight Bounds for Computing the Fréchet Distance in d-Dimensional Grid Graphs and the Implications for λ-low Dense Curves Conradi, Jacobus van der Hoog, Ivor Uldahl, Frederikke Rotenberg, Eva Computational Geometry Data Structures and Algorithms 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 $λ$. |
| title | Near-tight Bounds for Computing the Fréchet Distance in d-Dimensional Grid Graphs and the Implications for λ-low Dense Curves |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.24135 |