An Improved Algorithm for Shortest Paths in Weighted Unit-Disk Graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910511733407744 |
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| author | Brewer, Bruce W. Wang, Haitao |
| author_facet | Brewer, Bruce W. Wang, Haitao |
| contents | Let $V$ be a set of $n$ points in the plane. The unit-disk graph $G = (V, E)$ has vertex set $V$ and an edge $e_{uv} \in E$ between vertices $u, v \in V$ if the Euclidean distance between $u$ and $v$ is at most 1. The weight of each edge $e_{uv}$ is the Euclidean distance between $u$ and $v$. Given $V$ and a source point $s\in V$, we consider the problem of computing shortest paths in $G$ from $s$ to all other vertices. The previously best algorithm for this problem runs in $O(n \log^2 n)$ time [Wang and Xue, SoCG'19]. The problem has an $Ω(n\log n)$ lower bound under the algebraic decision tree model. In this paper, we present an improved algorithm of $O(n \log^2 n / \log \log n)$ time (under the standard real RAM model). Furthermore, we show that the problem can be solved using $O(n\log n)$ comparisons under the algebraic decision tree model, matching the $Ω(n\log n)$ lower bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03176 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Improved Algorithm for Shortest Paths in Weighted Unit-Disk Graphs Brewer, Bruce W. Wang, Haitao Computational Geometry Data Structures and Algorithms Let $V$ be a set of $n$ points in the plane. The unit-disk graph $G = (V, E)$ has vertex set $V$ and an edge $e_{uv} \in E$ between vertices $u, v \in V$ if the Euclidean distance between $u$ and $v$ is at most 1. The weight of each edge $e_{uv}$ is the Euclidean distance between $u$ and $v$. Given $V$ and a source point $s\in V$, we consider the problem of computing shortest paths in $G$ from $s$ to all other vertices. The previously best algorithm for this problem runs in $O(n \log^2 n)$ time [Wang and Xue, SoCG'19]. The problem has an $Ω(n\log n)$ lower bound under the algebraic decision tree model. In this paper, we present an improved algorithm of $O(n \log^2 n / \log \log n)$ time (under the standard real RAM model). Furthermore, we show that the problem can be solved using $O(n\log n)$ comparisons under the algebraic decision tree model, matching the $Ω(n\log n)$ lower bound. |
| title | An Improved Algorithm for Shortest Paths in Weighted Unit-Disk Graphs |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2407.03176 |