An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914078615666688 |
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| author | Brewer, Bruce W. Wang, Haitao |
| author_facet | Brewer, Bruce W. Wang, Haitao |
| contents | Given in the plane a set $S$ of $n$ points and a set of disks centered at these points, the disk graph $G(S)$ induced by these disks has vertex set $S$ and an edge between two vertices if their disks intersect. Note that the disks may have different radii. We consider the problem of computing shortest paths from a source point $s\in S$ to all vertices in $G(S)$ where the length of a path in $G(S)$ is defined as the number of edges in the path. The previously best algorithm solves the problem in $O(n\log^2 n)$ time. A lower bound of $Ω(n\log n)$ is also known for this problem under the algebraic decision tree model. In this paper, we present an $O(n\log n)$ time algorithm, which matches the lower bound and thus is optimal. Another virtue of our algorithm is that it is quite simple. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_05569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs Brewer, Bruce W. Wang, Haitao Computational Geometry Data Structures and Algorithms Given in the plane a set $S$ of $n$ points and a set of disks centered at these points, the disk graph $G(S)$ induced by these disks has vertex set $S$ and an edge between two vertices if their disks intersect. Note that the disks may have different radii. We consider the problem of computing shortest paths from a source point $s\in S$ to all vertices in $G(S)$ where the length of a path in $G(S)$ is defined as the number of edges in the path. The previously best algorithm solves the problem in $O(n\log^2 n)$ time. A lower bound of $Ω(n\log n)$ is also known for this problem under the algebraic decision tree model. In this paper, we present an $O(n\log n)$ time algorithm, which matches the lower bound and thus is optimal. Another virtue of our algorithm is that it is quite simple. |
| title | An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2507.05569 |