Shortest Paths on Convex Polyhedral Surfaces
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866911315691307008 |
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| author | Wang, Haitao |
| author_facet | Wang, Haitao |
| contents | Let $\mathcal{P}$ be the surface of a convex polyhedron with $n$ vertices. We consider the two-point shortest path query problem for $\mathcal{P}$: Constructing a data structure so that given any two query points $s$ and $t$ on $\mathcal{P}$, a shortest path from $s$ to $t$ on $\mathcal{P}$ can be computed efficiently. To achieve $O(\log n)$ query time (for computing the shortest path length), the previously best result uses $O(n^{8+ε})$ preprocessing time and space [Aggarwal, Aronov, O'Rourke, and Schevon, SICOMP 1997], where $ε$ is an arbitrarily small positive constant. In this paper, we present a new data structure of $O(n^{6+ε})$ preprocessing time and space, with $O(\log n)$ query time. For a special case where one query point is required to lie on one of the edges of $\mathcal{P}$, the previously best work uses $O(n^{6+ε})$ preprocessing time and space to achieve $O(\log n)$ query time. We improve the preprocessing time and space to $O(n^{5+ε})$, with $O(\log n)$ query time. Furthermore, we present a new algorithm to compute the exact set of shortest path edge sequences of $\mathcal{P}$, which are known to be $Θ(n^4)$ in number and have a total complexity of $Θ(n^5)$ in the worst case. The previously best algorithm for the problem takes roughly $O(n^6\log n\log^*n)$ time, while our new algorithm runs in $O(n^{5+ε})$ time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11299 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shortest Paths on Convex Polyhedral Surfaces Wang, Haitao Computational Geometry Data Structures and Algorithms Let $\mathcal{P}$ be the surface of a convex polyhedron with $n$ vertices. We consider the two-point shortest path query problem for $\mathcal{P}$: Constructing a data structure so that given any two query points $s$ and $t$ on $\mathcal{P}$, a shortest path from $s$ to $t$ on $\mathcal{P}$ can be computed efficiently. To achieve $O(\log n)$ query time (for computing the shortest path length), the previously best result uses $O(n^{8+ε})$ preprocessing time and space [Aggarwal, Aronov, O'Rourke, and Schevon, SICOMP 1997], where $ε$ is an arbitrarily small positive constant. In this paper, we present a new data structure of $O(n^{6+ε})$ preprocessing time and space, with $O(\log n)$ query time. For a special case where one query point is required to lie on one of the edges of $\mathcal{P}$, the previously best work uses $O(n^{6+ε})$ preprocessing time and space to achieve $O(\log n)$ query time. We improve the preprocessing time and space to $O(n^{5+ε})$, with $O(\log n)$ query time. Furthermore, we present a new algorithm to compute the exact set of shortest path edge sequences of $\mathcal{P}$, which are known to be $Θ(n^4)$ in number and have a total complexity of $Θ(n^5)$ in the worst case. The previously best algorithm for the problem takes roughly $O(n^6\log n\log^*n)$ time, while our new algorithm runs in $O(n^{5+ε})$ time. |
| title | Shortest Paths on Convex Polyhedral Surfaces |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2512.11299 |