An Optimal Algorithm for Computing Many Faces in Line Arrangements
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915835863367680 |
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| author | Wang, Haitao |
| author_facet | Wang, Haitao |
| contents | Given a set of $m$ points and a set of $n$ lines in the plane, we consider the problem of computing the faces of the arrangement of the lines that contain at least one point. In this paper, we present an $O(m^{2/3}n^{2/3}+(n+m)\log n)$ time algorithm for the problem. We also show that this matches the lower bound under the algebraic decision tree model and thus our algorithm is optimal. In particular, when $m=n$, the runtime is $O(n^{4/3})$, which matches the worst case combinatorial complexity $Ω(n^{4/3})$ of all output faces. This is the first optimal algorithm since the problem was first studied more than three decades ago [Edelsbrunner, Guibas, and Sharir, SoCG 1988]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04863 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Optimal Algorithm for Computing Many Faces in Line Arrangements Wang, Haitao Computational Geometry Given a set of $m$ points and a set of $n$ lines in the plane, we consider the problem of computing the faces of the arrangement of the lines that contain at least one point. In this paper, we present an $O(m^{2/3}n^{2/3}+(n+m)\log n)$ time algorithm for the problem. We also show that this matches the lower bound under the algebraic decision tree model and thus our algorithm is optimal. In particular, when $m=n$, the runtime is $O(n^{4/3})$, which matches the worst case combinatorial complexity $Ω(n^{4/3})$ of all output faces. This is the first optimal algorithm since the problem was first studied more than three decades ago [Edelsbrunner, Guibas, and Sharir, SoCG 1988]. |
| title | An Optimal Algorithm for Computing Many Faces in Line Arrangements |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2603.04863 |