An Output Sensitive Algorithm for Discrete Convex Hulls
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912800188661760 |
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| author | Har-Peled, Sariel |
| author_facet | Har-Peled, Sariel |
| contents | $\def\DD{\bf δ}\def\CH{\mathop{\mathrm{ConvexHull}}}\newcommand{\LL}{\cal {L}} \newcommand{\ZZ}{\mathbb{Z}} $ Given a convex body $C$ in the plane, its discrete hull is $C^0 = \CH( C \cap \LL )$, where $\LL = \ZZ \times \ZZ$ is the integer lattice. We present an $O( |C^0| \log \DD(C) )$-time algorithm for calculating the discrete hull of $C$, where $|C^0|$ denotes the number of vertices of $C^0$, and $\DD(C)$ is the diameter of $C$. Actually, using known combinatorial bounds, the running time of the algorithm is $O(\DD(C)^{2/3} \log{\DD(C)})$. In particular, this bound applies when $C$ is a disk. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_00392 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Output Sensitive Algorithm for Discrete Convex Hulls Har-Peled, Sariel Computational Geometry $\def\DD{\bf δ}\def\CH{\mathop{\mathrm{ConvexHull}}}\newcommand{\LL}{\cal {L}} \newcommand{\ZZ}{\mathbb{Z}} $ Given a convex body $C$ in the plane, its discrete hull is $C^0 = \CH( C \cap \LL )$, where $\LL = \ZZ \times \ZZ$ is the integer lattice. We present an $O( |C^0| \log \DD(C) )$-time algorithm for calculating the discrete hull of $C$, where $|C^0|$ denotes the number of vertices of $C^0$, and $\DD(C)$ is the diameter of $C$. Actually, using known combinatorial bounds, the running time of the algorithm is $O(\DD(C)^{2/3} \log{\DD(C)})$. In particular, this bound applies when $C$ is a disk. |
| title | An Output Sensitive Algorithm for Discrete Convex Hulls |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2601.00392 |