An Output Sensitive Algorithm for Discrete Convex Hulls

TL;DR

This paper introduces an output-sensitive algorithm for computing the discrete convex hull of a convex body in the plane, with a runtime depending on the size of the hull and the body's diameter.

Contribution

It presents a novel $O( |C^0| ext{log} ext{D}(C) )$-time algorithm for discrete convex hulls, improving efficiency based on geometric bounds.

Findings

Algorithm runs in $O( |C^0| ext{log} ext{D}(C) )$ time

Runtime is $O( ext{D}(C)^{2/3} ext{log} ext{D}(C))$ using known bounds

Efficient for convex bodies like disks

Abstract

$\def\DD{{{\bf \delta}}}\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.