Efficiently Testing Digital Convexity and Recognizing Digital Convex Polygons

Abstract

A set $S \subset \mathbb {Z}^2$ of integer points is digital convex if ${{\,\mathrm{conv}\,}}(S) \cap \mathbb {Z}^2 = S$, where ${{\,\mathrm{conv}\,}}(S)$ denotes the convex hull of S. In this paper, we consider the following two problems. The first one is to test whether a given set S of n lattice points is digital convex. If the answer to the first problem is positive, then the second problem is to find a polygon $P\subset \mathbb {Z}^2$ with minimum number of edges and whose intersection with the lattice $P\cap \mathbb {Z}^2$ is exactly S. We provide linear-time algorithms for these two problems. The algorithm is based on the well-known quickhull algorithm. The time to solve both problems is $O(n + h \log r) = O(n + n^{1/3} \log r)$, where $h = \min (|{{\,\mathrm{conv}\,}}(S)|, n^{1/3})$ and r is the diameter of S.