Efficient Algorithms to Test Digital Convexity

Abstract

A set $S\subset {\mathbb{Z}}^d$ is digital convex if $conv(S)\cap {\mathbb{Z}}^d=S$, where $conv(S)$ denotes the convex hull of $S$. In this paper, we consider the algorithmic problem of testing whether a given set $S$ of $n$ lattice points is digital convex. Although convex hull computation requires $\Omega (n\log n)$ time even for dimension $d=2$, we provide an algorithm for testing the digital convexity of $S\subset {\mathbb{Z}}^2$ in $O(n+h\log r)$ time, where $h$ is the number of edges of the convex hull and $r$ is the diameter of $S$. This main result is obtained by proving that if $S$ is digital convex, then the well-known quickhull algorithm computes the convex hull of $S$ in linear time. In fixed dimension $d$, we present the first polynomial algorithm to test digital convexity, as well as a simpler and more practical algorithm whose running time may not be polynomial in $n$ for certain inputs.