# Efficient Algorithms to Test Digital Convexity

## TL;DR

This paper introduces efficient algorithms for testing whether a set of lattice points is digitally convex, improving computational complexity especially in two dimensions and providing the first polynomial-time solution in fixed dimensions.

## Contribution

It presents a novel algorithm for digital convexity testing in 2D with improved runtime and the first polynomial algorithm for fixed dimensions, advancing computational geometry methods.

## Key findings

- Testing digital convexity in 2D can be done in $O(n + h \log r)$ time.
- The quickhull algorithm is linear time for digital convex sets.
- First polynomial-time algorithm for digital convexity testing in fixed dimensions.

## 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.