Enumerating Empty and Surrounding Polygons

Abstract

We are given a set S of n points in the Euclidean plane. We assume that S is in general position. A simple polygon P is an empty polygon of S if each vertex of P is a point in S and every point in S is either outside P or a vertex of P. In this paper, we consider the problem of enumerating all the empty polygons of a given point set. To design an efficient enumeration algorithm, we use a reverse search by Avis and Fukuda with child lists. We propose an algorithm that enumerates all the empty polygons of S in O(n² |ϵ(S) |)-time, where ϵ(S) is the set of empty polygons of S. Moreover, by applying the same idea to the problem of enumerating surrounding polygons of a given point set S, we propose an enumeration algorithm that enumerates them in O(n²)-delay, while the known algorithm enumerates in O(n² log n)-delay, where a surrounding polygon of S is a polygon such that each vertex of the polygon is a point in S and every point in S is either inside the polygon or a vertex of the polygon.