A modified Graham’s convex hull algorithm for finding the connected orthogonal convex hull of a finite planar point set

Abstract

Graham’s convex hull algorithm outperforms the others on those distributions where most of the points are on or near the boundary of the hull (Allison and Noga, 1984). To use this algorithm for finding an orthogonal convex hull of a finite planar point set, we introduce the concept of extreme points of a connected orthogonal convex hull of the set, and show that these points belong to the set. Then we prove that the connected orthogonal convex hull of a finite set of points is an orthogonal (x,y)-polygon where its convex vertices are its connected orthogonal convex hull’s extreme points. As a result, an efficient algorithm, based on the idea of Graham’s convex hull algorithm, for finding the connected orthogonal convex hull of a finite planar point set is presented. We also show that the lower bound of computational complexity of such algorithms is O(nlogn). Some numerical results for finding the connected orthogonal convex hulls of random sets are given.