Largest Convex Hulls for Constant Size, Convex-Hull Disjoint Clusters

Abstract

A cluster is a set of points, with a predefined similarity measure. In this paper, we study the problem of computing the largest possible convex hulls, measured by length and by area, of the points that are selected from a set of convex-hull disjoint clusters, one per cluster. We show that the largest convex hulls for convex-hull disjoint clusters of constant size, measured by length or area, can be computed in $O(n^4)$ time, where n is the sum of cardinalities of all clusters. Our solution of either considered problem is doubly founded on a structure of clusters, whose all points are in convex position. The restricted problem for the set of clusters, whose points are in convex position, can be reduced to a sequence of subproblems of computing the single-source shortest-paths in a weighted graph. Not only our results significantly improve upon the known time bound $O(n^9)$, but also our algorithms can be used to improve the known results on the problems of computing largest convex hulls for disjoint line segments or squares.

The work by Tan was partially supported by JSPS KAKENHI Grant Number 20K11683.