Dynamic minimum bichromatic separating circle

Abstract

In the Minimum Bichromatic Separating Circle problem, given a set R of n red points and a set B of m blue points, the goal is to find the smallest radius circle C that contains all red points and the smallest possible number of blue points. In this paper we study the dynamic version of the problem, which we call the Dynamic Minimum Bichromatic Separating Circle problem, and present data structures that allow to perform efficient reporting and updates under insertion and removal of blue points. We first describe a unified, linear size data structure that allows both insertions and deletions of blue points. Then, we present a data structure that allows to report an optimal circle in O(log⁡(mn)) time when only insertions of blue points are allowed, at the expense of an increase in update time and storage. Finally, we give a data structure that achieves O(log2⁡(mn)) reporting time when only deletions of blue points are allowed, with the same update time and storage as the insertions only data structure. Our results are the first known for the Dynamic Minimum Bichromatic Separating Circle problem.