Notes on the Dynamic Bichromatic All-Nearest-Neighbors Problem

Given a set S of n points in the plane, each point having one of c colors, the bichromatic all-nearestneighbors problem is the task to find (in the set S) a closest point of different color for each of the n points in S. We consider a dynamic variant of this problem where the points are fixed but can change color. More precisely, we consider restricted problem instances, which allow us to improve over the time needed for solving the problem from scratch after each color change. In these variants we maintain, in O(n) time per color change, a data structure of size O(cn) or O(n), with which the closest neighbor of different color of any point in S can be found in time O(log n), and the restrictions allow us to bound the number of look ups that are necessary in each step.