The Farthest Color Voronoi Diagram and Related Problems

Suppose there are k types of facilities, e. g. schools, post offices, supermarkets, modeled by n colored points in the plane, each type by its own color. One basic goal in choosing a residence location is in having at least one representative of each facility type in the neighborhood. In this paper we provide algorithms that may help to achieve this goal for various specifications of the term “neighborhood”. Several problems on multicolored point sets have been previously considered, such as the bichromatic closest pair, see e. g. Preparata and Shamos [14, Section 5.7], Agarwal et al. [1], and Graf and Hinrichs [8], the group Steiner tree, see Mitchell [11, Section 7.1], or the chromatic nearest neighbor search, see Mount et al. [12]. Let us call a set color-spanning if it contains at least one point of each color. A natural approach to the above location problem is to ask for the center of the smallest color-spanning circle. For k = n this amounts to finding the smallest circle enclosing n given points. This problem can be solved in time O(n log n) by means of the farthest site Voronoi diagram [3], in time O(n) using Megiddo’s linear programming method [10], or in randomized time O(n) by Welzl’s minidisk algorithm [16]. A special case is k = 2, then the solution is given by the bichromatic closest pair, see above. For 2 < k < n one can solve the problem as follows. Let us generalize Voronoi diagrams in the following way. If p denotes a site of color c, we put all points of the plane in the region of p for which c is the farthest color, and p the nearest c-colored site, i. e., z belongs to the