Round-Tour Voronoi Diagrams

This paper proposes a new generalization of the Voronoi diagram. Suppose that restaurants and bookstores are located in a city, and we want to visit both a restaurant and bookstore and return to our house. To each pair of a restaurant and a bookstore, we can assign a region such that a resident in this region can visit the restaurant and bookstore in a round tour that is shorter than that for a visit to any other pair. The city is partitioned into these regions according to which pair of a restaurant and bookstore permits the shortest round tour. We call this partitioning a ``round-tour Voronoi Diagram" for the restaurants and bookstores. We study the basic properties of this Voronoi diagram and consider an efficient algorithm for its approximate construction.