Dynamic Voronoi Diagram for Moving Disks

Abstract

Voronoi diagrams are powerful for understanding spatial properties. However, few reports have been made for moving generators despite their important applications. We present a topology-oriented event-increment (TOI-E) algorithm for constructing a Voronoi diagram of moving circular disks in the plane over the time horizon [0; t∞]. The proposed TOI-E algorithm computes the event history of the Voronoi diagram over the entire time horizon in O(kF log n þ κCnlog n) time with Oðnlog nÞ preprocessing time and O(n κF þ kC) memory for n disk generators, kF edge flips, and κC disk collisions during the time horizon. Given an event history, the Voronoi diagram of an arbitrary moment t < t∞ can be constructed in Oðκ þ nÞ time where κ represents the number of events in [; t∞]. An example of the collision avoidance problem among moving disks is given by predicting future conjunctions among the disks using the proposed algorithm. Dynamic Voronoi diagrams will be very useful as a platform for the planning and management of the traffics of unmanned vehicles such as cars on street, vessels on surface, drones and airplanes in air, and satellites in geospace.