Dynamic Voronoi Diagram for Moving Disks

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 $[0, t^{\infty })$. The proposed TOI-E algorithm computes the event history of the Voronoi diagram over the entire time horizon in $O(k_F \log n + k_C n \log n)$ time with $O(n \log n)$ preprocessing time and $O(n + k_F + k_C)$ memory for $n$ disk generators, $k_F$ edge flips, and $k_C$ disk collisions during the time horizon. Given an event history, the Voronoi diagram of an arbitrary moment $t^{\ast}<t^{\infty }$ can be constructed in $O(k^{\ast} + n)$ time where $k^{\ast}$ represents the number of events in $[0, t^{\ast})$. 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.