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.