Progress On A Perimeter Surveillance Problem

We consider a perimeter surveillance problem introduced by Kingston, Beard, and Holt in 2008 and studied by Davis, Humphrey, and Kingston in 2019. In this problem, n drones surveil a finite interval, moving at uniform speed and exchanging information only when they meet another drone. Kingston et al. described a particular online algorithm for coordinating their behavior and asked for an upper bound on how long it can take before the drones are fully synchronized. They divided the algorithm’s behavior into two phases and presented upper bounds on the length of each phase based on conjectured worst-case configurations. Davis et al. presented counterexamples to the conjecture for phase 1. We present sharp upper bounds on phase 2 which show that in this case the conjectured worst case is correct, and we report new lower bounds on phase 1.