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Progress on a perimeter surveillance problem

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Document pages: 15 pages

Abstract: We consider a perimeter surveillance problem introduced by Kingston, Beard,and Holt in 2008 and studied by Davis, Humphrey, and Kingston in 2019. In thisproblem, $n$ drones surveil a finite interval, moving at uniform speed andexchanging information only when they meet another drone. Kingston, Beard, andHolt described a particular online algorithm for coordinating their behaviorand asked for an upper bound on how long it can take before the drones arefully synchronized. They divided the algorithm s behavior into two phases, andpresented upper bounds on the length of each phase based on conjecturedworst-case configurations. Davis, Humphrey, and Kingston presentedcounterexamples to the conjecture for phase 1.We present sharp upper bounds on phase 2 which show that in this case theconjectured worst case is correct. We also present new lower bounds on phase 1and the total time to synchronization, and report partial progress towardsobtaining an upper bound.

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