Fence Patrolling with Two-speed Robots.

A fence, represented by a unit interval is to be patrolled collectively by n robots. At any moment a robot may move in one of the two possible states: walking or patrolling. Each state is associated with a maximal moving speed which cannot be exceeded. A robot may have a unique pair of speeds, but its patrolling speed is always smaller than its walking speed. Each robot is allowed to patrol while moving only in one of the two directions (not necessarily the same for all robots). We want to schedule the perpetual movements of the robots so as to minimize the idleness, defined as the smallest time interval within which every point is always visited by some robot. First, we give a centralized algorithm constructing schedules with optimal idleness, and subsequently we show a nice application to a transportation problem concerning Scheduling with Regular Delivery. Our main contribution is the study of distributed, dynamical schedules for patrolling robots with only primitive capabilities. Surprisingly we are able to design a dynamic schedule for very weak collections of two robots (silent, oblivious, passively mobile), achieving the optimal idleness. Our algorithm defines a dynamical system of memoryless robots moving back and forth in an interval. In general, analysis of the system dynamics is very complex. Part of our contribution is a very technical analysis of the dynamics of special families of dynamical systems of n robots that we call regular. For such systems we also propose a highly non-trivial O(n) algorithm to decide whether or not robots converge to a stable configuration thus verifying if the dynamic schedule is optimal. It turns out that a very natural family of dynamical systems that we call monotone can be shown to be regular. Further, for n ≤ 4, such monotone dynamical systems are shown to converge to stable configurations.