Time Complexity of Sensor-Based Vehicle Routing

In this paper, we study the following motion coordination problem: given n vehicles and n origin-destination pairs in the plane, what is the minimum time needed to transfer each vehicle from its origin to its destination, avoiding conflicts with other vehicles? The environment is free of obstacles and a conflict occurs when distance between any two vehicles is smaller than a velocity-dependent safety distance. In the case where the origin and destination points can be chosen arbitrarily, we show that the transfer takes Θ( √ nL̄) time to complete, where L̄ is the average distance between the origin and destination points. We also analyze the case in which origin and destination points are generated randomly according to a uniform distribution, and present an algorithm providing a constructive upper bound on the time needed to transfer vehicles from origins to their corresponding destination, proving that the transfer takes Θ( √ n) time for this case.