Transfer time complexity of conflict-free vehicle routing with no communications

The following motion coordination problem is studied: given n mobile vehicles and n source-destination pairs in the plane, what is the minimum time needed to transfer each vehicle from its source to its destination, avoiding conflicts with other vehicles? In the proposed model, vehicles do not explicitly communicate their intentions, and only have sensory information about the current position and velocity of their neighbors to ensure no conflicts. The environment is free of obstacles and a conflict occurs when the distance between any two vehicles is smaller than a velocity-dependent safety distance. The situation analyzed in which the vehicle size is such that at least a constant fraction of the n vehicles can befitted inside the environment simultaneously lit the best case in which the source and destination points can be chosen ideally to maximize the transfer efficiency, it is shown that the transfer takes Theta((L) over bar root(n) over bar) time to complete, where (L) over bar is the average distance between the source and destination points. It is shown that there exist a worst case distribution of the source and destination points, for which the tran, fer of vehicles takes at least Omega(n) nine. The case is also analyzed in which source and destination points are generated randomly according to a uniform distribution, and an algorithm is presented providing a constructive upper bound on the time needed to transfer vehicles from sources to their corresponding destinations, proving that the transfer takes Theta(root(n) over bar) time, with high probability, thus recovering the best case performance.