Offline variants of the “lion and man” problem: —Some problems and techniques for measuring crowdedness and for safe path planning—

Consider the following safe path planning problem: Given a set of trajectories (paths) of k point robots with maximum unit speed in a bounded region over a (long) time interval [0,T], find another trajectory (if it exists) subject to the same maximum unit speed limit, that avoids (that is, stays at a safe distance of) each of the other k trajectories over the entire time interval. We call this variant the continuous model of the safe path planning problem. The discrete model of this problem is: Given a set of trajectories (paths) of k point robots in a graph over a (long) time interval 0,1,2,…,T, find a trajectory (path) for another robot, that avoids each of the other k at any time instant in the given time interval.We introduce the notions of the avoidance number of a region, and that of a graph, respectively, as the maximum number of trajectories which can be avoided in the region (respectively, graph). We give the first estimates on the avoidance number of the n×n grid Gn, and also devise an efficient algorithm for the corresponding safe path planning problem in arbitrary graphs. We then show that our estimates on the avoidance number of Gn can be extended for the avoidance number of a bounded (fat) region. In the final part of our paper, we consider other related offline questions, such as the maximum number of men problem and the spy problem.