Offline variants of the "lion and man" problem

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