Abstract:
Consider the following survival problem:Given a set of k trajectories (paths) 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 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 another robot, 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 of trajectories which can be avoided in the region (resp. graph). We give the first estimates on the survival number of the n x 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 survival number of Gn on the number of paths that can be avoided in Gn can be extended for the survival number of a bounded (square) 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.