Offline variants of the "lion and man" problem

Adrian Dumitrescu, Ichiro Suzuki, Paweł Żyliński
2008 Theoretical Computer Science  
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
more » ... : 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 G n , 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 G n 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.
doi:10.1016/j.tcs.2008.02.039 fatcat:yb47lyztozftnly3gpwjdh3m7y