Cops and robber on grids and tori: basic algorithms and their extension to a large number of cops

Abstract

The studies of the classical cops and robber problem are generally aimed at determining the minimum number of cops needed to capture the robber, and proposing algorithms for the capture. This paper is a contribution to this problem, directed to two-dimensional grids, cylinders (i. e., grids with toroidal closure in one dimension), and tori, with a new extension to using teams of any number of cops. We discuss some new features of the problem and propose a new solution to the capture problem on grids that was already solved under a different approach, and then give efficient capture algorithms on cylinders and tori making use of these features. We examine the effect of using teams of any number k of cops and give efficient algorithms for this case, evaluating lower and upper bounds on the capture time $$t_k$$ , and compute the minimum value of k needed for any given capture time. To this aim we extend the concept of work $$w_k=k\cdot t_k$$ of an algorithm, inherited from parallel processing, and study a possible speed-up phenomenon using larger teams of cops.