Memory optimal distance-2-dispersion with termination

The aim of the dispersion problem is to place a set of (Formula presented.) mobile robots in the nodes of an unknown graph consisting of n nodes such that in the final configuration each node contains at most one robot, starting from any arbitrary initial configuration of the robots on the graph. In this work, we propose a variant of the dispersion problem, namely Distance-2-Dispersion, in short, D-2-D, where we start with any number of robots, and put an additional constraint that no two adjacent nodes contain robots in the final configuration. That is, the distance between any two nodes with robots must be at least 2. However, even if the number of robots k is less than n, it might be the case that it is not possible for each robot to find a distinct node to reside, maintaining our added constraint. More specifically, if a maximal independent set is already formed by the nodes containing a robot each, then any other unsettled robot will not find a node to settle. Hence we allow multiple robots to sit on some nodes only if there is no place to sit. If (Formula presented.), it is guaranteed that the nodes with robots form a maximal independent set of the underlying network. The graph (Formula presented.) is a port-labeled graph with n nodes and m edges, where nodes are anonymous. The robots have unique IDs in the range (Formula presented.), where (Formula presented.). Co-located robots can communicate among themselves. We provide an algorithm that solves D-2-D starting from a rooted configuration (i.e. initially all the robots are co-located) and terminates after (Formula presented.) synchronous rounds using (Formula presented.) memory per robot, without using any global knowledge of the graph parameters m, n and Δ, the maximum degree of the graph. We provide (Formula presented.) lower bound on the memory requirement by the robots to solve the D-2-D problem. Thus our algorithm is memory-optimal. We conjecture that the time required by the robots to solve the D-2-D problem, provided each of them have (Formula presented.) memory, is (Formula presented.) rounds. We further show that if the nodes are also equipped with a storage of (Formula presented.) bits, then even if the robots are arbitrarily positioned at the nodes of the graph in the initial configuration, they can solve the problem of D-2-D in (Formula presented.) rounds in the same setting.