Distance-2-Dispersion: Dispersion with Further Constraints

The aim of the dispersion problem is to place a set of k(≤n) 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. However, if a maximal independent set is already formed by the nodes which contain a robot each, then any other unsettled robot, if exists, 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 k≥ n, it is guaranteed that the nodes with robots form a maximal independent set of the underlying network. The graph G= (V, E) is a port-labelled graph having n nodes and m edges, where nodes are anonymous. The robots have unique ids in the range [1, L], where L≥ k. 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 2 Δ(8 m- 3 n+ 3 ) synchronous rounds using O(log Δ) memory per robot without using any global knowledge of the graph parameters m, n and Δ, the maximum degree of the graph. We also provide Ω(mΔ) lower bound on the number of rounds for the D-2-D problem.