Efficient dispersion of mobile robots on graphs

Abstract

The dispersion problem on graphs requires k robots placed arbitrarily at the n nodes of an anonymous graph, where k ≤ n, to coordinate with each other to reach a final configuration in which each robot is at a distinct node of the graph. The dispersion problem is important due to its relationship to graph exploration by mobile robots, scattering on a graph, and load balancing on a graph. In addition, an intrinsic application of dispersion has been shown to be the relocation of self-driven electric cars (robots) to recharge stations (nodes). We propose five algorithms to solve dispersion on graphs. The first three algorithms require O(k log Δ) bits at each robot and O(m) steps running time, where m is the number of edges and Δ is the degree of the graph. The algorithms differ in whether they address the synchronous or the asynchronous system model, and in what, where, and how data structures are maintained. The fourth algorithm, for the asynchronous model, has a space usage of O(D log Δ) bits at each robot and uses O(ΔD) steps, where D is the graph diameter. The fifth algorithm, for the asynchronous model, has a space usage of O(max(log k, log Δ)) bits at each robot and uses O((m - n)k) steps.