Efficient Dispersion of Mobile Robots on Graphs

The dispersion problem on graphs requires $k$ robots placed arbitrarily at the $n$ nodes of an anonymous graph, where $k \leq n$, 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 three efficient algorithms to solve dispersion on graphs. Our 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.