Dispersion of mobile robots on graphs in the asynchronous model - Kshemkalyani, Ajay D. (Theoretical computer science, 2025)
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. Prior work on solving dispersion assumed the synchronous model. We propose four algorithms to solve dispersion on graphs in the asynchronous model. The first two algorithms require O(klog⁡Δ) bits at each robot and O(min⁡(m,kΔ)) steps running time, where m is the number of edges and Δ is the maximum degree of the graph. The algorithms differ in what, where, and how data structures are maintained. The third algorithm has a space usage of O(max⁡(min⁡(D,k)⋅log⁡Δ,log⁡D)) bits at each robot and uses O(Δmin⁡(D,k)+1) steps, where D is the graph diameter. The fourth algorithm has a space usage of O(max⁡(log⁡k,log⁡Δ)) bits at each robot and uses O(min⁡(m,kΔ)⋅k) steps. In contrast with existing works which all assume the synchronous model, these are the first algorithms to solve dispersion in the weaker but more realistic asynchronous model.