Dispersion of mobile robots on directed anonymous graphs

Abstract

Given any arbitrary initial configuration of $k\le n$ robots positioned on the nodes of an n-node anonymous graph, the problem of dispersion is to autonomously reposition the robots such that each node will contain at most one robot. This problem gained significant interest due to its resemblance with several fundamental problems such as exploration, scattering, load balancing, relocation of electric cars to charging stations, etc. The objective is to solve dispersion simultaneously minimizing (or providing trade-off between) time and memory requirement at each robot. The literature mainly dealt with dispersion on undirected anonymous graphs. In this paper, we initiate the study of dispersion on directed anonymous graphs. We first show that it may not always be possible to solve dispersion when the directed graph is not strongly connected. We then establish some lower bounds on both time and memory requirement at each robot for solving dispersion on a strongly connected directed graph. Finally, we provide three deterministic algorithms solving dispersion on any strongly connected directed graph. Let D be the graph diameter, ${\Delta }_{out}$ be its maximum out-degree, and d be the deficiency (the minimum number of edges needed to add to the graph to make it Eulerian). The first algorithm solves dispersion in $O(d\cdot k^2)$ time with $O(k\cdot \log (k+{\Delta }_{out}\left)\right)$ bits at each robot. The second algorithm solves dispersion in $O(k^2\cdot {\Delta }_{out})$ time with $O(\log (k+{\Delta }_{out}\left)\right)$ bits at each robot. The third algorithm solves dispersion in $O(k\cdot D)$ time with $O(k\cdot \log (k+{\Delta }_{out}\left)\right)$ bits at each robot, provided that robots in the 1-hop neighborhood can communicate. All three algorithms extend to handle crash faults.