Efficient Dispersion of Mobile Robots on Dynamic Graphs

The dispersion problem on graphs asks k ≤n robots placed initially arbitrarily on the nodes of an n-node anonymous graph to reposition autonomously to reach a configuration in which each robot is on a distinct node of the graph. This problem is of significant interest due to its relationship to other fundamental robot coordination problems, such as exploration, scattering, load balancing, and relocation of self-driving electric cars (robots) to recharge stations (nodes). The objective is to simultaneously minimize (or provide trade-off between) two fundamental performance metrics: (i) time to achieve dispersion and (ii) memory requirement at each robot. This problem has been relatively well-studied on static graphs. In this paper, we investigate it for the very first time on dynamic graphs. Particularly, we show that, even with unlimited memory at each robot and 1-neighborhood knowledge, dispersion is impossible to solve on dynamic graphs in the local communication model, where a robot can only communicate with other robots that are present at the same node. We then show that, even with unlimited memory at each robot but without 1-neighborhood knowledge, dispersion is impossible to solve in the global communication model, where a robot can communicate with any other robot in the graph possibly at different nodes. We then consider the global communication model with 1-neighborhood knowledge and establish a tight bound of Θ(k) on the time complexity of solving dispersion in any n-node arbitrary anonymous dynamic graph with Θ(log k) bits memory at each robot. Finally, we extend the fault-free algorithm to solve dispersion for (crash) faulty robots under the global model with 1-neighborhood knowledge.