Dispersion of Mobile Robots: The Power of Randomness

We consider cooperation among insects, modeled as cooperation between mobile robots on a graph. Within this setting, we consider the problem of mobile robot dispersion on graphs. The study of mobile robots on a graph is an interesting paradigm with many interesting problems and applications. The problem of dispersion in this context, introduced by Augustine and Moses Jr. [4], asks that n robots, initially placed arbitrarily on an n node graph, work together to quickly reach a configuration with exactly one robot at each node. Previous work on this problem has looked at the trade-off between the time to achieve dispersion and the amount of memory required by each robot. However, the trade-off was analyzed for deterministic algorithms and the minimum memory required to achieve dispersion was found to be \(\varOmega (\log n)\) bits at each robot. In this paper, we show that by harnessing the power of randomness, one can achieve dispersion with \(O(\log \varDelta )\) bits of memory at each robot, where \(\varDelta \) is the maximum degree of the graph. Further, we show a matching lower bound of \(\varOmega (\log \varDelta )\) bits for any randomized algorithm to solve dispersion. We further extend the problem to a general k-dispersion problem where \(k> n\) robots need to disperse over n nodes such that at most \(\lceil k/n \rceil \) robots are at each node in the final configuration.