Efficient Dispersion of Mobile Robots on Arbitrary Graphs and Grids

The mobile robot dispersion problem on graphs asks $k\leq n$ 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-driven electric cars (robots) to recharge stations (nodes). In this paper, we provide two novel deterministic algorithms for dispersion, one for arbitrary graphs and another for grid graphs, in a synchronous setting where all robots perform their actions in every time step. Our algorithm for arbitrary graphs has $O(\min(m,k\Delta) \cdot \log k)$ steps runtime using $O(\log n)$ bits of memory at each robot, where $m$ is the number of edges and $\Delta$ is the maximum degree of the graph. This is an exponential improvement over the $O(mk)$ steps best previously known algorithm. In particular, the runtime of our algorithm is optimal (up to a $O(\log k)$ factor) in constant-degree arbitrary graphs. Our algorithm for grid graphs has $O(\min(k,\sqrt{n}))$ steps runtime using $\Theta(\log k)$ bits at each robot. This is the first algorithm for dispersion in grid graphs. Moreover, this algorithm is optimal for both memory and time when $k=\Omega(n)$.