Improved Dispersion of Mobile Robots on Arbitrary Graphs
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The dispersion problem on graphs asks k≤ n robots placed initially arbitrarily on the nodes of an n-node graph to reposition autonomously to reach a configuration in which each robot is at 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, etc. The goal is to develop both memory- and time-efficient algorithms. We provide an algorithm solving dispersion in O({m,Δ k}) time using O( ({Δ, k}))-bits at each robot in the robot-only memory model -- the robots have memory but not the nodes of the graph, where m is the number of edges and Δ is the maximum degree of any node in the graph. The runtime is optimal for bounded-degree graphs (Δ=O(1)) and improves significantly the O(mn) time of the best previously known algorithm. We provide two algorithms solving dispersion in the whiteboard model -- each node of the graph also has memory in addition to the memory at each robot. The first algorithm has O(m) time with O(Δ)-bits at each node and O( k)-bits at each robot. The second algorithm has O({m,Δ k}) time with O( ({Δ,k}))-bits at each node and O( k)-bits at each robot. These are the first terminating algorithms for dispersion. The second algorithm is time-optimal for bounded-degree graphs. We provide an algorithm solving dispersion in O(k) time in the whiteboard model enhanced with local messaging -- a robot visiting a node can send messages to the neighbors of that node, with O(Δ)-bits at each node and O( k)-bits at each robot. This is the first time-optimal algorithm for arbitrary graphs with termination guarantees.
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