Optimal gathering of robots in anonymous butterfly networks via leader election

Robots with very weak capabilities placed on the vertices of a graph are required to move toward a common vertex from where they do not move anymore. The task is known as the Gathering problem and it has been extensively studied in the last decade with respect to both general graphs and specific topologies. Most of the challenges faced are due to possible isometries observable from the placement of the robots with respect to the underlying topology. Rings, Grids, and Complete graphs are just a few examples of very regular topologies where the placement of the robots and suitable movements are crucial for succeeding in Gathering. Here we are interested in understanding what can be done in Butterfly graphs where really many isometries are present and most importantly unavoidable by any movement. We propose a Gathering algorithm for the so-called leader configurations, i.e., those where the initial placement of the robots admits the detection (and election) of one robot as the leader. We introduce a non-trivial technique to elect the leader which is of its own interest. We also prove that the proposed Gathering algorithm is asymptotically optimal in terms of synchronous rounds required.