Gathering robots in graphs: The central role of synchronicity

Abstract

The Gathering task for k robots disposed on the n vertices of a graph G requires robots to move toward a common vertex from where they do not move anymore.When dealing with very weak robots in terms of capabilities, considering synchronous or asynchronous settings may heavily affect the feasibility of the problem. In fact, even though dealing with asynchronous robots in general requires more sophisticated strategies with respect to the synchronous counterpart, sometimes it comes out that asynchronous robots simply cannot solve the problem whereas synchronous robots can.We study general properties of graphs that can be exploited in order to accomplish the gathering task in the synchronous setting, obtaining an interesting and innovative sufficient condition for the feasibility of the gathering task in graphs, regardless the topology.Furthermore, we consider dense and symmetric graphs like complete and complete bipartite graphs where in general the topology does not allow to distinguish vertices where to finalize the gathering. In such topologies, we fully characterize the solvability of the gathering task in the synchronous setting by suitably combining some strategies arising from the general approach with specific techniques dictated by the considered topologies.From the lower bound point of view in terms of number of synchronous time units required to accomplish the gathering task, in general nothing better than Ω(DG), with DG being the diameter of the input graph G, can be provided. For both complete and complete bipartite graphs, we prove a lower bound of Ω(logϕ⁡k), with ϕ being the well-known golden ratio. Combined with the provided algorithms, this reveals to be asymptotically tight for complete graphs while for complete bipartite graphs an additive factor of δ(k) is achieved, with δ(k) being the function that returns the number of divisors of the integer k.