The cost of probabilistic gathering in oblivious robot networks

In this paper we address the complexity issues of two agreement problems in oblivious robot networks namely gathering and scattering. These abstractions are fundamental coordination problems in cooperative mobile robotics. Moreover, their oblivious characteristics makes them appealing for self-stabilization since they are self-stabilizing with no extra-cost. Given a set of robots with arbitrary initial location and no initial agreement on a global coordinate system, gathering requires that all robots reach the exact same but not predetermined location while scattering aims at scatter robots such that no two robots share the same location. Both deterministic gathering and scattering have been proved impossible under arbitrary schedulers therefore probabilistic solutions have been recently proposed. The contribution of this paper is twofold. First, we propose a detailed complexity analysis of the existent probabilistic gathering algorithms in both fault-free and fault-prone environments. We consider both crash and byzantine-prone environments. Moreover, using Markov chains tools and additional assumptions on the environment we prove that the gathering convergence time can be reduced from O(n²⁾ (the best known tight bound) to O(nln(n)). Additionally, we prove that in crash-prone environments gathering is achieved in O(nln(n)+2f). Second, using the same technique we prove that the best known scattering strategy converges in fault-free systems is O(n) (which is one to optimal) while in crash-prone environments it needs O(n-f). Finally, we conclude the paper with a discussion related to different strategies to gather oblivious robots.