The cost of probabilistic agreement in oblivious robot networks
Résumé
In this paper, we look at the time complexity of two agreement problems in networks of oblivious mobile robots, namely, at the gathering and scattering problems. Given a set of robots with arbitrary initial locations and no initial agreement on a global coordinate system, gathering requires that all robots reach the exact same but not predetermined location. In contrast, scattering requires that no two robots share the same location. These two abstractions are fundamental coordination problems in cooperative mobile robotics. Oblivious solutions are appealing for self-stabilization since they are self-stabilizing at no extra cost. As neither gathering nor scattering can be solved deterministically under arbitrary schedulers, probabilistic solutions have been proposed recently.
The contribution of this paper is twofold. First, we propose a detailed time complexity analysis of a modified probabilistic gathering algorithm. Using Markov chains tools and additional assumptions on the environment, we prove that the convergence time of gathering can be reduced from $O(n^2)$ (the best known bound) to $O(1)$ or $O(\log{n} \log(\log{n}))$, depending on the model of multiplicity detection. Second, using the same technique, we prove that scattering can also be achieved in fault-free systems with the same bounds.