Randomized Gathering of Mobile Robots with Local-Multiplicity Detection

Abstract

Let us consider the gathering problem of n anonymous and oblivious mobile robots, which requires that all robots meet in finite time at a non-predefined point. While the gathering problem cannot be solved deterministically without any additional capability to robots, randomized approach easily allows it to be solvable. However, only the randomized solution taking expected round complexity exponential of n is currently known. Motivated by this fact, we investigate the feasibility of polynomial-expected-round randomized gathering in this paper. Our first contribution is to give a negative result about the round complexity of randomized gathering. It is proved that any algorithm without no additional assumption has \({\it \Omega}(\mathrm{exp}(n))\) expected-round lower bound. This lower bound yields a question: What additional assumptions are necessary to achieve gathering in polynomial expected rounds? We address this question from the aspect of multiplicity detection. This paper newly introduces two weaker variants of multiplicity detection capability, called local-strong and local-weak multiplicity, and investigates whether those capabilities permit polynomial-expected-round gathering or not. Our second contribution is to reveal the power of local (strong/weak) multiplicity by showing that local-strong multiplicity detection allows O(n)-expected round gathering but local-weak multiplicity detection takes an exponential-time lower bound. These results imply that those two kinds of multiplicity-detection capabilities have inherently large difference about their computational powers.