Optimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots

TL;DR

This paper demonstrates that four probabilistic robots can efficiently explore anonymous rings of any size without communication, overcoming symmetry issues faced by deterministic robots and removing previous constraints.

Contribution

It introduces a probabilistic approach to ring exploration, showing four robots suffice regardless of ring size and coprimality, unlike deterministic solutions.

Findings

Four probabilistic robots are necessary and sufficient for exploration.

Probabilistic algorithms remove symmetry constraints present in deterministic methods.

The approach works for rings of any size without communication.

Abstract

We consider a team of $k$ identical, oblivious, asynchronous mobile robots that are able to sense (\emph{i.e.}, view) their environment, yet are unable to communicate, and evolve on a constrained path. Previous results in this weak scenario show that initial symmetry yields high lower bounds when problems are to be solved by \emph{deterministic} robots. In this paper, we initiate research on probabilistic bounds and solutions in this context, and focus on the \emph{exploration} problem of anonymous unoriented rings of any size. It is known that $\Theta(\log n)$ robots are necessary and sufficient to solve the problem with $k$ deterministic robots, provided that $k$ and $n$ are coprime. By contrast, we show that \emph{four} identical probabilistic robots are necessary and sufficient to solve the same problem, also removing the coprime constraint. Our positive results are constructive.