Optimal deterministic ring exploration with oblivious asynchronous robots

TL;DR

This paper proves the minimum number of deterministic robots needed for exploring an anonymous ring, showing five robots are necessary and sufficient for certain ring sizes, and provides an optimal exploration protocol.

Contribution

It establishes the exact minimum number of deterministic robots required for ring exploration and introduces an optimal exploration protocol with five robots.

Findings

No deterministic exploration with fewer than five robots for even-sized rings.

Five robots suffice for rings coprime with five.

Exploration completes in O(n) robot moves, which is optimal.

Abstract

We consider the problem of exploring an anonymous unoriented ring of size $n$ by $k$ identical, oblivious, asynchronous mobile robots, that are unable to communicate, yet have the ability to sense their environment and take decisions based on their local view. Previous works in this weak scenario prove that $k$ must not divide $n$ for a deterministic solution to exist. Also, it is known that the minimum number of robots (either deterministic or probabilistic) to explore a ring of size $n$ is 4. An upper bound of 17 robots holds in the deterministic case while 4 probabilistic robots are sufficient. In this paper, we close the complexity gap in the deterministic setting, by proving that no deterministic exploration is feasible with less than five robots whenever the size of the ring is even, and that five robots are sufficient for any $n$ that is coprime with five. Our protocol completes exploration in O(n) robot moves, which is also optimal.