Oblivious Robots Under Round Robin: Gathering on Rings

TL;DR

This work investigates the Gathering problem for oblivious, anonymous robots on $n$-rings under a Round Robin sequential scheduler, without multiplicity detection and with possible initial multiplicities. It first proves impossibility under general $ ext{SEQ}$ schedulers, then delivers a complete $ ext{RR}$-based solution via a predicate-driven, multi-stage algorithm $ ext{GatheRRing}$ that identifies a gathering vertex and guides robots through two-island and two-vertex intermediates to full gathering, achieving a bound of at most $n-3$ epochs. The approach decomposes the global task into well-defined subproblems and tasks with explicit predicates and moves, and provides a rigorous correctness argument plus a running example. The results map the solvability landscape for rings with minimal sensing, and deliver a practically implementable, asymptotically optimal protocol for both Gathering and Distinct Gathering when multiplicities are restricted away from the initial configuration. This clarifies the impact of scheduler design on symmetry-breaking in ring topologies and informs future work on broader topologies under limited robot capabilities.

Abstract

Robots with very limited capabilities are placed on the vertices of a graph and are required to move toward a single, common vertex, where they remain stationary once they arrive. This task is referred to as the GATHERING problem. Most of the research on this topic has focused on feasibility challenges in the asynchronous setting, where robots operate independently of each other. A common assumption in these studies is that robots are equipped with multiplicity detection, the ability to recognize whether a vertex is occupied by more than one robot. Additionally, initial configurations are often restricted to ensure that no vertex hosts more than one robot. A key difficulty arises from the possible symmetries in the robots' placement relative to the graph's topology. This paper investigates the GATHERING problem on Rings under a sequential scheduler, where only one robot at a time is active. While this sequential activation helps to break symmetries, we remove two common assumptions: robots do not have multiplicity detection, and in initial configurations, vertices can be occupied by multiplicities. We prove that such a generalized GATHERING problem cannot be solved under general sequential schedulers. However, we provide a complete characterization of the problem when a sequential Round Robin scheduler is used, where robots are activated one at a time in a fixed cyclic order that repeats indefinitely. Furthermore, we fully characterize the DISTINCT GATHERING problem, the most used variant of GATHERING, in which the initial configurations do not admit multiplicities.

Figures (4)

• Figure 1: Configurations used for the proofs of Theorems \ref{['th:impseq']} and \ref{['th:imprr']}. The robots are represented by black circles inside vertices. A full black vertex represents a multiplicity. Edges are not drawn for clarity. Labels associated with vertices are used only for analysis purposes: robots are not aware of them as the rings are, in fact, anonymous.
• Figure 2: Configurations used for the proofs of Theorem \ref{['th:3/5']}, Lemma \ref{['lem:dist4/4']} and Lemma \ref{['lem:dist5/5']}, respectively.
• Figure 3: Transition graph derived from Table \ref{['tab:tasks2']}.
• Figure 4: Running example on a $6$-ring with $5$ robots and one multiplicity at $v_6$.

Theorems & Definitions (33)

• Definition 3.1: Gathering
• Definition 3.2: Distinct Gathering
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Definition 3.3: Unsolvable Configuration
• Theorem 3.2
• proof
• Theorem 3.3