Optimal Dispersion of Silent Robots in a Ring

TL;DR

The paper addresses dispersion of $k$ silent robots starting from multiple sources on an oriented ring. It introduces AlgorithmMultiStart, a phase-based deterministic approach that combines leader election, chain merging, and dispersion to ensure at most one robot per node. The authors prove correctness and derive an upper bound of $O(\log L + k)$ rounds with $O(\log L)$ memory per robot, and establish a matching lower bound of $\Omega(\log L + k)$, demonstrating optimality. The results advance understanding of silent robot dispersion in ring topologies with multiple sources and orientation constraints, with potential implications for distributed deployment and resource allocation in ring-like networks.

Abstract

Given a set of co-located mobile robots in an unknown anonymous graph, the robots must relocate themselves in distinct graph nodes to solve the dispersion problem. In this paper, we consider the dispersion problem for silent robots \cite{gorain2024collaborative}, i.e., no direct, explicit communication between any two robots placed in the nodes of an oriented $n$ node ring network. The robots operate in synchronous rounds. The dispersion problem for silent mobile robots has been studied in arbitrary graphs where the robots start from a single source. In this paper, we focus on the dispersion problem for silent mobile robots where robots can start from multiple sources. The robots have unique labels from a range $[0,\;L]$ for some positive integer $L$. Any two co-located robots do not have the information about the label of the other robot. The robots have weak multiplicity detection capability, which means they can determine if it is alone on a node. The robots are assumed to be able to identify an increase or decrease in the number of robots present on a node in a particular round. However, the robots can not get the exact number of increase or decrease in the number of robots. We have proposed a deterministic distributed algorithm that solves the dispersion of $k$ robots in an oriented ring in $O(\log L+k)$ synchronous rounds with $O(\log L)$ bits of memory for each robot. A lower bound $Ω(\log L+k)$ on time for the dispersion of $k$ robots on a ring network is presented to establish the optimality of the proposed algorithm.

Figures (6)

• Figure 1: Representation of the status of robots positioned on three consecutive nodes $v_i$, $v_{i+1}$, and $v_{i+2}$ (a) $passive$ (pink), $activedisperse$ (green) and $passive$ (pink), (b) $activedisperse$, $passive$, and $activedisperse$, (c) $activedisperse$, $passive$ and $wait$ (turquoise) and $jump$ (violet), (d) $wait$ and $jump$, $passive$ and $activedisperse$, (e) $wait$ and $jump$, $passive$ and $wait$ and $jump$.
• Figure 2: Representation of the status of robots positioned on three consecutive nodes $v_i$, $v_{i+1}$, and $v_{i+2}$ (a) $passive$, $wait$ and $jump$ and $passive$, (b) $activedisperse$, $passive$, and $wait$ and $jump$, (c) $wait$ and $jump$, $passive$, and $wait$ and $jump$.
• Figure 3: Representation of the status of robots positioned on three consecutive nodes $v_i$, $v_{i+1}$, and $v_{i+2}$ (a) $activedisperse$, $passive$, and $activedisperse$, (b) $passive$, $activedisperse$ and $passive$, (c) $passive$, $wait$ and $jump$, and $passive$.
• Figure 4: Representation of the status of robots positioned on three consecutive nodes $v_i$, $v_{i+1}$, and $v_{i+2}$ (a) $activedisperse$, $passive$, and $wait$ and $jump$, (b) $passive$, $activedisperse$ and $passive$, (c) $passive$, $wait$ and $jump$, and $passive$.
• Figure 5: Representation of the status of robots positioned on three consecutive nodes $v_i$, $v_{i+1}$, and $v_{i+2}$ (a) $wait$ and $jump$, $passive$, and $activedisperse$, (b) $passive$, $wait$ and $jump$, and $passive$.

Theorems & Definitions (29)

• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6