Stand-Up Indulgent Gathering on Lines for Myopic Luminous Robots

TL;DR

The paper tackles the stand-up indulgent gathering problem (SUIG) for mobile robots on line-shaped networks with limited visibility and lights, under crash faults and in the FSYNC model. It first identifies impossibility regimes, showing that symmetry and certain initial configurations prevent gathering, which motivates focusing on parity-based solvable cases where either $M_{init}$ or $O_{init}$ is odd. It then proves constructive algorithms: for $M_{init}$ odd, gathering is achievable without lights; for $O_{init}$ odd, gathering is achieved with a three-color luminous protocol, both in $O(M_{init})$ rounds and robust to crashes. The work provides the first SUIG solutions for myopic luminous robots on lines, and highlights open questions about even-parity cases, crashes at different nodes, and color-optimal variants. These results extend crash-tolerant gathering to resource-constrained robots on simple topologies, with implications for robust coordination in constrained environments.

Abstract

We consider a strong variant of the crash fault-tolerant gathering problem called stand-up indulgent gathering (SUIG), by robots endowed with limited visibility sensors and lights on line-shaped networks. In this problem, a group of mobile robots must eventually gather at a single location, not known beforehand, regardless of the occurrence of crashes. Differently from previous work that considered unlimited visibility, we assume that robots can observe nodes only within a certain fixed distance (that is, they are myopic), and emit a visible color from a fixed set (that is, they are luminous), without multiplicity detection. We consider algorithms depending on two parameters related to the initial configuration: $M_{init}$, which denotes the number of nodes between two border nodes, and $O_{init}$, which denotes the number of nodes hosting robots. Then, a border node is a node hosting one or more robots that cannot see other robots on at least one side. Our main contribution is to prove that, if $M_{init}$ or $O_{init}$ is odd, SUIG can be solved in the fully synchronous model.

Figures (5)

• Figure 1.1: Execution of border robots before the number of occupied nodes becomes three.
• Figure 1.2: Execution of Case 1.
• Figure 1.3: Execution of Case 2.
• Figure 1.4: Execution of Case 3.
• Figure 1.5: Execution of Case 4.

Theorems & Definitions (17)

• theorem 1: REA2008
• corollary thmcountercorollary: BramasKLT23
• corollary thmcountercorollary: OPODIS2019
• lemma thmcounterlemma: BramasKLT23
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof