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Tolerance to Asynchrony of an Algorithm for Gathering Myopic Robots on an Infinite Triangular Grid

Arya Tanmay Gupta, Sandeep S Kulkarni

TL;DR

The paper addresses gathering distance-1 myopic robots on an infinite triangular grid under asynchronous execution. It proves the Goswami et al. algorithm is lattice-linear, hence correct without a scheduler and compatible with a unidirectional camera, while enabling the gathering point to be determinable from initial state. It also tightens the convergence bound to $2n$ rounds and introduces a simplified revised algorithm that preserves all performance guarantees. Together, these results strengthen asynchronous tolerance for distributed robot gathering and demonstrate the practical value of lattice-linear approaches for grid-based multi-robot systems.

Abstract

In this paper, we study the problem of gathering distance-1 myopic robots on an infinite triangular grid. We show that the algorithm developed by Goswami et al. (SSS, 2022) is lattice-linear (cf. Gupta and Kulkarni, SRDS 2023). This implies that a distributed scheduler, assumed therein, is not required for this algorithm: it runs correctly in asynchrony. It also implies that the algorithm works correctly even if the robots are equipped with a unidirectional \textit{camera} to see the neighbouring robots (rather than an omnidirectional one, which would be required under a distributed scheduler). Due to lattice-linearity, we can predetermine the point of gathering. We also show that this algorithm converges in $2n$ rounds, which is lower than the complexity ($2.5(n+1)$ rounds) that was shown in Goswami et al.

Tolerance to Asynchrony of an Algorithm for Gathering Myopic Robots on an Infinite Triangular Grid

TL;DR

The paper addresses gathering distance-1 myopic robots on an infinite triangular grid under asynchronous execution. It proves the Goswami et al. algorithm is lattice-linear, hence correct without a scheduler and compatible with a unidirectional camera, while enabling the gathering point to be determinable from initial state. It also tightens the convergence bound to rounds and introduces a simplified revised algorithm that preserves all performance guarantees. Together, these results strengthen asynchronous tolerance for distributed robot gathering and demonstrate the practical value of lattice-linear approaches for grid-based multi-robot systems.

Abstract

In this paper, we study the problem of gathering distance-1 myopic robots on an infinite triangular grid. We show that the algorithm developed by Goswami et al. (SSS, 2022) is lattice-linear (cf. Gupta and Kulkarni, SRDS 2023). This implies that a distributed scheduler, assumed therein, is not required for this algorithm: it runs correctly in asynchrony. It also implies that the algorithm works correctly even if the robots are equipped with a unidirectional \textit{camera} to see the neighbouring robots (rather than an omnidirectional one, which would be required under a distributed scheduler). Due to lattice-linearity, we can predetermine the point of gathering. We also show that this algorithm converges in rounds, which is lower than the complexity ( rounds) that was shown in Goswami et al.
Paper Structure (18 sections, 10 theorems, 1 equation, 2 figures, 2 algorithms)

This paper contains 18 sections, 10 theorems, 1 equation, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions of the paper
  3. Organization of the paper
  4. Preliminaries
  5. Execution without Synchronization
  6. Embedding a $\prec$-lattice among Global States
  7. Lattice-Linear Problems
  8. Lattice-Linear Algorithms
  9. Gathering of Distance-1 Myopic Robots on Infinite Triangular Grid (GMRIT)
  10. Problem Statement
  11. Problem Specific Definitions
  12. General Idea of the Algorithm
  13. GSGS Algorithm Goswami2022 for GMRIT problem
  14. Lattice-Linearity
  15. Time Complexity Properties
  16. ...and 3 more sections

Key Result

Lemma 1

The predicate $\forall i:\lnot\textsc{Impedensable\xspace-GSGS}(i)$ is a lattice-linear predicate on $n$ robots, and the visibility graph does not get disconnected by the actions under algorithm:gsgs-algo.

Figures (2)

  • Figure 1: Robots on an infinite triangular grid: one on every round highlighted vertex.
  • Figure 2: (a) Naming conventions for neighbourhood of a robot. (b) Cases where a node is impedensable. Note that the mirror images of these local states are also impedensable.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 11 more