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Min-Max Gathering on Infinite Grid

Abhinav Chakraborty, Pritam Goswami, Satakshi Ghosh

TL;DR

This paper addresses the optimal gathering problem on an infinite grid, aiming to improve the energy efficiency by minimizing the maximum distance any robot must travel.

Abstract

Gathering is a fundamental coordination problem in swarm robotics, where the objective is to bring robots together at a point not known to them at the beginning. While most research focuses on continuous domains, some studies also examine the discrete domain. This paper addresses the optimal gathering problem on an infinite grid, aiming to improve the energy efficiency by minimizing the maximum distance any robot must travel. The robots are autonomous, anonymous, homogeneous, identical, and oblivious. We identify all initial configurations where the optimal gathering problem is unsolvable. For the remaining configurations, we introduce a deterministic distributed algorithm that effectively gathers $n$ robots ($n\ge 9$). The algorithm ensures that the robots gathers at one of the designated min-max nodes in the grid. Additionally, we provide a comprehensive characterization of the subgraph formed by the min-max nodes in this infinite grid model.

Min-Max Gathering on Infinite Grid

TL;DR

This paper addresses the optimal gathering problem on an infinite grid, aiming to improve the energy efficiency by minimizing the maximum distance any robot must travel.

Abstract

Gathering is a fundamental coordination problem in swarm robotics, where the objective is to bring robots together at a point not known to them at the beginning. While most research focuses on continuous domains, some studies also examine the discrete domain. This paper addresses the optimal gathering problem on an infinite grid, aiming to improve the energy efficiency by minimizing the maximum distance any robot must travel. The robots are autonomous, anonymous, homogeneous, identical, and oblivious. We identify all initial configurations where the optimal gathering problem is unsolvable. For the remaining configurations, we introduce a deterministic distributed algorithm that effectively gathers robots (). The algorithm ensures that the robots gathers at one of the designated min-max nodes in the grid. Additionally, we provide a comprehensive characterization of the subgraph formed by the min-max nodes in this infinite grid model.

Paper Structure

This paper contains 86 sections, 48 theorems, 11 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Works
  4. Our Contributions
  5. Outline
  6. Models and Definitions
  7. Models
  8. Terminologies
  9. Problem Definition
  10. Problem Definition
  11. Technical Difficulties and Overview of our Strategy
  12. Characterization of the min-max nodes in Infinite Grids
  13. Case-1
  14. Case-2
  15. Impossibility Results
  16. ...and 71 more sections

Key Result

Lemma 1

For a finite set of grid points in $\mathcal{R}$, if $\mathcal{M}$$\in$$\mathcal{MED}(\mathcal{R})$, then $\mathcal{C}(\mathcal{M})$$\in V_M(\mathcal{R})$.

Figures (12)

  • Figure 1: Figure demonstrating the largest lexicographic strings
  • Figure 2: More than one min-max node.
  • Figure 3: Min-max node is not invariant under the movement of robot towards itself.
  • Figure 4: New min-max node is created after the movement of a robot towards a min-max node.
  • Figure 5: Example of a configuration where $G_M(\mathcal{R})$ is a step-graph.
  • ...and 7 more figures

Theorems & Definitions (107)

  • Definition 1: Grid lines passing through a node ($L_H(v)$ and $L_V(v)$)
  • Definition 2: Diagonals of a grid node ($D(v)$)
  • Definition 3: Neighbours of a grid node ($N(v)$)
  • Definition 4: Diamond
  • Definition 5: Perimeter of a diamond $\mathcal{M}$ ($\mathcal{P}(\mathcal{M})$)
  • Definition 6: Enclosing Diamond ($\mathcal{ED}(R)$)
  • Definition 7: Minimal Enclosing Diamond($\mathcal{MED}(\mathcal{R})$)
  • Definition 8: Min-Max Node
  • Definition 9
  • Definition 10
  • ...and 97 more