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

TL;DR

This paper proves that a robot gathering algorithm on an infinite triangular grid is robust to asynchrony and limited vision, converging faster than previously established, with implications for simpler robot designs.

Contribution

It demonstrates that the algorithm is lattice-linear, allowing correct operation without a distributed scheduler and under unidirectional vision, and improves convergence time bounds.

Findings

Algorithm is lattice-linear, enabling asynchrony tolerance.

Works with unidirectional camera vision.

Converges in 2n rounds, faster than previous bounds.

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.