Move and Time Optimal Arbitrary Pattern Formation by Asynchronous Robots on Infinite Grid

TL;DR

This paper introduces two algorithms for arbitrary pattern formation by asynchronous robots on an infinite grid, achieving move and time optimality under minimal assumptions and without global coordinate access.

Contribution

It presents the first algorithms that solve the APF problem with asymptotic move and time optimality on an infinite grid for asynchronous robots.

Findings

First move optimal algorithm with O(𝔻') complexity

First time optimal algorithm with O(𝔻') epochs

Algorithms are asymptotically optimal in both move and time

Abstract

The \textsc{Arbitrary Pattern Formation} (\textsc{Apf}) is a widely studied in distributed computing for swarm robots. This problem asks to design a distributed algorithm that allows a team of identical, autonomous mobile robots to form any arbitrary pattern given as input. This paper considers that the robots are operating on a two-dimensional infinite grid. Robots are initially positioned on distinct grid points forming an asymmetric configuration (no two robots have the same snapshot). They operate under a fully asynchronous scheduler and do not have any access to a global coordinate system, but they will align the axes of their local coordinate systems along the grid lines. The previous work dealing with \textsc{Apf} problem solved it in $O(\mathcal{D}^2k)$ robot movements under similar conditions, where $\mathcal{D}$ is the side of the smallest square that can contain both initial and target configuration and, $k$ is the number of robots. Let $\mathcal{D}'=\max\{\mathcal{D},k\}$. This paper presents two algorithms of \textsc{Apf} on an infinite grid. The first algorithm solves the \textsc{Apf} problem using $O(\mathcal{D}')$ asymptotically move optimal. The second algorithm solves the \textsc{Apf} problem in $O(\mathcal{D}')$ epochs, which we show is asymptotically optimal.