Space and Move-optimal Arbitrary Pattern Formation on Infinite Rectangular Grid by Oblivious Robot Swarm

TL;DR

This paper presents an asymptotically move-optimal algorithm for arbitrary pattern formation on an infinite grid by oblivious, anonymous robots with full visibility, achieving near-minimal space complexity in an asynchronous setting.

Contribution

It introduces the first asymptotically move-optimal APF algorithm for OBLOT robots on an infinite grid with minimal space complexity.

Findings

Algorithm is asymptotically move-optimal.

Space complexity is within 4 of the lower bound.

First to achieve near-optimal space and move efficiency for OBLOT robots.

Abstract

Arbitrary Pattern Formation (APF) is a fundamental coordination problem in swarm robotics. It requires a set of autonomous robots (mobile computing units) to form an arbitrary pattern (given as input) starting from any initial pattern. This problem has been extensively investigated in continuous and discrete scenarios, with this study focusing on the discrete variant. A set of robots is placed on the nodes of an infinite rectangular grid graph embedded in the euclidean plane. The movements of each robot is restricted to one of the four neighboring grid nodes from its current position. The robots are autonomous, anonymous, identical, and homogeneous, and operate Look-Compute-Move cycles. In this work, we adopt the classical $\mathcal{OBLOT}$ robot model, meaning the robots have no persistent memory or explicit communication methods, yet they possess full and unobstructed visibility. This work proposes an algorithm that solves the APF problem in a fully asynchronous scheduler assuming the initial configuration is asymmetric. The considered performance measures of the algorithm are space and number of moves required for the robots. The algorithm is asymptotically move-optimal. Here, we provide a definition of space complexity that takes the visibility issue into consideration. We observe an obvious lower bound $\mathcal{D}$ of the space complexity and show that the proposed algorithm has the space complexity $\mathcal{D}+4$. On comparing with previous related works, we show that this is the first proposed algorithm considering $\mathcal{OBLOT}$ robot model that is asymptotically move-optimal and has the least space complexity which is almost optimal.