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

TL;DR

The paper addresses APF for k oblivious, anonymous robots on an infinite rectangular grid under an asynchronous scheduler within the classical OBLOT model. It introduces a seven-phase APF algorithm that first establishes a global coordinate system via a leading corner derived from lexicographic binary strings on the SER, then uses a line-based subroutine (ApfLine) to reposition inner robots along a spanning line, while the tail expands the current SER and the head anchors toward target positions. The main contributions are a space complexity bound of at most $\\mathcal{D}+4$ (with enclosure $(M+4)\\times(N+1)$) and a move complexity of $O(k\\mathcal{D})$, making it the first deterministic OBLOT APF algorithm on a grid that is asymptotically move-optimal and near space-optimal. This work advances scalable, collision-free swarm coordination under strict memoryless and communication-free constraints on unbounded grids, with potential implications for robust distributed pattern formation in robotics.

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.

Figures (4)

• Figure 1: $ABCD$ is the SER of the configuration. $\lambda_{AB}=01101101010011010100$ is the largest lexicographic string, and $r_h$ and $r_t$ are respectively the head and tail robots of the configuration.
• Figure 2: Path joining the nodes $A$ and $C$ mentioned in bold edges
• Figure 3: An image related to Phase V
• Figure 4: Flow of the Algorithm ApfLine

Theorems & Definitions (21)

• Definition 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Proposition 6
• Proposition 7
• Lemma 8
• Lemma 9
• Lemma 10