Pattern Formation for Fat Robots with Memory

TL;DR

This work tackles Pattern Formation for $n$ fat, unit-disk robots under obstructed visibility in the classical Look-Compute-Move model with limited memory and no inter-robot communication. It introduces a three-phase approach: first achieve Mutual Visibility by enlarging the convex hull so every robot lies on a hull corner, then elect a single Leader via a centroid-based boundary-circle protocol with probabilistic contention, and finally perform Pattern Formation by a Leader-guided, scaled placement of robots into the target pattern. The algorithm is collision-free and achieves the goal in $O(n) + O(q \log n)$ rounds with probability at least $1 - n^{-q}$ in the fully synchronous model while requiring only $O(1)$ memory per robot. This represents a significant extension of Pattern Formation to fat robots with memory, addressing obstructed visibility without lights or axis agreement, and establishing a foundation for future work in semi-synchronous and asynchronous regimes.

Abstract

Given a set of $n\geq 1$ autonomous, anonymous, indistinguishable, silent, and possibly disoriented mobile unit disk (i.e., fat) robots operating following Look-Compute-Move cycles in the Euclidean plane, we consider the Pattern Formation problem: from arbitrary starting positions, the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. We assume that a robot's movement cannot be interrupted by an adversary and that robots have a small $O(1)$-sized memory that they can use to store information, but that cannot be communicated to the other robots. To solve this problem, we present an algorithm that works in three steps. First it establishes mutual visibility, then it elects one robot to be the leader, and finally it forms the required pattern. The whole algorithm runs in $O(n) + O(q \log n)$ rounds with probability at least $1 - n^{-q}$. The algorithms are collision-free and do not require the knowledge of the number of robots.

Figures (5)

• Figure 1: An example of the Mutual Visibility phase: (a) an initial configuration and (b) the end configuration of this phase. Throughout the paper the robots are shown as dots for simplicity.
• Figure 2: An example configuration where a corner robot $r_i$ believes all robots are in convex position when there is an interior robot $r_j$ left which blocks visibility to robot $r_k$.
• Figure 3: Robot $r_j$ blocks visibility between $r_i$ and $r_k$: (a) $r_i$ concludes that $r_j$ is an interior robot, (b) $r_i$ concludes that the robots are not yet in convex position, and (c) $r_i$ incorrectly concludes that the robots are in convex position, but $r_k$ still correctly concludes that this is not the case.
• Figure 4: An example of the Leader Election phase: (a) centroid $c$ and distance $d$ are computed and used to form a circle, (b) an unsuccessful iteration where two robots remain on the boundary of the circle, and (c) a successful iteration where a leader is elected and the phase ends.
• Figure 5: An example of the Pattern Formation phase (leader is colored red): (a) the situation after a leader has been elected, (b) the leader moves next to $r_1$, (c) the leader moves $r_1$ to $p_1$, (d) the leader moves next to the next robot, and (e) the pattern is built and the phase ends.

Theorems & Definitions (24)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof