Pattern Formation for Fat Robots with Lights

TL;DR

This work studies Pattern Formation for a set of $n$ unit-disk fat robots with obstructed visibility in the fully synchronous, lights-enabled model. It introduces a four-phase framework (Mutual Visibility, Leader Election, Line Formation, Pattern Formation) and provides two pattern-formation algorithms: one allowing pattern scaling and achieving a tight color bound of $7$, and another without scaling achieving $8$ colors, both with a running time of $O(n) + O(q \log n)$ rounds and high probability. By reusing colors across phases, it further reduces the color requirements from prior $10$–$11$-color schemes to $7$–$8$ while maintaining collision-free operation and no reliance on global coordinate systems. These results advance practical pattern formation for opaque, extended robots and open avenues for lower bounds on color usage and asynchronous/semi-synchronous extensions.

Abstract

Given a set of $n\geq 1$ unit disk robots in the Euclidean plane, we consider the Pattern Formation problem, i.e., 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. Recently, this problem was solved in the asynchonous model for fat robots that agree on at least one axis in the robots with lights model where each robot is equipped with an externally visible persistent light that can assume colors from a fixed set of colors [K. Bose, R. Adhikary, M. K. Kundu, and B. Sau. Arbitrary pattern formation by opaque fat robots with lights. CALDAM, pages 347-359, 2020]. In this work, we reduce the number of colors needed and remove the axis-agreement requirement in the fully synchronous model. In particular, we present an algorithm requiring 7 colors when scaling the target pattern is allowed and an 8-color algorithm if scaling is not allowed. Our algorithms run in $O(n) + O(q \log n)$ rounds with probability at least $1 - n^{-q}$.

Figures (5)

• Figure 1: An example of the Mutual Visibility phase: (a) an initial configuration and (b) the end configuration. Throughout the paper the robots are shown as dots for simplicity.
• Figure 2: An example of the Leader Election phase: (a) initially all robots are competing (orange), (b) an unsuccessful iteration with competing and non-competing (gray) robots, and (c) a successful iteration, where a single robot is elected leader (purple).
• Figure 3: An example of the Line Formation phase (numbers indicate order of operations): (a) the leader (purple) moves to the first position on the line and signals (yellow) the closest robot to move there in the next round before moving out of the way and setting its color to do not follow (lightblue), (b) the first robot changes its color to on the line (green) and the leader moves next to the next robot to be moved, (c) the leader moves the next robot to its position on the line while the following robot has its color set to following the leader (brown), after which the leader moves to guide the next robot, and (d) the result of the Line Formation phase.
• Figure 4: An example of the Pattern Formation phase when scaling is allowed (numbers indicate order of operations): (a) the leader moves to the position above the topmost robot $r_1$ on the line and signals that it should follow (yellow), which causes $r_1$ to change its color to following the leader (brown) and move accordingly, (b) the leader moves out of the way so robot $r_1$ can reach its final position, which the leader indicates by changing its color to do not follow (lightblue) and $r_1$ sets its color to at final position (blue), after which the leader moves to be immediately above the next robot on the line, (c) the leader guides the next robot to its final position, and (d) all robots have reached their final position.
• Figure 5: An example of the Pattern Formation phase when scaling is not allowed (numbers indicate order of operations): (a) the leader moves to the position on the line with $y$-coordinate equal to where the first robot $r_1$ needs to be placed and signals that $r_1$ should follow (yellow), which causes $r_i$ to change its color to following the leader (brown) and move accordingly while the leader moves away preparing to push $r_1$, (b) the leader changes its color to push (gold) and $r_1$ moves distance equal to its current distance to the leader away from the leader to reach its final position and sets its color to at final position (blue), after which the leader moves to pull the next robot on the line, (c) the leader first pulls and then pushes the next robot to its final position, and (d) all robots have reached their final position.

Theorems & Definitions (21)

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