Optimal Arbitrary Pattern Formation on a Grid by Asynchronous Autonomous Robots

Abstract

We consider the distributed setting of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> autonomous mobile robots that operate in Look-Compute-Move (LCM) cycles following either the robots with lights model or the classical oblivious robots model. For the lights model, we assume obstructed visibility so that a robot cannot see another robot if a third robot is positioned between them on the straight line connecting them. In contrast, we assume unobstructed visibility in the classical model so that a robot sees all others irrespective of their positions. In addition, we consider a grid-based terrain embedded in the 2-dimensional Euclidean plane that restricts each robot's movement to one of the four neighboring grid points from its current position. This grid setting is a natural discretization of the 2-dimensional real plane and extends the robot swarm model in directions of greater applicability. The Arbitrary Pattern Formationproblem is to relocate the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> robots (starting at arbitrary but distinct initial positions on a grid) to form an arbitrary target pattern given as input. In this paper, we provide two asynchronous algorithms for Arbitrary Pattern Formation, one on the lights model and another on the classical model. Key measures of the algorithms' performance include the time taken and the number of moves by each robot. Both algorithms run in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\max\{D^{i}, D^{p}\})$</tex> time with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\max\{D^{i}, D^{p}\})$</tex> moves by each robot, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$D^{i}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$D^{p}$</tex> , respectively, are the diameters of the initial and pattern configurations. The algorithm for the lights model uses <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(1)$</tex> colors. We also prove a lower bound of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega(\max\{D^{i}, D^{p}\})$</tex> for time for any Arbitrary Pattern Formationalgorithm if scaling is not allowed on the target pattern. Therefore, our algorithms are optimal w.r.t. time. Furthermore, our algorithms are also optimal w.r.t. the number of moves given the existing lower bound of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega(\max\{D^{i}, D^{p}\})$</tex> on the number of moves. In sum, our results show that having lights provides a trade-off on the unobstructed visibility requirement in the classical model for Arbitrary Pattern Formation.