Optimal Convex Hull Formation on a Grid by Asynchronous Robots with Lights

Abstract

We consider the distributed setting of n autonomous mobile robots that operate in Look-Compute-Move (LCM) cycles and communicate with other robots using a constant number of colored lights (the robots with lights model). We assume obstructed visibility where a robot cannot see another robot if a third robot is positioned between them on the straight line connecting them. In addition, we consider a grid-based terrain embedded in the 2-dimensional Euclidean plane that restricts each robot 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 Convex Hull Formation problem is to relocate the n robots (starting at arbitrary, but distinct, initial positions) so that each robot is positioned on a vertex of a convex hull. In this paper, we provide two asynchronous algorithms for Convex Hull Formation, both using a constant number of colors. Key measures of the algorithms’ performance include the time taken and the space occupied (measured as the perimeter of the smallest rectangle enclosing the convex hull formed). The first O(max{n2, D})-time and O(n2)-perimeter algorithm serves to introduce key ideas, where D is the diameter of the initial configuration. The second algorithm runs in $O\left( {\max \left\{ {{n^{\frac{3}{2}}},D} \right\}} \right)$ time with a perimeter of $O\left( {{n^{\frac{3}{2}}}} \right)$. We also prove lower bounds of $\Omega \left( {{n^{\frac{3}{2}}}} \right)$ for both the time and perimeter for any Convex Hull Formation algorithm; that is, we establish our second algorithm as optimal in both time and perimeter.