The Complexity of Grid Coverage by Swarm Robotics

In this paper we discuss the task of efficiently using ant-like robotic agents for covering a connected region on the Z grid, whose shape and size are unknown in advance, and which expands at a given rate. This is done using myopic robots, with no ability to directly communicate with each other, where each robot is equipped with only O(1) memory. We show that regardless of the algorithm used, and the robots’ hardware and software specifications, the minimal number of robots required in order to enable such coverage is Ω( √ n) (where n is the initial size of the connected region). In addition, we show that when the region expands at a sufficiently slow rate, a team of Θ( √ n) robots could cover it in at most O(n lnn) time. Regarding the coverage of nonexpanding regions in the grid, we improve the current best known result of O(n) by demonstrating an algorithm of worse case completion time of O( 1 k n + n), and faster for shapes of perimeter length which is shorter than O(n).