The Search Complexity of Collaborative Swarms in Expanding \(\mathbf{Z}^{2}\) Grid Regions

In this paper we study the strengths and limitations of collaborative swarms of simple agents. In particular, we discuss the efficient use of simple drones, or “ant robots” for covering a connected region on the \(\mathbf{Z}^{2}\) grid, whose area is unknown in advance, and which expands at a given rate, where n is the initial size of the connected region. We show that regardless of the algorithm used, and the robots’ hardware and software specifications, the minimal number of robots required in order for such coverage to be possible is \(\varOmega ({\sqrt{n}})\). In addition, we show that when the region expands at a sufficiently slow rate, a team of \(\varTheta (\sqrt{n})\) robots could cover it in at most \(O(n^{2} \ln n)\) time. This completion time can even be achieved by myopic robots, with no ability to directly communicate with each other, and where each robot is equipped with a memory of size O(1) bits w.r.t the size of the region (therefore, the robots cannot maintain maps of the terrain, nor plan complete paths). Regarding the coverage of non-expanding regions in the grid, we improve the current best known result of \(O(n^{2})\) by demonstrating an algorithm that guarantees such a coverage with completion time of \(O(\frac{1}{k} n^{1.5} + n)\) in the worst case, and faster for shapes of perimeter length which is shorter than O(n).