Gathering Anonymous, Oblivious Robots on a Grid

We consider a swarm of $n$ autonomous mobile robots, distributed on a 2-dimensional grid. A basic task for such a swarm is the gathering process: all robots have to gather at one (not predefined) place. The work in this paper is motivated by the following insight: On one side, for swarms of robots distributed in the 2-dimensional Euclidean space, several gathering algorithms are known for extremely simple robots that are oblivious, have bounded viewing radius, no compass, and no "flags" to communicate a status to others. On the other side, in case of the 2-dimensional grid, the only known gathering algorithms for robots with bounded viewing radius without compass, need to memorize a constant number of rounds and need flags. In this paper we contribute the, to the best of our knowledge, first gathering algorithm on the grid, which works for anonymous, oblivious robots with bounded viewing range, without any further means of communication and without any memory. We prove its correctness and an $O(n^2)$ time bound. This time bound matches those of the best known algorithms for the Euclidean plane mentioned above.