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. A common local model for extremely simple robots is the following: The robots do not have a common compass, only have a constant viewing radius, are autonomous and indistinguishable, can move at most a constant distance in each step, cannot communicate, are oblivious and do not have flags or states. The only gathering algorithm under this robot model, with known runtime bounds, needs \(\mathcal {O}(n^2)\) rounds and works in the Euclidean plane. The underlying time model for the algorithm is the fully synchronous \(\mathcal {FSYNC}\) model. On the other side, in the case of the 2-dimensional grid, the only known gathering algorithms for the same time and a similar local model additionally require a constant memory, states and “flags” to communicate these states to neighbors in viewing range. They gather in time \(\mathcal {O}(n)\).