Pattern Formation for Fat Robots with Memory

Given a set of \(n\geq 1\) autonomous, anonymous, indistinguishable, silent, and possibly disoriented mobile unit disk (i.e., fat) robots operating following Look-Compute-Move cycles in the Euclidean plane, we consider the Pattern Formation problem: from arbitrary starting positions, the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. We assume that a robot's movement cannot be interrupted by an adversary and that robots have a small \(O(1)\)-sized memory that they can use to store information, but that cannot be communicated to the other robots. To solve this problem, we present an algorithm that works in three steps. First it establishes mutual visibility, then it elects one robot to be the leader, and finally it forms the required pattern. The whole algorithm runs in \(O(n) + O(q \log n)\) rounds, where \(q>0\) is related to leader election, which takes \(O(q \log n)\) rounds with probability at least \(1-n^{-q}\). The algorithms are collision-free and do not require the knowledge of the number of robots.