Conic Formation in Presence of Faulty Robots

Pattern formation is one of the most fundamental problems in distributed computing, which has recently received much attention. In this paper, we initiate the study of distributed pattern formation in situations when some robots can be \textit{faulty}. In particular, we consider the well-established \textit{look-compute-move} model with oblivious, anonymous robots. We first present lower bounds and show that any deterministic algorithm takes at least two rounds to form simple patterns in the presence of faulty robots. We then present distributed algorithms for our problem which match this bound, \textit{for conic sections}: in at most two rounds, robots form lines, circles and parabola tolerating $f=2,3$ and $4$ faults, respectively. For $f=5$, the target patterns are parabola, hyperbola and ellipse. We show that the resulting pattern includes the $f$ faulty robots in the pattern of $n$ robots, where $n \geq 2f+1$, and that $f < n < 2f+1$ robots cannot form such patterns. We conclude by discussing several relaxations and extensions.