Abstract
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 faulty. In particular, we consider the well-established 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, 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 \(n \ge 2f+1\), and that \(f< n < 2f+1\) robots cannot form such patterns. We conclude by discussing several relaxations and extensions.
D. Pattanayak—Visit to University of Vienna is supported by the Overseas Visiting Doctoral Fellowship, 2018 Award No. ODF/2018/001055 by the Science and Engineering Research Board (SERB), Government of India.
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Notes
- 1.
This assumption allows us to have a unique ordering of the robots [8].
- 2.
As any set of non-faulty robots that share a position will always perform the same actions from then on and be indistinguishable from each other.
- 3.
That is a point for \(f=1\), a line for \(f=2\), a circle for \(f=3\), a parabola for \(f=4\), and an ellipse or parabola or hyperbola for \(f=5\).
- 4.
Two parabolas intersect at 4 points, which can be the common points between two parabola patterns.
- 5.
The latus rectum is the line that passes through the focus of the parabola and parallel to the directrix.
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Pattanayak, D., Foerster, KT., Mandal, P.S., Schmid, S. (2020). Conic Formation in Presence of Faulty Robots. In: Pinotti, C.M., Navarra, A., Bagchi, A. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2020. Lecture Notes in Computer Science(), vol 12503. Springer, Cham. https://doi.org/10.1007/978-3-030-62401-9_12
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