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International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics

ALGOSENSORS 2020: Algorithms for Sensor Systems pp 154–169Cite as

Efficient Dispersion on an Anonymous Ring in the Presence of Weak Byzantine Robots

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Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 12503))

Abstract

The problem of dispersion of mobile robots on a graph asks that n robots initially placed arbitrarily on the nodes of an n-node anonymous graph, autonomously move to reach a final configuration where exactly each node has at most one robot on it. This problem has been relatively well-studied when robots are non-faulty. In this paper, we introduce the notion of Byzantine faults to this problem, i.e., we formalize the problem of dispersion in the presence of up to f Byzantine robots. We then study the problem on a ring while simultaneously optimizing the time complexity of algorithms and the memory requirement per robot. Specifically, we design deterministic algorithms that attempt to match the time lower bound (\(\varOmega (n)\) rounds) and memory lower bound (\(\varOmega (\log n)\) bits per robot).

Our main result is a deterministic algorithm that is both time and memory optimal, i.e., O(n) rounds and \(O(\log n)\) bits of memory required per robot, subject to certain constraints. We subsequently provide results that require less assumptions but are either only time or memory optimal but not both. We also provide a primitive that takes robots initially gathered at a node of the ring and disperses them in a time and memory optimal manner without additional assumptions required .

The work of W. K. Moses Jr. was supported in part by a Technion fellowship. A. R. Molla is supported, in part, by DST INSPIRE Faculty Research Grant DST/INSPIRE/04/2015/002801, Govt. of India.

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Notes

  1. 1.

    There are mainly two types of faults–one is “crash fault” which means that once a robot crashes, it is dead and will not be active again thereafter; another one is “Byzantine fault” which means that a robot is alive throughout and may behave maliciously. Note that the Byzantine fault subsumes the crash fault.

  2. 2.

    In the algorithms, we mention a robot resets its sense of direction, i.e., it resets its notion of clockwise and counter-clockwise. That refers to the robot performing this check again and redefining clockwise and counter-clockwise accordingly.

  3. 3.

    Note that it is not necessary for a robot to know the value of n in order to maintain a counter using \(\log n\) bits of memory given that the robot’s total memory is \(c \log n\) bits of memory, where c is a sufficiently large constant.

  4. 4.

    Notice that we say that other co-located robots are to follow \(R_1\), instead of just staying put. This is to ensure that all robots initially co-located with \(R_1\) continue to stay with \(R_1\), even if \(R_1\) is a Byzantine robot and moves around during the \(n+1\) rounds.

  5. 5.

    Possibly some Byzantine robots may also be gathered as well, but the presence of these robots does not cause problems as the subsequent procedure, Rooted-Ring-Dispersion is correct even in the presence of \(n-1\) Byzantine robots.

  6. 6.

    This could happen at the beginning of the stage, if the two robots are co-located on the same node.

  7. 7.

    If r terminates prior to the end of round n, it becomes invisible to other robots. Thus, there is the risk of another non-Byzantine robot settling at the same node as r if r terminates early.

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Author information

Authors and Affiliations

  1. Computer and Communication Sciences, Indian Statistical Institute, Kolkata, India

    Anisur Rahaman Molla

  2. Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, India

    Kaushik Mondal

  3. Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology, Haifa, Israel

    William K. Moses Jr.

Authors
  1. Anisur Rahaman Molla
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  2. Kaushik Mondal
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  3. William K. Moses Jr.
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Correspondence to William K. Moses Jr. .

Editor information

Editors and Affiliations

  1. Mathematics and Computer Science, University of Perugia, Perugia, Italy

    Cristina M. Pinotti

  2. Mathematics and Computer Science, University of Perugia, Perugia, Italy

    Alfredo Navarra

  3. Computer Science and Engineering, Indian Institute of Technology Delhi, New Delhi, India

    Amitabha Bagchi

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Molla, A.R., Mondal, K., Moses, W.K. (2020). Efficient Dispersion on an Anonymous Ring in the Presence of Weak Byzantine 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_11

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  • DOI: https://doi.org/10.1007/978-3-030-62401-9_11

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