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Memory Optimal Dispersion by Anonymous Mobile Robots

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Algorithms and Discrete Applied Mathematics (CALDAM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12601))

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Abstract

Consider a team of \(k \le n\) autonomous mobile robots initially placed at a node of an arbitrary graph G with n nodes. The dispersion problem asks for a distributed algorithm that allows the robots to reach a configuration in which each robot is at a distinct node of the graph. If the robots are anonymous, i.e., they do not have any unique identifiers, then the problem is not solvable by any deterministic algorithm. However, the problem can be solved even by anonymous robots if each robot is given access to a fair coin which they can use to generate random bits. In this setting, it is known that the robots require \(\varOmega (\log {\varDelta })\) bits of memory to achieve dispersion, where \(\varDelta \) is the maximum degree of G. On the other hand, the best known memory upper bound is \(min \{\varDelta , max\{\log {\varDelta }, \log {D}\}\}\) (D = diameter of G), which can be \(\omega (\log {\varDelta })\), depending on the values of \(\varDelta \) and D. In this paper, we close this gap by presenting an optimal algorithm requiring \(O(\log {\varDelta })\) bits of memory.

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Acknowledgement

We would like to thank Pritam Goswami for valuable discussions. The first two authors are supported by UGC, Govt. of India, and NBHM DAE, Govt. of India respectively. We would like to thank the anonymous reviewers for their valuable comments which helped us to improve the quality and presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, India

    Archak Das, Kaustav Bose & Buddhadeb Sau

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  1. Archak Das
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  2. Kaustav Bose
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  3. Buddhadeb Sau
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Correspondence to Kaustav Bose .

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Editors and Affiliations

  1. Indian Institute of Technology Ropar, Rupnagar, India

    Apurva Mudgal

  2. Institute of Mathematical Sciences, Chennai, India

    C. R. Subramanian

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Das, A., Bose, K., Sau, B. (2021). Memory Optimal Dispersion by Anonymous Mobile Robots. In: Mudgal, A., Subramanian, C.R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2021. Lecture Notes in Computer Science(), vol 12601. Springer, Cham. https://doi.org/10.1007/978-3-030-67899-9_34

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  • DOI: https://doi.org/10.1007/978-3-030-67899-9_34

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