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Minimum Algorithm Sizes for Self-stabilizing Gathering and Related Problems of Autonomous Mobile Robots (Extended Abstract)

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Stabilization, Safety, and Security of Distributed Systems (SSS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14310))

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Abstract

We investigate swarms of autonomous mobile robots in the Euclidean plane. Each robot has a target function to determine a destination point from the robots’ positions. All robots in a swarm conventionally take the same target function. We allow the robots in a swarm to take different target functions, and investigate the effects of the number of distinct target functions on the problem-solving ability. Specifically, we are interested in how many distinct target functions are necessary and sufficient to solve some well-known problems which are not solvable when all robots take the same target function, regarding target function as a resource, like time and message, to solve a problem. The number of distinct target functions necessary and sufficient to solve a problem \(\varPi \) is called the minimum algorithm size (MAS) for \(\varPi \). (The MAS is \(\infty \), if \(\varPi \) is not solvable even for the robots with unique target functions.) We establish the MASs for solving the gathering and related problems from any initial configuration, i.e., in a self-stabilizing manner. Our results include: There is a family of the scattering problems cSCT \((1 \le c \le n)\) such that the MAS for the cSCAT is c, where n is the size of the swarm. The MAS for the gathering problem is 2. It is 3, for the problem of gathering all non-faulty robots at a single point, regardless of the number \((< n)\) of crash failures. It is however \(\infty \), for the problem of gathering all robots at a single point, in the presence of at most one crash failure.

Due to space limitation, some proofs and contributions are deferred to full version [7].

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Notes

  1. 1.

    Here, we abuse a term “algorithm.” Despite that an algorithm must have a finite description conventionally, a target function (and hence a set of target functions) may not, as defined in Sect. 2. To compensate the abuse, when we will show the existence of an algorithm, we insist on giving a finite procedure to compute it.

  2. 2.

    We say that one object is similar to another, if the latter is obtained from the former by a combination of scaling, translation, and rotation (but not by a reflection).

  3. 3.

    Since \(Z_i\) is self-centric, \((0,0) \not \in P\) means an error of eye sensor, which we assume will not occur.

  4. 4.

    We use the same notation < to denote the lexicographic order on \(R^2\) and the order on R to save the number of notations.

  5. 5.

    We assume \(\boldsymbol{p}_0 = \boldsymbol{q}_0\).

  6. 6.

    Since \(dist(\boldsymbol{o}_P, \boldsymbol{o}_P) = 0\), \(V_P({\boldsymbol{q}})\) is not compared with \(V_P(\boldsymbol{o}_P)\) with respect to \(\sqsupset \).

  7. 7.

    Note that Z is uniquely determined, and the unit distance of Z is \(dist(\boldsymbol{p}_1,\boldsymbol{p}_3)/31\).

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Acknowledgments

This work is supported in part by JSPS KAKENHI Grant Numbers JP17K00024 and JP22K11915.

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Authors and Affiliations

  1. Kyushu Sangyo University, Fukuoka, Japan

    Yuichi Asahiro

  2. Kyushu University, Fukuoka, Japan

    Masafumi Yamashita

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  1. Yuichi Asahiro
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  2. Masafumi Yamashita
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Correspondence to Yuichi Asahiro .

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Editors and Affiliations

  1. Ben-Gurion University of the Negev, Be'er-Sheva, Israel

    Shlomi Dolev

  2. New Jersey Institute of Technology, Newark, NJ, USA

    Baruch Schieber

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Asahiro, Y., Yamashita, M. (2023). Minimum Algorithm Sizes for Self-stabilizing Gathering and Related Problems of Autonomous Mobile Robots (Extended Abstract). In: Dolev, S., Schieber, B. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2023. Lecture Notes in Computer Science, vol 14310. Springer, Cham. https://doi.org/10.1007/978-3-031-44274-2_23

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  • DOI: https://doi.org/10.1007/978-3-031-44274-2_23

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