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Compatibility of Convergence Algorithms for Autonomous Mobile Robots (Extended Abstract)

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Structural Information and Communication Complexity (SIROCCO 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13892))

Abstract

We investigate autonomous mobile robots in the Euclidean plane. A robot has a function called target function to decide the destination from the robots’ positions, and operates in Look-Compute-Move cycles, i.e., identifies the robots’ positions, computes the destination by the target function, and then moves there. Robots can have different target functions. Let \(\varPhi \) and \(\varPi \) be a set of target functions and a problem, respectively. If the robots whose target functions are chosen from \(\varPhi \) always solve \(\varPi \), we say that \(\varPhi \) is compatible with respect to \(\varPi \). Suppose that \(\varPhi \) is compatible with respect to \(\varPi \). Then two swarms controlled by (possibly different) target functions in \(\varPhi \) can merge to form a larger swarm, and a broken robot can be replaced with another robot with any target function in \(\varPhi \), keeping the correctness of solving \(\varPi \). We investigate the convergence, the gathering, and some fault tolerant convergence problems, assuming crash failures, from the view point of compatibility.

Due to the space limitation, we omit most of the proofs and some contributions. The full version of the paper [4] contains them.

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Notes

  1. 1.

    Roughly, a target function is a function from \((R^2)^n\) to \(R^2\), where R is the set of real numbers and n is the number of robots, i.e., given a snapshot in \((R^2)^n\), it returns a destination point in \(R^2\). Later, we define a target function a bit more carefully.

  2. 2.

    Let P, D, and \(\boldsymbol{o}\) be the multiset of robots’ positions, the axes aligned minimum box containing P, and its center, respectively. Define \(\delta * D = \{ (1 - 2 \delta ) \boldsymbol{x} + 2 \delta \boldsymbol{o} : \boldsymbol{x} \in D \}\). A function \(\phi \) is \(\delta \)-inner, if \(\phi (P)\) is included in \(\delta * D\) for any P.

  3. 3.

    Here, we abuse term “algorithm,” since an algorithm must have a finite description. A target function may not. To compensate the abuse, we insist on giving a finite procedure when we show the existence of a target function.

  4. 4.

    That \((0,0) \not \in P\) means an error of eye sensor, which we assume will not occur, in this paper.

  5. 5.

    For the sake of completeness, we assume that \(\alpha (\phi ,P) = 0\) when \(\phi (P) = \bot \).

  6. 6.

    Since \((0,0) \in Q^{(i)}_t\) by definition, \(\boldsymbol{y} \not = \bot \).

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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

Authors
  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. National Autonomous University of Mexico, Mexico, Mexico

    Sergio Rajsbaum

  2. Gran Sasso Science Institute, L’Aquila, Italy

    Alkida Balliu

  3. Arizona State University, Tempe, AZ, USA

    Joshua J. Daymude

  4. Gran Sasso Science Institute, L’Aquila, Italy

    Dennis Olivetti

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Asahiro, Y., Yamashita, M. (2023). Compatibility of Convergence Algorithms for Autonomous Mobile Robots (Extended Abstract). In: Rajsbaum, S., Balliu, A., Daymude, J.J., Olivetti, D. (eds) Structural Information and Communication Complexity. SIROCCO 2023. Lecture Notes in Computer Science, vol 13892. Springer, Cham. https://doi.org/10.1007/978-3-031-32733-9_8

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  • DOI: https://doi.org/10.1007/978-3-031-32733-9_8

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