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Uniform Deployment of Mobile Robots in Complete Bipartite Graphs

Authors Masahiro Shibata , Naoki Kitamura , Ryota Eguchi , Yuichi Sudo , Junya Nakamura , Yonghwan Kim , Yoshiaki Katayama , Toshimitsu Masuzawa , Quentin Bramas , Sébastien Tixeuil

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Subject Classification

ACM Subject Classification
  • Theory of computation → Self-organization
Keywords
  • mobile robots
  • uniform deployment
  • complete bipartite graphs

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Abstract

In this paper, we address the problem of uniformly deploying mobile robots in complete bipartite graphs. Specifically, when n robots are positioned arbitrarily at distinct nodes in a complete bipartite graph K_{n,n}, which consists of two n-node sets V_L and V_R, the uniform deployment problem requires the robots to achieve one of the following configurations: (a) each node in V_L is occupied by exactly one robot, with no robots in V_R, or (b) each node in V_R is occupied by exactly one robot, with no robots in V_L. In either configuration, the distance between any two robots is 2, ensuring that the robots are uniformly deployed. In this paper, we explore the relationship between the visibility range of robots and the solvability of the uniform deployment problem. First, we characterize solvable and unsolvable initial configurations under the assumption that robots have an infinite visibility range. Next, we demonstrate that visibility range 1 (meaning robots can only observe nodes at a distance of 1 and the robots positioned on them) is insufficient, proving the impossibility of solving the problem under this constraint. Conversely, we show that visibility range Θ(log n) is sufficient by presenting an algorithm that solves the uniform deployment problem in O(1) rounds, starting from any solvable initial configuration. Finally, we briefly introduce an example showing that robots with a constant visibility range (which is 3 in this example) cannot solve the problem in a native way.

Cite As Get BibTex

Masahiro Shibata, Naoki Kitamura, Ryota Eguchi, Yuichi Sudo, Junya Nakamura, Yonghwan Kim, Yoshiaki Katayama, Toshimitsu Masuzawa, Quentin Bramas, and Sébastien Tixeuil. Uniform Deployment of Mobile Robots in Complete Bipartite Graphs. In 29th International Conference on Principles of Distributed Systems (OPODIS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 361, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.OPODIS.2025.34

Author Details

Masahiro Shibata
  • Kyushu Institute of Technology, Fukuoka, Japan
Naoki Kitamura
  • The University of Osaka, Japan
Ryota Eguchi
  • Nara Institute of Science and Technology, Japan
Yuichi Sudo
  • Hosei University, Tokyo, Japan
Junya Nakamura
  • Toyohashi University of Technology, Aichi, Japan
Yonghwan Kim
  • Nagoya Institute of Technology, Aichi, Japan
Yoshiaki Katayama
  • Nagoya Institute of Technology, Aichi, Japan
Toshimitsu Masuzawa
  • Notre Dame Seishin University, Okayama, Japan
Quentin Bramas
  • University of Strasbourg, ICube, CNRS, France
Sébastien Tixeuil
  • Sorbonne University, CNRS, LIP6, IUF, Paris, France

Funding

  • Shibata, Masahiro: JSPS KAKENHI 23K28037 and 25K14995
  • Kitamura, Naoki: JSPS KAKENHI 23K16838
  • Sudo, Yuichi: JSPS KAKENHI 25K03078 and 25K03079, and JST FOREST Program JPMJFR226U
  • Nakamura, Junya: JSPS KAKENHI 22K11971 and 25K03101
  • Katayama, Yoshiaki: JSPS KAKENHI 25K14991
  • Masuzawa, Toshimitsu: JSPS KAKENHI 20KK0232 and 25K15057
  • Tixeuil, Sébastien: ANR SAPPORO (ANR-19-CE25-0005)

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