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Partial Gathering of Mobile Agents in Dynamic Tori

Authors Masahiro Shibata , Naoki Kitamura , Ryota Eguchi , Yuichi Sudo , Junya Nakamura , Yonghwan Kim



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Author Details

Masahiro Shibata
  • Kyushu Institute of Technology, Fukuoka, Japan
Naoki Kitamura
  • Osaka University, Japan
Ryota Eguchi
  • NAIST, Nara, Japan
Yuichi Sudo
  • Hosei University, Tokyo, Japan
Junya Nakamura
  • Toyohashi University of Technology, Aichi, Japan
Yonghwan Kim
  • Nagoya Institute of Technology, Aichi, Japan

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Masahiro Shibata, Naoki Kitamura, Ryota Eguchi, Yuichi Sudo, Junya Nakamura, and Yonghwan Kim. Partial Gathering of Mobile Agents in Dynamic Tori. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 2:1-2:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SAND.2023.2

Abstract

In this paper, we consider the partial gathering problem of mobile agents in synchronous dynamic tori. The partial gathering problem is a generalization of the (well-investigated) total gathering problem, which requires that all k agents distributed in the network terminate at a non-predetermined single node. The partial gathering problem requires, for a given positive integer g (< k), that agents terminate in a configuration such that either at least g agents or no agent exists at each node. So far, in almost cases, the partial gathering problem has been considered in static graphs. As only one exception, it is considered in a kind of dynamic rings called 1-interval connected rings, that is, one of the links in the ring may be missing at each time step. In this paper, we consider partial gathering in another dynamic topology. Concretely, we consider it in n× n dynamic tori such that each of row rings and column rings is represented as a 1-interval connected ring. In such networks, when k = O(gn), focusing on the relationship between the values of k, n, and g, we aim to characterize the solvability of the partial gathering problem and analyze the move complexity of the proposed algorithms when the problem can be solved. First, we show that agents cannot solve the problem when k = o(gn), which means that Ω (gn) agents are necessary to solve the problem. Second, we show that the problem can be solved with the total number of O(gn³) moves when 2gn+2n-1 ≤ k ≤ 2gn + 6n +16g -12. Finally, we show that the problem can be solved with the total number of O(gn²) moves when k ≥ 2gn + 6n +16g -11. From these results, we show that our algorithms can solve the partial gathering problem in dynamic tori with the asymptotically optimal number Θ (gn) of agents. In addition, we show that agents require a total number of Ω(gn²) moves to solve the partial gathering problem in dynamic tori when k = Θ(gn). Thus, when k ≥ 2gn+6n+16g -11, our algorithm can solve the problem with asymptotically optimal number O(gn²) of agent moves.

Subject Classification

ACM Subject Classification
  • Theory of computation → Self-organization
Keywords
  • distributed system
  • mobile agents
  • partial gathering
  • dynamic tori

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