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Near-Optimal Dispersion on Arbitrary Anonymous Graphs

Authors Ajay D. Kshemkalyani , Gokarna Sharma

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Ajay D. Kshemkalyani
  • University of Illinois at Chicago, IL, USA
Gokarna Sharma
  • Kent State University, OH, USA

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Ajay D. Kshemkalyani and Gokarna Sharma. Near-Optimal Dispersion on Arbitrary Anonymous Graphs. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.OPODIS.2021.8

Abstract

Given an undirected, anonymous, port-labeled graph of n memory-less nodes, m edges, and degree Δ, we consider the problem of dispersing k ≤ n robots (or tokens) positioned initially arbitrarily on one or more nodes of the graph to exactly k different nodes of the graph, one on each node. The objective is to simultaneously minimize time to achieve dispersion and memory requirement at each robot. If all k robots are positioned initially on a single node, depth first search (DFS) traversal solves this problem in O(min{m,kΔ}) time with Θ(log(k+Δ)) bits at each robot. However, if robots are positioned initially on multiple nodes, the best previously known algorithm solves this problem in O(min{m,kΔ}⋅ log 𝓁) time storing Θ(log(k+Δ)) bits at each robot, where 𝓁 ≤ k/2 is the number of multiplicity nodes in the initial configuration. In this paper, we present a novel multi-source DFS traversal algorithm solving this problem in O(min{m,kΔ}) time with Θ(log(k+Δ)) bits at each robot, improving the time bound of the best previously known algorithm by O(log 𝓁) and matching asymptotically the single-source DFS traversal bounds. This is the first algorithm for dispersion that is optimal in both time and memory in arbitrary anonymous graphs of constant degree, Δ = O(1). Furthermore, the result holds in both synchronous and asynchronous settings.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Computing methodologies → Distributed algorithms
  • Computer systems organization → Robotics
Keywords
  • Distributed algorithms
  • Multi-agent systems
  • Mobile robots
  • Local communication
  • Dispersion
  • Exploration
  • Time and memory complexity

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