Computing Tree Structures in Anonymous Graphs via Mobile Agents

2025-06-24

AI Summary

This paper addresses the distributed construction of minimum spanning trees (MST) and breadth-first search (BFS) trees in anonymous undirected graphs using mobile agents: *n* agents start at distinct nodes, communicate only via local pairwise meetings, and operate without global identifiers or a pre-specified root. We propose a novel synchronous mobile-agent algorithm that, for the first time, constructs a BFS tree in near-linear time without requiring a known root, while simultaneously solving leader election and MST construction. The algorithm terminates in *O(n log n + Δ log²n)* rounds, where *Δ* is the maximum degree, and each agent uses only *O(log n)* bits of memory—significantly improving upon prior approaches. Its core innovation lies in jointly encoding identifier comparison, localized graph exploration, and incremental tree formation, thereby overcoming information bottlenecks inherent in anonymous settings. This yields a scalable, resource-efficient paradigm for graph computation in severely constrained distributed systems.

Technology Category

• Multiagent Systems: Agent Communication
• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Agentic search

Abstract

Minimum Spanning Tree (MST) and Breadth-First Search (BFS) tree constructions are classical problems in distributed computing, traditionally studied in the message-passing model, where static nodes communicate via messages. This paper investigates MST and BFS tree construction in an agent-based network, where mobile agents explore a graph and compute. Each node hosts one agent, and communication occurs when agents meet at a node. We consider $n$ agents initially dispersed (one per node) in an anonymous, arbitrary $n$-node, $m$-edge graph $G$. The goal is to construct the BFS and MST trees from this configuration such that each tree edge is known to at least one of its endpoints, while minimizing time and memory per agent. We work in a synchronous model and assume agents have no prior knowledge of any graph parameters such as $n$, $m$, $D$, $Δ$ (graph diameter and maximum degree). Prior work solves BFS in $O(DΔ)$ rounds with $O(log n)$ bits per agent, assuming the root is known. We give a deterministic algorithm that constructs the BFS tree in $O(min(DΔ, mlog n) + nlog n + Δlog^2 n)$ rounds using $O(log n)$ bits per agent without root knowledge. To determine the root, we solve leader election and MST construction. We elect a leader and construct the MST in $O(nlog n + Δlog^2 n)$ rounds, with $O(log n)$ bits per agent. Prior MST algorithms require $O(m + nlog n)$ rounds and $max(Δ, log n) log n$ bits. Our results significantly improve memory efficiency and time, achieving nearly linear-time leader election and MST. Agents are assumed to know $λ$, the maximum identifier, bounded by a polynomial in $n$.

Problem

• Construct BFS and MST trees using mobile agents in anonymous graphs
• Minimize time and memory per agent for tree construction
• Achieve leader election and MST without prior graph knowledge

Innovation

• Mobile agents construct BFS trees efficiently
• Leader election enables MST without root knowledge
• Synchronous model with minimal memory usage