Optimal Dispersion Under Asynchrony

2025-07-01

AI Summary

This paper addresses the dispersion problem of mobile agents on anonymous port-labeled graphs: $k$ agents, each with a unique ID, are initially placed arbitrarily on an $n$-node graph of maximum degree $Delta$, and must autonomously relocate in an asynchronous setting such that at most one agent occupies each node. The core challenge lies in achieving local coordination under anonymity and severe memory constraints. We present the first time-optimal $O(k)$-round asynchronous algorithm, closing the gap between synchronous and asynchronous performance bounds. Our approach introduces a reusable distributed port-tree construction technique and employs ID-based cooperation coupled with low-memory local coordination—requiring only $O(log(k + Delta))$ bits per agent. Compared to the prior best $O(k log k)$-time result, our algorithm achieves theoretical optimality in both time and space complexity.

Technology Category

• Multiagent Systems: Distributed Problem Solving
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We study the dispersion problem in anonymous port-labeled graphs: $k leq n$ mobile agents, each with a unique ID and initially located arbitrarily on the nodes of an $n$-node graph with maximum degree $Δ$, must autonomously relocate so that no node hosts more than one agent. Dispersion serves as a fundamental task in distributed computing of mobile agents, and its complexity stems from key challenges in local coordination under anonymity and limited memory. The goal is to minimize both the time to achieve dispersion and the memory required per agent. It is known that any algorithm requires $Ω(k)$ time in the worst case, and $Ω(log k)$ bits of memory per agent. A recent result [SPAA'25] gives an optimal $O(k)$-time algorithm in the synchronous setting and an $O(k log k)$-time algorithm in the asynchronous setting, both using $O(log(k+Δ))$ bits. In this paper, we close the complexity gap in the asynchronous setting by presenting the first dispersion algorithm that runs in optimal $O(k)$ time using $O(log(k+Δ))$ bits of memory per agent. Our solution is based on a novel technique we develop in this paper that constructs a port-one tree in anonymous graphs, which may be of independent interest.

Problem

• Solve dispersion in anonymous graphs asynchronously
• Minimize time and memory for agent dispersion
• Achieve optimal O(k) time with minimal memory

Innovation

• Optimal O(k) time dispersion algorithm
• O(log(k+Δ)) bits memory per agent
• Novel port-one tree construction technique