Improved Linear-Time Construction of Minimal Dominating Set via Mobile Agents

TL;DR

This paper introduces two new linear-time algorithms for computing minimal dominating sets in anonymous graphs using mobile agents, significantly improving efficiency while also constructing spanning trees and electing leaders.

Contribution

The paper presents the first linear-time algorithms for minimal dominating set computation in anonymous graphs with mobile agents, requiring only logarithmic memory and no global knowledge.

Findings

Minimal dominating set computed in O(n) rounds

Constructs spanning tree and elects leader in O(n) rounds

Uses only O(log n) bits of memory per agent

Abstract

Mobile agents have emerged as a powerful framework for solving fundamental graph problems in distributed settings in recent times. These agents, modelled as autonomous physical or software entities, possess local computation power, finite memory and have the ability to traverse a graph, offering efficient solutions to a range of classical problems. In this work, we focus on the problem of computing a \emph{minimal dominating set} (mDS) in anonymous graphs using mobile agents. Building on the recently proposed optimal dispersion algorithm on the synchronous mobile agent model, we design two new algorithms that achieve a \emph{linear-time} solution for this problem in the synchronous setting. Specifically, given a connected $n$-node graph with $n$ agents initially placed in either rooted or arbitrary configurations, we show that an mDS can be computed in $O(n)$ rounds using only $O(\log n)$ bits of memory per agent, without using any prior knowledge of any global parameters. This improves upon the best-known complexity results in the literature over the same model. In addition, as natural by-products of our methodology, our algorithms also construct a spanning tree and elect a unique leader in $O(n)$ rounds, which are also important results of independent interest in the mobile-agent framework.