Agent-based Leader Election, MST, and Beyond

TL;DR

This work studies leader election and minimum spanning tree construction in the agent-based distributed model on anonymous, port-labeled graphs without any a priori graph knowledge. It develops a two-stage, parameter-free leader election (local then global) that works under dispersed, rooted, or general initial configurations, with tight time/memory guarantees, and uses the elected leader to realize an MST in $O(m+n\log n)$ rounds and per-agent memory of $O(\Delta \log n)$ in the general case (or $O(\log^2 n)$ when dispersed). Beyond MST, the paper shows improved time/memory bounds for gathering, MIS, and MDS by leveraging the leader election framework, all without assuming known $n$ or $\Delta$. The approach blends a careful DFS-based global coordination, padding to ensure meetings among similarly-degreed nodes, and MOE-based component merging, demonstrating the practical viability of parameter-free computation for key graph problems in mobile-agent systems. This work thus advances the understanding of how to perform fundamental distributed tasks with relocatable devices in parameter-free, synchronous environments, with potential impact on distributed robotics and network management.

Abstract

Leader election is one of the fundamental and well-studied problems in distributed computing. In this paper, we initiate the study of leader election using mobile agents. Suppose $n$ agents are positioned initially arbitrarily on the nodes of an arbitrary, anonymous, $n$-node, $m$-edge graph $G$. The agents relocate themselves autonomously on the nodes of $G$ and elect an agent as a leader such that the leader agent knows it is a leader and the other agents know they are not leaders. The objective is to minimize time and memory requirements. Following the literature, we consider the synchronous setting in which each agent performs its operations synchronously with others and hence the time complexity can be measured in rounds. The quest in this paper is to provide solutions without agents knowing any graph parameter, such as $n$, a priori. We first establish that, without agents knowing any graph parameter a priori, there exists a deterministic algorithm to elect an agent as a leader in $O(m)$ rounds with $O(n\log n)$ bits at each agent. Using this leader election result, we develop a deterministic algorithm for agents to construct a minimum spanning tree of $G$ in $O(m+n\log n)$ rounds using $O(n \log n)$ bits memory at each agent, without agents knowing any graph parameter a priori. Finally, using the same leader election result, we provide improved time/memory results for other fundamental distributed graph problems, namely, gathering, maximal independent set, and minimal dominating sets, removing the assumptions on agents knowing graph parameters a priori.

Figures (2)

• Figure 1: The components $C_{r_w}$ and $C_{r_u}$ before $C_{r_w}$ merges with $C_{r_u}$ due to the MOE connecting node $r_u\in C_{r_u}$ with node $r_w\in C_{r_w}$ and $rank(C_{r_u})<rank(C_{r_w})$ (if $rank(C_{r_w})<rank(C_{r_u})$, then $C_{r_u}$ merges with $C_{r_w}$ due to the MOE connecting $r_w\in C_{r_w}$ with node $r_u\in C_{r_u}$). Both $C_{r_u}$ and $C_{r_w}$ are rooted trees with roots $r_a$ and $r_y$, respectively. In the figure, $r_A \rightarrow r_B$ denotes $r_A$ is the parent of $r_B$.
• Figure 2: The resulting component $C^{new}_{r_u}$ after merging $C_{r_w}$ with $C_{r_u}$. Component $C^{new}_{r_u}$ gets rank $rank(C^{new}_{r_u})\leftarrow rank(C_{r_u})$ since $C_{r_w}$ merged with $C_{r_u}$ to become $C^{new}_{r_u}$ due to $rank(C_{r_u})<rank(C_{r_w})$. Two parent-child pointers in $C_{r_w}$ are reversed to keep $C^{new}_{r_u}$ a rooted tree.

Theorems & Definitions (38)

• Theorem 1: Simulation
• Theorem 2: Leader election
• Theorem 3: MST
• Lemma 4
• proof
• proof
• Lemma 5
• proof
• Lemma 6
• proof