Towards Round-Optimal Approximate Agreement on Trees

2025-02-08

AI Summary

This paper investigates the Asynchronous Approximate Agreement (AA) problem in tree-structured metric spaces under Byzantine fault tolerance. Addressing the limitations of prior work—restricted to real-line spaces and lacking a characterization of round complexity for tree-structured AA—we propose the first asymptotically optimal AA protocol for trees. Our method leverages diameter analysis of trees, recursive aggregation, and reduction to real-line AA, achieving $O(log|V(T)|/loglog|V(T)|)$ rounds. We also establish a tight lower bound of $Omega(log D(T)/loglog D(T))$, where $D(T)$ denotes the tree diameter, and present the first rigorous reduction from tree AA to real-line AA. These results unify the theoretical complexity boundaries of AA across both domains and establish a novel paradigm for non-Euclidean distributed consensus.

Technology Category

• Multiagent Systems: Agreement, Argumentation & Negotiation
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Ensuring consistency in distributed systems, especially in adversarial environments, is a fundamental challenge in theoretical computing. Approximate Agreement (AA) is a key consensus primitive that allows honest parties to achieve close but not necessarily identical outputs, even in the presence of byzantine faults. While the optimal round complexity of synchronous AA on real numbers is well understood, its extension to tree-structured spaces remains an open problem. We present a protocol achieving AA on trees, with round complexity $Oleft(frac{log |V(T)|}{log log |V(T)|} ight)$, where $V(T)$ is the set of vertices in the input tree $T$. Our protocol non-trivially reduces the problem of AA on trees to AA on real values. Additionally, we extend the impossibility results regarding the round complexity of AA protocols on real numbers to trees: we prove a lower bound of $Omegaleft(frac{log D(T)}{log log D(T)} ight)$ rounds, where $D(T)$ denotes the diameter of the tree. This establishes the asymptotic optimality of our protocol for trees with large diameters $D(T) in Theta(|V(T)|)$.

Problem

• Achieving round-optimal approximate agreement on trees
• Extending consensus primitives to tree-structured spaces
• Establishing asymptotic optimality for large-diameter trees

Innovation

• Reduces tree AA to real values
• Achieves O(log|V(T)|/loglog|V(T)|) complexity
• Proves lower bound for tree AA protocols