Disconnected Agreement in Networks Prone to Link Failures

TL;DR

This paper studies algorithms for achieving agreement in networks with unreliable links, introducing concepts like stretch and providing algorithms with different time and message complexities based on network knowledge.

Contribution

It introduces the concept of stretch for disconnected agreement and presents algorithms with optimal and near-optimal time complexities under various assumptions.

Findings

An algorithm working in $O( ext{stretch})$ time with $O( ext{log } n)$ bits messages.

A general $n+1$ rounds algorithm with $O( ext{log } n)$ bits messages.

An optimal $O( ext{stretch})$ time algorithm with larger message size.

Abstract

We consider deterministic distributed algorithms for reaching agreement in synchronous networks of arbitrary topologies. Links are bi-directional and prone to failures while nodes stay non-faulty at all times. A faulty link may omit messages. Agreement among nodes is understood as holding in each connected component of a network obtained by removing faulty links. We call ``disconnected agreement'' the algorithmic problem of reaching such agreement. We introduce the concept of stretch, which is the number of connected components of a network, obtained by removing faulty links, minus~$1$ plus the sum of diameters of connected components. We define the concepts of ``fast'' and ``early-stopping'' algorithms for disconnected agreement by referring to stretch. A network has $n$ nodes and $m$ links. Nodes are normally assumed to know their own names and ability to associate communication with local ports. If we additionally assume that a bound~$\Lambda$ on stretch is known to all nodes, then there is an algorithm for disconnected agreement working in time $O(\Lambda)$ using messages of $O(\log n)$ bits. We give a general disconnected agreement algorithm operating in~$n+1$ rounds that uses messages of $O(\log n)$ bits. Let~$\lambda$ be an unknown stretch occurring in an execution; we give an algorithm working in time~$(\lambda+2)^3$ and using messages of $O(n\log n)$ bits. We show that disconnected agreement can be solved in the optimal $O(\lambda)$ time, but at the cost of increasing message size to~$O(m\log n)$. We also design an algorithm that uses only~$O(n)$ non-faulty links and works in time~$O(n m)$, while nodes start with their ports mapped to neighbors and messages carry $O(m\log n)$ bits. We prove lower bounds on the performance of disconnected-agreement solutions that refer to the parameters of evolving network topologies and the knowledge available to nodes.