Revisiting Lower Bounds for Two-Step Consensus

TL;DR

This paper revisits lower bounds for two-step consensus, showing that practical protocols can operate with fewer processes under more realistic conditions, and establishes tight bounds depending on implementation type.

Contribution

It introduces a pragmatic condition for two-step consensus bounds, refining classical lower bounds based on implementation as an object or a task.

Findings

For consensus as an object, $ ext{max}\{2e+f-1,2f+1 ight\}$ processes are necessary and sufficient.

For consensus as a task, the tight bound is $ ext{max}\{2e+f, 2f+1 ight\}$ processes.

Classical bounds are overly conservative for practical scenarios.

Abstract

A seminal result by Lamport shows that at least $\max\{2e+f+1,2f+1\}$ processes are required to implement partially synchronous consensus that tolerates $f$ process failures and can furthermore decide in two message delays under $e$ failures. This lower bound is matched by the classical Fast Paxos protocol. However, more recent practical protocols, such as Egalitarian Paxos, provide two-step decisions with fewer processes, seemingly contradicting the lower bound. We show that this discrepancy arises because the classical bound requires two-step decisions under a wide range of scenarios, not all of which are relevant in practice. We propose a more pragmatic condition for which we establish tight bounds on the number of processes required. Interestingly, these bounds depend on whether consensus is implemented as an atomic object or a decision task. For consensus as an object, $\max\{2e+f-1,2f+1\}$ processes are necessary and sufficient for two-step decisions, while for a task the tight bound is $\max\{2e+f, 2f+1\}$.