Lattice Agreement in Message Passing Systems

TL;DR

This paper introduces improved algorithms for lattice agreement in distributed systems, achieving better round and time complexities in both synchronous and asynchronous models, and extends to the generalized lattice agreement for replicated state machines.

Contribution

It presents new algorithms with reduced round and time complexities for lattice agreement and generalized lattice agreement in message passing systems, outperforming prior solutions.

Findings

Synchronous algorithms run in $ ext{log} f$ and $ ext{min} ext{ } ext{O}( ext{log}^2 h(L)), ext{O}( ext{log}^2 f)$ rounds.

Asynchronous lattice agreement achieved with $2 imes ext{min} ext{ } ext{O}(h(L)), f+1$ message delays.

Generalized lattice agreement guarantees liveness with $ ext{min} ext{ } ext{O}(h(L)), ext{O}(f)$ time, better than previous $O(n)$.

Abstract

This paper studies the lattice agreement problem and the generalized lattice agreement problem in distributed message passing systems. In the lattice agreement problem, given input values from a lattice, processes have to non-trivially decide output values that lie on a chain. We consider the lattice agreement problem in both synchronous and asynchronous systems. For synchronous lattice agreement, we present two algorithms which run in $\log f$ and $\min \{O(\log^2 h(L)), O(\log^2 f)\}$ rounds, respectively, where $h(L)$ denotes the height of the {\em input sublattice} $L$, $f < n$ is the number of crash failures the system can tolerate, and $n$ is the number of processes in the system. These algorithms have significant better round complexity than previously known algorithms. The algorithm by Attiya et al. \cite{attiya1995atomic} takes $\log n$ synchronous rounds, and the algorithm by Mavronicolasa \cite{mavronicolasabound} takes $\min \{O(h(L)), O(\sqrt{f})\}$ rounds. For asynchronous lattice agreement, we propose an algorithm which has time complexity of $2 \cdot \min \{h(L), f + 1\}$ message delays which improves on the previously known time complexity of $O(n)$ message delays. The generalized lattice agreement problem defined by Faleiro et al in \cite{faleiro2012generalized} is a generalization of the lattice agreement problem where it is applied for the replicated state machine. We propose an algorithm which guarantees liveness when a majority of the processes are correct in asynchronous systems. Our algorithm requires $\min \{O(h(L)), O(f)\}$ units of time in the worst case which is better than $O(n)$ units of time required by the algorithm of Faleiro et al. \cite{faleiro2012generalized}.