$t$-Resilient $k$-Immediate Snapshot and its Relation with Agreement Problems

TL;DR

This paper introduces the $k$-resilient immediate snapshot, explores its impossibility in certain models, and establishes its relation to agreement problems, showing how it enhances computational power in asynchronous systems with crash failures.

Contribution

It defines the $k$-resilient immediate snapshot, proves its impossibility for certain parameters, and links it to agreement problems, extending understanding of distributed computing objects.

Findings

$k$-resilient immediate snapshot is impossible for $k,t<n-1$.

Systems with $k$-immediate snapshot can solve $x$-set agreement for specific $x$.

$k$-resilient immediate snapshot and consensus are equivalent under certain conditions.

Abstract

An immediate snapshot object is a high level communication object, built on top of a read/write distributed system in which all except one processes may crash. It allows a process to write a value and obtain a set of values that represent a snapshot of the values written to the object, occurring immediately after the write step. Considering an $n$-process model in which up to $t$ processes may crash, this paper introduces first the $k$-resilient immediate snapshot object, which is a natural generalization of the basic immediate snapshot (which corresponds to the case $k=t=n-1$). In addition to the set containment properties of the basic immediate snapshot, a $k$-resilient immediate snapshot object requires that each set returned to a process contains at least $(n-k)$ pairs. The paper first shows that, for $k,t<n-1$, $k$-resilient immediate snapshot is impossible in asynchronous read/write systems. %Then the paper investigates the space of objects that %are impossible to solve in $n$-process $t$-crash read/write systems. Then the paper investigates a model of computation where the processes communicate with each other by accessing $k$-immediate snapshot objects, and shows that this model is stronger than the $t$-crash model. Considering the space of $x$-set agreement problems (which are impossible to solve in systems such that $x\leq t$), the paper shows then that $x$-set agreement can be solved in read/write systems enriched with $k$-immediate snapshot objects for $x=\max(1,t+k-(n-2))$. It also shows that, in these systems, $k$-resilient immediate snapshot and consensus are equivalent when $1\leq t<n/2$ and $t\leq k\leq (n-1)-t$. Hence, %thanks to the problem map it provides, the paper establishes strong relations linking fundamental distributed computing objects (one related to communication, the other to agreement), which are impossible to solve in pure read/write systems.