A Speedup Theorem for Asynchronous Computation with Applications to Consensus and Approximate Agreement

TL;DR

This paper introduces the asynchronous speedup theorem, a novel method for proving lower bounds in distributed computing, demonstrating its application to consensus and approximate agreement problems.

Contribution

It establishes a new speedup theorem for asynchronous models and applies it to prove impossibility results and analyze the power of additional objects in distributed tasks.

Findings

The speedup theorem reduces the number of rounds needed to solve tasks.

Consensus remains impossible with certain objects, as shown by closure analysis.

Additional objects do not speed up approximate agreement despite increased computational power.

Abstract

We study two fundamental problems of distributed computing, consensus and approximate agreement, through a novel approach for proving lower bounds and impossibility results, that we call the asynchronous speedup theorem. For a given $n$-process task $\Pi$ and a given computational model $M$, we define a new task, called the closure of $\Pi$ with respect to $M$. The asynchronous speedup theorem states that if a task $\Pi$ is solvable in $t\geq 1$ rounds in $M$, then its closure w.r.t. $M$ is solvable in $t-1$ rounds in $M$. We prove this theorem for iterated models, as long as the model allows solo executions. We illustrate the power of our asynchronous speedup theorem by providing a new proof of the wait-free impossibility of consensus using read/write registers, and a new proof of the wait-free impossibility of solving consensus using registers and test\&set objects for $n>2$. The proof is merely by showing that, in each case, the closure of consensus (w.r.t. the corresponding model) is consensus itself. Our main application is the study of the power of additional objects, namely test\&set and binary consensus, for wait-free solving approximate agreement faster. By analyzing the closure of approximate agreement w.r.t. each of the two models, we show that while these objects are more powerful than read/write registers from the computability perspective, they are not more powerful as far as helping solving approximate agreement faster is concerned.