Doing-it-All with Bounded Work and Communication

TL;DR

This paper introduces new algorithms for the Do-All problem in a crash-prone, message-passing system, optimizing work and communication efficiency with near-optimal effort bounds.

Contribution

It presents both constructive and nonconstructive algorithms with improved work and effort bounds for the Do-All problem under crash failures.

Findings

Nonconstructive algorithm approaches near lower bound on work

Effort proportional to work when crashes are limited

Algorithms achieve efficient completion with bounded crashes

Abstract

We consider the Do-All problem, where $p$ cooperating processors need to complete $t$ similar and independent tasks in an adversarial setting. Here we deal with a synchronous message passing system with processors that are subject to crash failures. Efficiency of algorithms in this setting is measured in terms of work complexity (also known as total available processor steps) and communication complexity (total number of point-to-point messages). When work and communication are considered to be comparable resources, then the overall efficiency is meaningfully expressed in terms of effort defined as work + communication. We develop and analyze a constructive algorithm that has work $O( t + p \log p\, (\sqrt{p\log p}+\sqrt{t\log t}\, ) )$ and a nonconstructive algorithm that has work $O(t +p \log^2 p)$. The latter result is close to the lower bound $\Omega(t + p \log p/ \log \log p)$ on work. The effort of each of these algorithms is proportional to its work when the number of crashes is bounded above by $c\,p$, for some positive constant $c < 1$. We also present a nonconstructive algorithm that has effort $O(t + p ^{1.77})$.