Doing-it-All with Bounded Work and Communication

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 $O(t + p ^{1.77})$

The Complexity of Synchronous Iterative Do-All with Crashes

Do-All is the problem of performing N tasks in a distributed system of P failure-prone processors [9]. Many distributed and parallel algorithms have been developed for this basic problem and several algorithm simulations have been developed by iterating Do-All algorithms. The eciency of the solutions for Do-All is measured in terms of work complexity where all processing steps taken by the processors are counted. Work is ideally expressed as a function of N , P , and f , the number of processor crashes. However the known lower bounds and the upper bounds for extant algorithms do not adequately show how work depends on f . We present the rst non-trivial lower bounds for Do-All that capture the dependence of work on N , P and f . For the model of computation where processors are able to make perfect load-balancing decisions locally, we also present matching upper bounds. Thus we give the rst complete analysis of DoAll for this model. We dene the r-iterative Do-All problem that abstracts the repeated use of Do-All such as found in algorithm simulations. Our f-sensitive analysis enables us to derive a tight bound for r-iterative Do-All work (that is stronger than the r-fold work complexity of a single Do-All). Our approach that models perfect load-balancing allows for the analysis of specic algorithms to be divided into two parts: (i) the analysis of the cost of tolerating failures while performing work, and (ii) the analysis of the cost of implementing load-balancing. We demonstrate the utility and generality of this approach by improving the analysis of two known ecient algorithms. We give an improved analysis of an ecient message-passing algorithm (algorithm AN [5]). We also derive a new and complete analysis of the best known Do-All algorithm for..

The complexity of synchronous iterative Do-All with crashes

Abstract. Do-All is the problem of performing N tasks in a distributed system of P failure-prone processors [8]. Many distributed and parallel algorithms have been developed for this problem and several algorithm simulations have been developed by iterating Do-All algorithms. The efficiency of the solutions for Do-All is measured in terms of work complexity where all processing steps taken by the processors are counted. We present the first non-trivial lower bounds for Do-All that capture the dependence of work on N, P and f, the number of processor crashes. For the model of computation where processors are able to make perfect load-balancing decisions locally, we also present matching upper bounds. We define the r-iterative Do-All problem that abstracts the repeated use of Do-All such as found in algorithm simulations. Our f-sensitive analysis enables us to derive a tight bound for r-iterative Do-All work (that is stronger than the r-fold work complexity of a single Do-All). Our approach that models perfect load-balancing allows for the analysis of specific algorithms to be divided into two parts: (i) the analysis of the cost of tolerating failures while performing work, and (ii) the analysis of the cost of implementing load-balancing. We demonstrate the utility and generality of this approach by improving the analysis of two known efficient algorithms. Finally we present a new upper bound on simulations of synchronous shared-memory algorithms on crash-prone processors.