Performing work with asynchronous processors: Message-delay-sensitive bounds

Abstract

This paper considers the problem of performing tasks in asynchronous distributed settings. This problem, called Do-All, has been substantially studied in synchronous models, but there is a dearth of efficient algorithms for asynchronous message-passing processors. Do-All can be trivially solved without any communication by an algorithm where each processor performs all tasks. Assuming p processors and t tasks, this requires work Θ ( p · t ). Thus, it is important to develop subquadratic solutions (when p and t are comparable) by trading computation for communication. Following the observation that it is not possible to obtain subquadratic work when the message delay d is substantial, e.g., d = Θ ( t ), this work pursues a message-delay-sensitive approach. Here, the upper bounds on work and communication are given as functions of p , t , and d , the upper bound on message delays, however, algorithms have no knowledge of d and they cannot rely on the existence of an upper bound on d . This paper presents two families of asynchronous algorithms achieving, for the first time, subquadratic work as long as d = o ( t ). The first family uses as its basis a shared-memory algorithm without having to emulate atomic registers assumed by that algorithm. These deterministic algorithms have work O ( tp ε + pd ⌈ t / d ⌉ ε ) for any ε > 0. The second family uses specific permutations of tasks, with certain combinatorial properties, to sequence the work of the processors. These randomized (deterministic) algorithms have expected (worst-case) work O ( t log p + pd log (2 + t / d )). Another important contribution in this work is the first delay-sensitive lower bound for this problem that helps explain the behavior of our algorithms: any randomized (deterministic) algorithm has expected (worst-case) work of Ω ( t + pd log d +1 t ).