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

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 (tpε + 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 logd+1t).