Writing-all deterministically and optimally using a non-trivial number of asynchronous processors

• Authors: Dariusz R. Kowalski;Alexander A. Shvartsman
• Venue: Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures
• Year: 2004

The problem of performing n tasks on p asynchronous or undependable processors is a basic problem in distributed computing. This paper considers an abstraction of this problem called Write-All: using p processors write 1's into all locations of an array of size n. In this problem writing 1 abstracts the notion of performing a simple task. Despite substantial research, there is a dearth of efficient deterministic asynchronous algorithms for Write-All. Efficiency of algorithms is measured in terms of work that accounts for all local steps performed by the processors in solving the problem. Thus an optimal algorithm would have work Θ(n), however it is known that optimality cannot be achieved when p=Ω(n). The quest then is to obtain work-optimal solutions for this problem using a non-trivial, compared to n, number of processors p. Recently it was shown that optimality can be achieved using a non-trivial number M of processors, where M=4√n/log n. The new result in this paper significantly extends the range of processors for which optimality is achieved. The result shows that optimality can be achieved using close to M2 processors; more precisely, using (M log M)2-ε processors, for any ε 0. Additionally, the new result uses only the atomic read/write memory, without resorting to using the test-and-set primitive that was necessary in the previous solution. This paper presents the algorithm and gives its analysis showing that the work complexity of the algorithm is O(n+p2+ε), which is optimal when p = O(n1/(2+ε)), while all prior deterministic algorithms require super-linear work when p=Ω(n1/4).