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

Abstract

The problem of performing n tasks on p asynchronous or undependable processors is a basic problem in distributed computing. This article considers an abstraction of this problem called WriteAll: 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 (n), however it is known that optimality cannot be achieved when p = (n). The quest then is to obtain work-optimal solutions for this problem using a nontrivial, compared to n, number of processors p. The algorithm presented in this article has work complexity of O(n + p 2+ ), and it achieves work optimality for p = O(n 1/(2+ε) ) for any ε > 0, while the previous best result achieved optimality for p ≤ 4 √ n/ log n. 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.

Solving the at-most-once problem with nearly optimal effectiveness

“…Another related problem 3 is process renaming, see Attiya et al [4] where each process identifier should be assigned to at most one process. The at-most-once problem has also many similarities with the Write-All problem for the shared memory model [3,9,18,23,28,36]. First presented by Kanellakis and Shvartsman [23], the Write-All problem is concerned with performing each job at-least-once.…”

“…The algorithm presented by Malewicz [36] has work O(n + m 4 log n) and uses test-and-set operations. Later Kowalski and Shvartsman [28] presented a solution for the Write-All problem that for any constant ǫ has work O(n + m 2+ǫ ). Their algorithm uses a collection of q permutations with contention O(q log q) for a properly chosen constant q and does not rely on test-and-set operations.…”

How to Allocate Tasks Asynchronously

“…There have been many subsequent papers on the topic, e.g. [4], [14], [16], [23], [25], [26]. Asynchronous task allocation is also related to distributed collect [2], in which p processors need to aggregate values from m registers.…”

“…The last twenty years have seen a long-standing quest to establish the complexity of asynchronous task allocation [4], [14], [16], [23], [25], [26]. Various structures such as low-contention permutations [4] and expander-based progress graphs [16] were developed to this end.…”

“…The most efficient deterministic task allocation algorithm for general m and p, given by Kowalski and Shvartsman [23], uses O(m+p 2+ ) total work. It uses a collection of permutations with low contention; to date, it is not known how to construct such permutations in polynomial time, therefore their algorithm is not explicit.…”