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Optimal computation of prefix sums on a binary tree of processors. (English) Zbl 0639.68032
Given n numbers $$a_ 0,a_ 1,...,a_{n-1}$$, it is required to compute all sums of the form $$a_ 0+a_ 1+...+a_ i$$, for $$i=0,1,...,n-1$$. This problem arises in many applications and is trivial to solve sequentially in O(n) time. Besides its practical importance, the problem gains an additional theoretical interest in parallel computation. A technique known as recursive doubling allows all sums to be computed in O(log n) time on a model of computation where n processors communicate through an inverse perfect shuffle interconnection network.
In this paper we show how the problem can be solved on a simple network, namely a binary tree of processors. In addition, we show how to extend our solution to obtain an optimal-cost algorithm. The algorithm uses p processors and runs in $$O((n/p)+\log p)$$ time, for a cost of $$O(n+p \log p)$$. This cost is optimal when p log p$$=O(n)$$. Finally, two applications of our results are illustrated, namely job scheduling with deadlines and the knapsack problem.

##### MSC:
 68M20 Performance evaluation, queueing, and scheduling in the context of computer systems 68N25 Theory of operating systems 68Q25 Analysis of algorithms and problem complexity 90C35 Programming involving graphs or networks
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