An Optimal Parallel Algorithm for Selection
An Optimal Parallel Algorithm for Selection
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L. G. That log^n is an integer. Problem . Compute their sum. Algorithm . "Plant" a balanced binary tree with n leaves. Every node of the tree is denoted [h, j]. See Fig. 1. Leaf [0, j] corresponds to A(j). Associate a number B[h, j] with every node of the tree. Initialization . For all 1 < j < n pardo B [, j ] : = A( j ) . For h := 1 to^ log n for all 1 < j < 2^o§ ^ ~ ^ pardo B[h, j] := B[h-l, 2j-l] + B[h-l, 2j]. It is easy to verify that B[log n, I] holds the desired sum. Think first about an n processor implementation of this summation algorithm. It runs in O(log n) time. Then apply the proof of Brent's theorem to get an alternative implementation that uses only n/log n -7- processors and runs in O(log n) time. This summation algorithm can be extended to solve the following partial-sum problem. Input . As for the summation problem. Problem . Compute / A(j) for all 1 < i < n. Algorithm . Perform the summation algorithm given above. An additional "down-sweep" of the tree (from the root to the leaves), which roughly amounts to reversing the operation of the summation algorithm, will complete the job: Associate another number C[h, j] with each node [h, j].
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To quickly assess the difficulty of the text, read a short excerpt:
L. G. That log^n is an integer. Problem . Compute their sum. Algorithm . "Plant" a balanced binary tree with n leaves. Every node of the tree is denoted [h, j]. See Fig. 1. Leaf [0, j] corresponds to A(j). Associate a number B[h, j] with every node of the tree. Initialization . For all 1 < j < n pardo B [, j ] : = A( j ) . For h := 1 to^ log n for all 1 < j < 2^o§ ^ ~ ^ pardo B[h, j] := B[h-l, 2j-l] + B[h-l, 2j]. It is easy to verify that B[log n, I] holds the desired sum. Think first about an n processor implementation of this summation algorithm. It runs in O(log n) time. Then apply the proof of Brent's theorem to get an alternative implementation that uses only n/log n -7- processors and runs in O(log n) time. This summation algorithm can be extended to solve the following partial-sum problem. Input . As for the summation problem. Problem . Compute / A(j) for all 1 < i < n. Algorithm . Perform the summation algorithm given above. An additional "down-sweep" of the tree (from the root to the leaves), which roughly amounts to reversing the operation of the summation algorithm, will complete the job: Associate another number C[h, j] with each node [h, j].
What to read after An Optimal Parallel Algorithm for Selection?
You can find similar books in the "Read Also" column, or choose other free books by U Vishkin to read online
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