The additive and logical complexities of linear and bilinear arithmetic algorithms

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Abstract

Let CA(±) be the additive complexity of a (bi)linear algorithm A for a given problem; D(A) and D(A) are two acyclic diagraphs that represent A, each of them is obtained from another one by reversing directions of all edges; ir(D) and do(D) are two numbers that are introduced to measure the structural deficiencies of an acyclic digraph D. K and Q are the numbers of outputs and input-variables. do(D(A)), do(D(A)), and ir(D(A)) characterize the logical complexity of A. It is shown that CA(±) + do(D(A)) + ir(D(A)) = ω(K + Q)log(K + Q) and CA(±) + do(D(A)) = ω(K + Q)log(K + Q) in the cases of DFT, vector convolution, and matrix multiplication. Also lower bounds on CA(±) + do(D(A)) and on CA(±) are expressed in terms of algebraic quantities such as the ranks of matrices and of multidimensional tensors associated with the problems.

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This research has been supported by National Science Foundation Grant MSC-8003347, Institute for Advanced Study; by National Science Foundation grant MCS-77-23738 and Office of Naval Research Contract N00014-81-K-0269, Stanford University. Reproduction in whole or in part is permitted for any purpose of the United States government.

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