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**1 - 1**of**1**### Communication Lower Bounds for Tensor Contraction Algorithms

"... Contractions of nonsymmetric tensors are reducible to matrix mul-tiplication, however, ‘fully symmetric contractions ’ in which the tensors are symmetric and the result is symmetrized can be done with fewer operations. The ‘direct evaluation algorithm ’ for fully symmetric contractions exploits equi ..."

Abstract
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Contractions of nonsymmetric tensors are reducible to matrix mul-tiplication, however, ‘fully symmetric contractions ’ in which the tensors are symmetric and the result is symmetrized can be done with fewer operations. The ‘direct evaluation algorithm ’ for fully symmetric contractions exploits equivalence between terms in the contraction equation to obtain a lower computation cost than the cost associated with nonsymmetric contractions. The ‘symmetry preserving algorithm ’ lowers the cost even further via an algebraic reorganization of the contraction equation. We derive vertical (be-tween memory and cache) and horizontal (interprocessor) commu-nication lower bounds for both of these algorithms. We demon-strate that any load balanced parallel schedule of the direct evalu-ation algorithm requires asymptotically more horizontal communi-cation for some fully symmetric contractions than matrix multipli-cation for nonsymmetric contractions of the same size. Instances of such fully symmetric contractions arise in quantum chemistry calculations. Further, we prove that any schedule of the symme-try preserving algorithm requires asymptotically more vertical and horizontal communication than the direct evaluation algorithm for some fully symmetric contractions. However, for the instances of fully symmetric contractions that arise in quantum chemistry cal-culations, our lower bounds are asymptotically the same for both of these algorithms. 1.