Faster Tensor Canonicalization

02/27/2017 ∙ by Benjamin E. Niehoff, et al. ∙ 0

The Butler-Portugal algorithm for obtaining the canonical form of a tensor expression with respect to slot symmetries and dummy-index renaming suffers, in certain cases with a high degree of symmetry, from O(n!) explosion in both computation time and memory. We present a modified algorithm which alleviates this problem in the most common cases---tensor expressions with subsets of indices which are totally symmetric or totally antisymmetric---in polynomial time. We also present an implementation of the label-renaming mechanism which improves upon that of the original Butler-Portugal algorithm, thus providing a significant speed increase for the average case as well as the highly-symmetric special case. The worst-case behavior remains O(n!), although it occurs in more limited situations unlikely to appear in actual computations. We comment on possible strategies to take if the nature of a computation should make these situations more likely.

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