# Faster Tensor Canonicalization

## TL;DR

This paper introduces a modified tensor canonicalization algorithm that significantly improves performance for common symmetric or antisymmetric tensor expressions, reducing complexity from factorial to polynomial time in typical cases.

## Contribution

A new algorithm and implementation that optimize tensor canonicalization for symmetric cases, reducing computational complexity and increasing speed in practical scenarios.

## Key findings

- Polynomial time performance for common symmetric tensor cases
- Significant speed improvements over previous algorithms
- Limited worst-case factorial complexity in rare scenarios

## Abstract

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.