An O(1) Space Algorithm for N-Dimensional Tensor Rotation: A Generalization of the Reversal Method

TL;DR

This paper generalizes the three-reversal in-place array rotation algorithm to N-dimensional tensors, enabling efficient, O(1) space rotation with linear time complexity for high-dimensional data structures.

Contribution

It introduces the $2^n+1$ reversal algorithm, extending in-place rotation techniques to N-dimensional tensors with formal proofs of correctness.

Findings

Achieves in-place tensor rotation with O(1) auxiliary space.

Provides a formal proof of the algorithm's correctness.

Demonstrates the pattern of reversals for any number of dimensions.

Abstract

The rotation of multi-dimensional arrays, or tensors, is a fundamental operation in computer science with applications ranging from data processing to scientific computing. While various methods exist, achieving this rotation in-place (i.e., with O(1) auxiliary space) presents a significant algorithmic challenge. The elegant three-reversal algorithm provides a well-known O(1) space solution for one-dimensional arrays. This paper introduces a generalization of this method to N dimensions, resulting in the "$2^n+1$ reversal algorithm". This algorithm achieves in-place tensor rotation with O(1) auxiliary space and a time complexity linear in the number of elements. We provide a formal definition for N-dimensional tensor reversal, present the algorithm with detailed pseudocode, and offer a rigorous proof of its correctness, demonstrating that the pattern observed in one dimension ($2^1+1=3$ reversals) and two dimensions ($2^2+1=5$ reversals) holds for any arbitrary number of dimensions.