N-Dimensional Tensor Reversal

• N-dimensional tensor reversal is a technique that inverts tensor indices along each axis to achieve complete in-place reversal of multidimensional data.
• The 2^n+1 algorithm efficiently performs cyclic rotation by partitioning the tensor into binary-indexed blocks using O(1) auxiliary space and linear time complexity.
• This method underpins various applications such as data augmentation, scientific simulations, and optimized multidimensional array manipulation.

N-dimensional tensor reversal refers to the in-place rearrangement of elements within an $n$-dimensional tensor according to an axis-reversal operation or to effect a cyclic shift (rotation) by a prescribed vector. This operation is foundational for array manipulation algorithms, including multidimensional data rotation, data augmentation, and low-level scientific computing routines. Achieving this task using $O(1)$ auxiliary space and with time complexity linear in the number of tensor elements represents a significant algorithmic challenge, especially for high-dimensional data structures. An $O(1)$ space, $2^n+1$ reversal algorithm for in-place $n$-dimensional tensor rotation establishes a systematic generalization of the classical three-reversal method for arrays to arbitrary tensor order, grounded in recursive block-wise permutation and inversion properties (Chen, 27 Nov 2025).

1. Formal Definition and Properties of N-Dimensional Tensor Reversal

Given a tensor $T$ of order $n$ with shape $(d_0, d_1, \ldots, d_{n-1})$ and multi-index $\mathbf{i} = (i_0, \ldots, i_{n-1})$ where $0 \le i_\ell < d_\ell$, the $n$-dimensional reversal operator $\mathit{Rev}$ is defined as: $$\bigl(\mathit{Rev}(T)\bigr)_{i_0,\ldots,i_{n-1}} = T_{d_0-1-i_0,\, d_1-1-i_1,\,\ldots,\, d_{n-1}-1-i_{n-1}}.$$ This operator negates each per-axis index with respect to the axis range. Equivalently, for start and end vectors $\mathbf{s} = (0,\ldots,0)$, $\mathbf{e}= (d_0-1, \ldots, d_{n-1}-1)$,

$$\bigl(\mathit{Rev}(T)\bigr)_{\mathbf{i}} = T_{\mathbf{s} + \mathbf{e} - \mathbf{i}}.$$

A fundamental property is involution: $\mathit{Rev} \circ \mathit{Rev} = I,$ implying that applying the reversal operation twice restores the original tensor.

2. The $2^n+1$ Reversal Algorithm for In-Place Rotation

The tensor cyclic rotation problem is to shift all elements by an index vector $\mathbf{k} = (k_0, \ldots, k_{n-1})$, modulo the size of each dimension. The $2^n+1$ reversal algorithm proceeds as follows:

1. Normalization: Each $k_\ell$ is reduced modulo $d_\ell$.
2. Global Tensor Reversal: Invoke $\mathit{Rev}$ over the full tensor, interchanging each block labeled by a binary block index $\mathbf{b} \in \{0,1\}^n$ with its complement.
3. Block-Wise Reversals: Enumerate all $2^n$ blocks—each corresponding to a binary vector $\mathbf{b}$ that selects, per axis, whether the index is less (0) or greater/equal (1) to $k_\ell$. For each block, reverse the sub-tensor indexed by that range if it is non-empty.

The algorithm utilizes only $O(n)$ auxiliary scalars (for index vectors and temporary counters) and achieves $O(N)$ total data movement, where $N=\prod_\ell d_\ell$.

Pseudocode

Sub-Tensor Reversal

function ReverseND(T, s[0..n-1], e[0..n-1]) i ← s loop j ← s + e − i if i lex≥ j then break swap T[i] and T[j] d ← n−1 while d ≥ 0 do if i[d] < e[d] then i[d] ← i[d] + 1 break else i[d] ← s[d] d ← d − 1 end if end while if d < 0 then break end loop end function

Rotation by Vector $\mathbf{k}$

function RotateND(T, k[0..n-1]) n ← dimensionality of T d[ℓ] ← shape of T in dimension ℓ for ℓ in 0..n-1 do k[ℓ] ← k[ℓ] mod d[ℓ] end for s_global ← (0,0,…,0) e_global ← (d[0]-1, …, d[n-1]-1) ReverseND(T, s_global, e_global) for b in 0..(2^n−1) do for ℓ in 0..n-1 do if the ℓ-th bit of b is 0 then s_block[ℓ] ← 0; e_block[ℓ] ← k[ℓ]−1 else s_block[ℓ] ← k[ℓ]; e_block[ℓ] ← d[ℓ]−1 end if end for if ∀ℓ: s_block[ℓ] ≤ e_block[ℓ] then ReverseND(T, s_block, e_block) end if end for end function

(Chen, 27 Nov 2025)

3. Algorithmic Invariants and Correctness

The $2^n+1$ reversal algorithm exploits the involutive property of $\mathit{Rev}$ and recursive block decomposition. The set of all tensor index vectors is partitioned into $2^n$ blocks, each demarcated by comparisons to the corresponding $k_\ell$. Under the global reversal, block $\mathbf{b}$ receives the element-wise reversed content of block $\bar{\mathbf{b}}$ (bitwise complement). Each inner block reversal undoes this internal order, enabling data from block $\bar{\mathbf{b}}$ to be efficiently mapped to block $\mathbf{b}$—effecting a cyclic shift. Applying $\mathit{Rev}$ twice (via the full reversal and block reversal) ensures each datum's correct placement via the composability of the involutive operator: $$\mathit{Rev}(\mathit{Rev}(B_{\bar{\mathbf{b}}})) = B_{\bar{\mathbf{b}}}.$$ This satisfies the correct data movement corresponding to a $k$-shift in all axes, implemented entirely in-place with only pairwise swaps.

4. Computational Complexity

Time and space complexity characteristics are determined by the linear cost of elementary reversals and the block decomposition structure. A single reversal of a block of cardinality $N_{\mathrm{sub}}$ requires $O(N_{\mathrm{sub}})$ swaps. The global reversal and all $2^n$ block reversals collectively cover the tensor domain exactly, resulting in total work of $O(N)$, where $N = \prod_{\ell=0}^{n-1} d_\ell$. The memory overhead comprises $O(n)$ scalars for indices and counters, making auxiliary space $O(1)$ under the data-size model—a critical property for large-scale or memory-constrained contexts.

5. Illustrative Cases Across Multiple Dimensions

The $2^n+1$ reversal framework generalizes naturally from classical one-dimensional rotations to higher-order arrays with analogous block structures.

Dimensionality ($n$)	Number of Reversals	Block Decomposition
1	3 ($2^1+1$)	2 segments (A, B)
2	5 ($2^2+1$)	4 quadrants $(B_{00}, B_{01}, B_{10}, B_{11})$
3	9 ($2^3+1$)	8 octants labeled by $(b_0b_1b_2)$

• For $n=1$, a vector rotated by $k$ is reversed globally, then both left and right segments are reversed in-place: $$[1\,2\,3\,4\,5\,6\,7] \xrightarrow{\mathit{Rev}} [7\,6\,5\,4\,3\,2\,1]$$, then two segment reversals yield $[5\,6\,7\,1\,2\,3\,4]$—the desired rotation.
• For $n=2$, a $d_0 \times d_1$ matrix is split at $(k_0, k_1)$ into quadrants, reversed globally, then each quadrant reversed, resulting in a $(k_0, k_1)$ block-wise cyclic permutation.
• For $n=3$, eight octants are manipulated analogously, with all blockwise mapping following bitwise complement labeling.

In each case, data in any block $B_{\mathbf{b}}$ is mapped, via double reversal, into the location for a cyclic shift by $\mathbf{k}$.

6. Applications and Context

N-dimensional tensor reversal and in-place cyclic rotation are central to array manipulation libraries, data augmentation pipelines, scientific simulations involving periodic boundaries, and bijective data reshaping. The $O(1)$ space requirement is particularly relevant for devices with constrained memory or in high-throughput environments where allocation of large auxiliary workspace is infeasible. The method generalizes classical array and matrix reversal tools to higher-order data, preserving strict in-place semantics. When compared to alternative tensor permutation strategies involving auxiliary buffers, the $2^n+1$ reversal method minimizes temporal and spatial overhead.

7. Significance, Limitations, and Extensions

The $2^n+1$ reversal algorithm formalizes and unifies a family of in-place tensor cyclic permutation algorithms, extending the reach of elementary reversal-based methods to arbitrary dimension while optimizing for constant auxiliary space and linear time. Based on elementary index manipulations and swap operations, the approach is broadly applicable independent of the underlying data type or tensor storage layout, provided index arithmetic is feasible. A plausible implication is that further optimizations or parallel variants could extend this approach to distributed or multi-core hardware with care to avoid conflicting swaps and maintain in-place guarantees. The algorithm's dependence on $2^n$ block enumerations may render it less suitable for extremely high $n$ in practice, but for the most common cases ($n \leq 6$) its efficiency and robustness are particularly compelling (Chen, 27 Nov 2025).