Integer multiplication is at least as hard as matrix transposition (2503.22848v1)

Published 28 Mar 2025 in cs.CC and cs.SC

Abstract: Working in the multitape Turing model, we show how to reduce the problem of matrix transposition to the problem of integer multiplication. If transposing an $n \times n$ binary matrix requires $\Omega(n² \log n)$ steps on a Turing machine, then our reduction implies that multiplying $n$-bit integers requires $\Omega(n \log n)$ steps. In other words, if matrix transposition is as hard as expected, then integer multiplication is also as hard as expected.

Summary

The paper demonstrates a reduction showing that complexity lower bounds for matrix transposition extend to integer multiplication.
It uses discrete Fourier transform algorithms like Bluestein’s and Cooley-Tukey to map transposition as a convolution process relevant to multiplication.
The findings suggest that efficient transposition algorithms could lead to breakthroughs in multiplication complexity analysis.

Integer Multiplication and Matrix Transposition Complexity

Introduction

The paper discusses the relationship between integer multiplication and matrix transposition complexity within the multitape Turing machine model. Both operations are fundamental in computational complexity theory but possess distinct computational complexities.

Background

In the multitape Turing model, the time complexity represents the steps executed by the machine. Integer multiplication concerns computing the product of two $m$ -bit numbers, and matrix transposition involves rearranging an $n_1 \times n_2$ matrix into its transpose, with each operation having specific upper bounds, i.e., $O(m \lg m)$ for integer multiplication and $O(n_1 n_2 b \lg \min(n_1, n_2))$ for matrix transposition.

Reduction from Transposition to Multiplication

The core contribution of the paper is a reduction technique, showing that lower bounds for matrix transposition imply lower bounds for integer multiplication. For example, if matrix transposition is conjectured to require $\Omega(m \lg m)$ steps, then integer multiplication requires at least as much. This implies that transposition, although seemingly simpler, imposes a similar computational burden as multiplication when considered under these assumptions.

Key Results

Main Reduction: The reduction involves interpreting a matrix as a vector and performing discrete Fourier transforms (DFTs) through algorithms like Bluestein’s and Cooley-Tukey methods, relating matrix transposition to convolution processes that map onto integer multiplication problems.
Complexity Claims:
- If no transposition machine can complete within $O(m \lg m)$ , no multiplication machine can either.
- Transposition problems might inherently share the deterministic-time complexity with integer multiplication under specific conditions.

Generalized Transposition Scenarios

For broader application, the paper considers a generalized scenario with dimensions $l_1 \times n_1 \times n_2 \times l_2$ and effectively applies the reduction strategy, ensuring adaptability to various matrix sizes and dimensions.

Implications and Future Directions

This research posits that attacking transposition complexities may lead to breakthroughs in understanding multiplication limits. Open questions remain, particularly concerning practical transposition bounds within realistic computational models.

Moreover, possible extensions include exploring deterministic bounds in alternative models and broadening the reduction approach to other fundamental permutation problems.

Conclusion

The findings underscore a profound theoretical link between integer multiplication and matrix transposition within algorithmic complexity. By demonstrating that matrix transposition is at least as hard as integer multiplication under reasonable conjectures, this paper opens pathways for future research to explore lower bounds and efficiency improvements in computational arithmetic operations.