Hybrid optimization technique for matrix chain multiplication using Strassen's algorithm.

Q2 Pharmacology, Toxicology and Pharmaceutics
F1000Research Pub Date : 2025-05-27 eCollection Date: 2025-01-01 DOI:10.12688/f1000research.162848.2
Srinivasarao Thota, Thulasi Bikku, Rakshitha T
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引用次数: 0

Abstract

Background: Matrix Chain Multiplication (MCM) is a fundamental problem in computational mathematics and computer science, often encountered in scientific computing, graphics, and machine learning. Traditional MCM optimization techniques use Dynamic Programming (DP) with Memoization to determine the optimal parenthesization for minimizing the number of scalar multiplications. However, standard matrix multiplication still operates in O(n 3) time complexity, leading to inefficiencies for large matrices.

Methods: In this paper, we propose a hybrid optimization technique that integrates Strassen's algorithm into MCM to further accelerate matrix multiplication. Our approach consists of two key phases: (i) matrix chain order optimization, using a top-down memoized DP approach, we compute the best multiplication sequence, and (ii) hybrid multiplication strategy, we selectively apply Strassen's algorithm for large matrices (n ≥ 128), reducing the complexity from O(n 3) to O(n 2.81), while using standard multiplication for smaller matrices to avoid recursive overhead. We evaluate the performance of our hybrid method through computational experiments comparing execution time, memory usage, and numerical accuracy against traditional MCM and Strassen's standalone multiplication.

Results: Our results demonstrate that the proposed hybrid method achieves significant speedup (4x-8x improvement) and reduces memory consumption, making it well-suited for large-scale applications. This research opens pathways for further optimizations in parallel computing and GPU-accelerated matrix operations.

Conclusion: This study presents a hybrid approach to Matrix Chain Multiplication by integrating Strassen's algorithm, reducing execution time and memory usage. By selectively applying Strassen's method for large matrices, the proposed technique improves efficiency while preserving accuracy. Future work can focus on parallel computing and GPU acceleration for further optimization.

基于Strassen算法的矩阵链乘法混合优化技术。
背景:矩阵链乘法(Matrix Chain Multiplication, MCM)是计算数学和计算机科学中的一个基本问题,在科学计算、图形学和机器学习中经常遇到。传统的MCM优化技术使用带记忆的动态规划(DP)来确定最优的圆括号,以最小化标量乘法的数量。然而,标准矩阵乘法仍然在O(n 3)时间复杂度中运行,导致大型矩阵的效率低下。方法:本文提出了一种将Strassen算法集成到MCM中的混合优化技术,以进一步加速矩阵乘法。我们的方法包括两个关键阶段:(i)矩阵链顺序优化,使用自顶向下的记忆DP方法,我们计算最佳乘法序列;(ii)混合乘法策略,我们有选择性地对大矩阵(n≥128)应用Strassen算法,将复杂度从O(n 3)降低到O(n 2.81),同时对较小的矩阵使用标准乘法以避免递归开销。我们通过计算实验来评估我们的混合方法的性能,将执行时间、内存使用和数值精度与传统MCM和Strassen的独立乘法进行比较。结果:我们的结果表明,所提出的混合方法实现了显着的加速(4 -8倍的改进),并减少了内存消耗,使其非常适合大规模应用。这项研究为并行计算和gpu加速矩阵运算的进一步优化开辟了道路。结论:本研究提出了一种结合Strassen算法的矩阵链乘法混合方法,减少了执行时间和内存使用。通过选择性地对大矩阵应用Strassen方法,所提出的技术在保持精度的同时提高了效率。未来的工作可以集中在并行计算和GPU加速上,以进一步优化。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
F1000Research
F1000Research Pharmacology, Toxicology and Pharmaceutics-Pharmacology, Toxicology and Pharmaceutics (all)
CiteScore
5.00
自引率
0.00%
发文量
1646
审稿时长
1 weeks
期刊介绍: F1000Research publishes articles and other research outputs reporting basic scientific, scholarly, translational and clinical research across the physical and life sciences, engineering, medicine, social sciences and humanities. F1000Research is a scholarly publication platform set up for the scientific, scholarly and medical research community; each article has at least one author who is a qualified researcher, scholar or clinician actively working in their speciality and who has made a key contribution to the article. Articles must be original (not duplications). All research is suitable irrespective of the perceived level of interest or novelty; we welcome confirmatory and negative results, as well as null studies. F1000Research publishes different type of research, including clinical trials, systematic reviews, software tools, method articles, and many others. Reviews and Opinion articles providing a balanced and comprehensive overview of the latest discoveries in a particular field, or presenting a personal perspective on recent developments, are also welcome. See the full list of article types we accept for more information.
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