Following is simple Divide and Conquer method to multiply two square matrices. 1) Divide matrices A and B in 4 sub-matrices of size N/2 x N/2 as shown in the below diagram. 2) Calculate following values recursively. Addition of two matrices takes O(NGenerally Strassen’s Method is not preferred for practical applications for following reasons.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed aboveAttention reader! Don’t stop learning now. However let’s get again on what’s behind the divide and conquer approach.Unlike the dynamic programming where we “expand” the solutions of sub-problems in order to get the final solution, here we are talking more on joining sub-solutions together. Before jumping to Strassen's algorithm, it is necessary that you should be familiar with matrix multiplication using the Divide and Conquer method.Consider two matrices A and B with 4x4 dimension each as shown below,The matrix multiplication of the above two matrices A and B is Matrix C,Now, let's look at the Divide and Conquer approach to multiply two matrices.Take two submatrices from the above two matrices A and B each as ($A_{11}$ & $A_{12}$) and ($B_{11}$ & $B_{21}$) as shown below,And the matrix multiplication of the two 2x2 matrices A11 and B11 is,Also, the matrix multiplication of two 2x2 matrices A12 and B21 is as follows,$A_{11}*B_{11} + A_{12}*B_{21} = \begin{bmatrix}And $c_{11}$, $c_{12}$, $c_{21}$ and $c_{22}$ are equal to equations 1, 2, 3 and 4 respectively.So the idea is to recursively divide n x n matrices into n/2 x n/2 matrices until they are small enough to be multiplied in the naive way, more specifically into 8 multiplications and 4 matrix additions as shown below in the code.Note: Here the dimension n is of the power of 2. Get hold of all the important DSA concepts with the If you like GeeksforGeeks and would like to contribute, you can also write an article using Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below. Before jumping to Strassen's algorithm, it is necessary that you should be familiar with matrix multiplication using the Divide and Conquer method. However, let’s get again on what’s behind the divide and conquer approach and implement it. I'm a frequent speaker at tech conferences and events.World's No 1 Animated self learning Website with Informative tutorials explaining the code and the choices behind it all.Strassen’s Matrix Multiplication-Divide and Conquer-Given two square matrices A and B of size n x n each, find their multiplication . In the above divide and conquer method, the main component for high time complexity is 8 recursive calls.
The usual matrix multiplication of two \(n \times n\) matrices has a time-complexity of \(\mathcal{O}(n^3)\).This means, if \(n\) doubles, the time for the computation increases by a factor of 8. Strassen’s method of matrix multiplication is a typical divide and conquer algorithm. We use cookies to ensure you have the best browsing experience on our website.
Introduction. Submit your Article Strassen’s algorithm makes use of the same divide and conquer approach as above, but instead uses only 7 recursive calls rather than 8. Strassen’s method of matrix multiplication is a typical divide and conquer algorithm. You can find a tip somewhere at the end of the article on how to generalize this algorithm for any value of n.For multiplying two matrices of size n x n, we make 8 recursive calls above, each on a matrix/subproblem with size n/2 x n/2.
In the above method, we do 8 multiplications for matrices of size N/2 x N/2 and 4 additions.
ae + bg, af + bh, ce + dg and cf + dh. My role as the CEO of Wikitechy, I help businesses build their next generation digital platforms and help with their product innovation and growth strategy. I will use the I am also happy if you post some of your solutions with running times ☺If you know other languages, you could create a script for these. Divide and Conquer Method. Consider two matrices A and B with 4x4 dimension each as shown below, The matrix multiplication of the above two matrices A and B is Matrix C, Strassen’s Matrix Multiplication algorithm is the first algorithm to prove that matrix multiplication can be done at a time faster than O(N^3). I focus on Python, Java and C++.I have implemented only the Strassen algorithm for this post. See the following quote from Cormen: The key to Strassen’s method is to make the recursion tree slightly less bushy.
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strassen's matrix multiplication program in python using divide and conquer