matrix multiplication divide and conquer algorithm

The constants used in Strassen’s method are high and for a typical application Naive method works better.įor Sparse matrices, there are better methods especially designed for them. Generally Strassen’s Method is not preferred for practical applications for following reasons. */Ĭout << 'martrix multiplication not possible' * the matrices must be resized and padded with zeroes to meet this criteria. * Check to see if these matrices are already square and have dimensions of a power of 2. dividing the matrices into sub-matrices:įor (i = 0 i > &A, vector> &B, int m, int n, int a, int b) P5(new_size, z), p6(new_size, z), p7(new_size, z),ĪResult(new_size, z), bResult(new_size, z) Worst case time complexity: Θ(n^2.8074).Combine these submatricies into our new matrix C.Calculate the 7 matrix multiplications recursively.Divide matrices A and B in 4 sub-matrices of size N/2 x N/2 as shown in the above diagram.This comes out to approxiamtely O(n^2.8074) which is better than O(n^3) The time completxity using the Master Theorem. We can finish the call with 7 recursive calls and a little bit of addition and subtraction. Strassen's insight was that we don't actually need 8 recursive calls to complete this process. Using the Master Theorem with T(n) = 8T(n/2) + O(n^2) we still get a runtime of O(n^3). The above strategy is the basic O(N^3) strategy. We then use these results to compute C's submatricies. This strategy is called Divide and Conquer. The basic idea behind Strassen's algorithm is to split A & B into 8 submatricies and then recursively compute the submatricies of C. It was the first algorithm to prove that the basic O(n^3) runtime was not optiomal. Volker Strassen first published his algorithm in 1969. Since then, we have come a long way to better and clever matrix multiplication algorithms. In general, multipling two matrices of size N X N takes N^3 operations. Multiplication is one of the most fundamental operation in Machine Learning and optimizing it is the key to several optimizations.

M7 = (x - x) * (y+y) Ĭout<<' Product achieved using Strassen's algorithm ' Product achieved using Strassen's algorithm Printf(' Product achieved using Strassen's algorithm ') Let us look at the implementation of Strassen's Matrix Multiplication in various programming languages: C, C++, Java, Python. Order of both of the matrices are n × n.ĭivide X, Y and Z into four (n/2)×(n/2) matrices as represented below − Strassen’s Matrix multiplication can be performed only on square matrices where n is a power of 2. In this context, using Strassen’s Matrix multiplication algorithm, the time consumption can be improved a little bit. It uses the Divide and Conquer approach to multiply the matrices. Strassen’s Matrix Multiplication Algorithm Detailed Solution It is a method used for matrix multiplication. Hence, the algorithm takes O(n 3) time to execute. There are three for loops in this algorithm and one is nested in other. Here, we assume that integer operations take O(1) time. The following pseudocode describes the naïve multiplication − Algorithm: Matrix-Multiplication (X, Y, Z) Using Naïve method, two matrices ( X and Y) can be multiplied if the order of these matrices are p × q and q × r and the resultant matrix will be of order p × r. Naïve Methodįirst, we will discuss naïve method and its complexity. Strassen’s method was introduced to reduce the time complexity from O(n 3) to O(n log 7). The time complexity taken by this approach is O(n 3), since it takes two loops to multiply. The usual matrix multiplication method multiplies each row with each column to achieve the product matrix. Strassen’s Matrix Multiplication is the divide and conquer approach to solve the matrix multiplication problems. Travelling Salesperson Approximation Algorithm.Travelling Salesman Problem | Dynamic Programming.Asymptotic Notations & Apriori Analysis.

Matrix multiplication divide and conquer algorithm