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1 Attendance (2) Performance (3) Oral (5) Total (10) Dated Sign of Subject Teacher Date of Performance:... Actual Date of Completion:... Expected Date of Completion: Assignment No: Title of the Assignment: STRASSEN'S MATRİX MULTİPLİCATİON Objective of the Assignment: Write a java program to implement matrix multiplication using Strassen's method. (Divide and Conquer) Prerequisite: 1. Basic Algorithm for Matrix Multiplication 2. Algorithm Analysis techniques Theory: Divide-and-Conquer Strategy: Given a function that has to compute on n input the divide and conquer strategy suggest splitting k distinct subsets, 1 < K n, that yields K sub problems. Each of these sub problems must be solved and the solutions are obtained. A method must be found to combine the sub solutions into a solution of a whole. If the sub problems are still relatively large, the divide and conquer strategy can be reapplied. The sub problems are of the same type as the original problem. We can write a control abstraction that gives a flow of the control but hose primary operations are specified by other procedures. The meaning of these procedures is left undefined.

2 D and C (P) where P is the problem to be solved. A Boolean valued function that determines if the input size is small enough that the answer can be computed without splitting. If it is so the function S is invoked else the problem P is divided into smaller sub problems. The sub problems P1, P2,...Pk can be solved by recursive applications of D and C. Combine is a function that determines the solution to P, using the solution the K sub problems. Algorithm D and C (P) if small (p) then return S (P); else divide P into smaller instances P1, P2,...Pk K 1 Apply D and C to each to these sub problems. return combine (D and C (P1), D and C (P2)...D and C (PK)); if the size of P is n and the size of K sub problems are n1, n2...nk respectively the computing time of D and C is described by the recurrence relation. T(n) = g(n) where n is the small T(n1) + T(n2)...+t(nk) + f(n) T(n) is the time for D and C is any input n. g(n) is the time to compute the sum directly for small inputs. f(n) is the time for dividing D and for combining the solutions for the sub problems. When the sub problems are of the same type as the original problems we describe the algorithm using recursion. Strassen's Matrix Multiplication Let A and B be two n x n matrices. The product matrix C = AB is also an n x n matrix. Whose i th and j th elements is formed by taking [i, j] the elements in the i th column of B and multiplying them to give c [i j] = A [i k] B [k,j], 1 k < n for all i and j between 1 and n. To compute c[i, j] using this formula we require n 3 multiplications.

3 The divide and conquer strategy suggest another way to compute the product of two n x n matrices. We will assume that n is a power of 2. in case n is not a power of 2 then enough rows and columns of zeros may be added to both A and B so that the resulting dimensions are a power of 2. Imagine that A and B are each partitioned into 4 2 sub matrices each having dimension n/2 x n/2 then the product AB can be computed by using the above formula for the product of 2 x 2 matrices. To calculate the matrix product C = AB, Strassen's algorithm partitions the data to reduce the number of multiplications performed. This algorithm requires M, N and P to be powers of 2. The straightforward method needs 8 multiplications of numbers and 4 additions of numbers. In contrast, the Strassen's idea needs only 7 multiplications of numbers while it needs 18 additions of numbers. The idea consists of the following steps. 1. Partition A, B and and C into 4 equal parts: 2. Evaluate the intermediate matrices 3. Construct C using the intermediate metrices:

4 Algorithm: Algorithm Matrix Strassen(Matrix A, Matrix B, int n) // Input: Two n x n matrices A = [a_ij] and B = [b_ij] // Output: n x n matrix C = [c_ij] = AB Matrix C; Matrix A11; Matrix A12; Matrix A21; Matrix A22; // partition of A Matrix B11; Matrix B12; Matrix B21; Matrix B22; // partition of B Matrix C11; Matrix C12; Matrix C21; Matrix C22; // partition of C Matrix M1; Matrix M2; Matrix M3; Matrix M4; Matrix M5; Matrix M6; Matrix M7; if (n is not a power of 2) expand A and B by padding rows and columns of all 0's until their size becomes the next power of 2; update n accordingly; ; if (n=1) // the simplest case return a 1 x 1 matrix C = [ a_11 * b_11 ]; else M1 = Strassen(subtract(A12,A22,n/2),add(B21,B22,n/2),n/2); M2 = Strassen(add(A11,A22,n/2),add(B11,B22,n/2),n/2); M3 = Strassen(subtract(A11,A21,n/2),add(B11,B12,n/2),n/2); M4 = Strassen(add(A11,A12,n/2),B22,n/2); M5 = Strassen(A11,subtract(B12,B22,n/2),n/2); M6 = Strassen(A22,subtract(B21,B11,n/2),n/2); M7 = Strassen(add(A21,A22,n/2),B11,n/2); C11 = add(subtract(add(m1,m2,n/2),m4,n/2),m6,n/2);

5 C12 = add(m4,m5,n/2); C21 = add(m6,m7,n/2); C22 = subtract(add(subtract(m2,m3,n/2),m5,n/2),m7,n/2); return C = [Cij] (1<=i<=2 and 1<=j<=2); Algorithm Matrix add(matrix A, Matrix B, int n) // Input: Two n x n matrices A = [a_ij] and B = [b_ij] // Output: n x n matrix C = [c_ij] = A + B Matrix C; for (i=1; i<=n; i++) for (j=1; j<=n; j++) c_ij = a_ij + b_ij; return C = [c_ij]; Algorithm Matrix subtract(matrix A, Matrix B, int n) // Input: Two n x n matrices A = [a_ij] and B = [b_ij] // Output: n x n matrix C = [c_ij] = A - B Matrix C; for (i=1; i<=n; i++) for (j=1; j<=n; j++) c_ij = a_ij - b_ij; return C = [c_ij];

6 Analysis of the Time Complexity: The numbers of n/2 n/2 matrix multiplication may set back to 7 through the Strassen algorithm, when compared with the general algorithm for matrix multiplication. This meantime, there are 18 matrix addition or matrix subtraction. As n is a power of 2, if let k be nonnegative integer, then n is k-th power of 2. If k=0, then the time complexity of matrix multiplication is 1. If k>0, generally we can explain time complexity of matrix multiplication through the following recursive expression. by solving this O(n lg 7 ) = O(n ). Conclusion: Strassen's algorithm definitely performs better than the traditional matrix multiplication algorithm due to the reduced number of multiplications and better memory separation. FAQs: 1. Write down control abstraction for Divide and Conquer Method. 2. What are the limitations of Strassen s algorithm? Find out and explain. 3. Compare the Time Complexity of Streasen s Multiplication with the Traditional Matrix Multiplication Algorithm.