Strassen's algorithm is a divide and conquer approach for matrix multiplication that reduces the number of recursive calls from 8 to 7, resulting in a more efficient algorithm.
The standard matrix multiplication algorithm has a time complexity of O(n³), while Strassen's algorithm achieves O(n^2.81).
The key insight of Strassen's algorithm is that it's possible to multiply two 2×2 matrices using only 7 multiplications instead of the 8 required by the standard algorithm.
This reduction in the number of multiplications leads to a more efficient algorithm when applied recursively to larger matrices.
Strassen's algorithm computes the following seven products:
These products are then combined to form the four quadrants of the result matrix:
Here's an implementation of Strassen's matrix multiplication algorithm:
O(n^log₂7) ≈ O(n^2.81), where n is the size of the matrices.
O(n²) for storing the matrices and intermediate results.
Learn the matrix/numerical pattern of divide and conquer through Strassen's matrix multiplication algorithm.
Strassen's algorithm is a divide and conquer approach for matrix multiplication that reduces the number of recursive calls from 8 to 7, resulting in a more efficient algorithm.
The standard matrix multiplication algorithm has a time complexity of O(n³), while Strassen's algorithm achieves O(n^2.81).
Example:
Input: Two 2×2 matrices A and B
Standard Algorithm: 8 multiplications
Strassen's Algorithm: 7 multiplications
Explanation: By cleverly combining additions and subtractions with multiplications, Strassen reduced the number of required multiplications.
Developed by Volker Strassen in 1969, this algorithm was the first to break the O(n³) barrier for matrix multiplication, showing that the naive algorithm is not optimal.
Strassen's algorithm demonstrated that asymptotic complexity can be improved through clever algorithmic design, inspiring further research in fast matrix multiplication.
While theoretically faster, Strassen's algorithm is often not used in practice for small matrices due to its overhead and numerical stability issues. However, it's important for large matrices and theoretical computer science.
The standard approach to multiply two 2×2 matrices requires 8 scalar multiplications:
C11 = A11 × B11 + A12 × B21
C12 = A11 × B12 + A12 × B22
C21 = A21 × B11 + A22 × B21
C22 = A21 × B12 + A22 × B22
Strassen's key insight was that it's possible to compute the same result using only 7 multiplications by cleverly combining additions and subtractions:
P1 = (A11 + A22) × (B11 + B22)
P2 = (A21 + A22) × B11
P3 = A11 × (B12 - B22)
P4 = A22 × (B21 - B11)
P5 = (A11 + A12) × B22
P6 = (A21 - A11) × (B11 + B12)
P7 = (A12 - A22) × (B21 + B22)
C11 = P1 + P4 - P5 + P7
C12 = P3 + P5
C21 = P2 + P4
C22 = P1 + P3 - P2 + P6
Here's an implementation of Strassen's matrix multiplication algorithm:
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899function strassenMultiply(A, B) { const n = A.length; // Base case: 1x1 matrices if (n === 1) { return [[A[0][0] * B[0][0]]]; } // Ensure n is a power of 2 (for simplicity) // In a real implementation, you would pad the matrices with zeros // Divide matrices into quadrants const mid = n / 2; // Create submatrices const a11 = submatrix(A, 0, 0, mid); const a12 = submatrix(A, 0, mid, mid); const a21 = submatrix(A, mid, 0, mid); const a22 = submatrix(A, mid, mid, mid); const b11 = submatrix(B, 0, 0, mid); const b12 = submatrix(B, 0, mid, mid); const b21 = submatrix(B, mid, 0, mid); const b22 = submatrix(B, mid, mid, mid); // Compute the seven products (recursively) const p1 = strassenMultiply(add(a11, a22), add(b11, b22)); const p2 = strassenMultiply(add(a21, a22), b11); const p3 = strassenMultiply(a11, subtract(b12, b22)); const p4 = strassenMultiply(a22, subtract(b21, b11)); const p5 = strassenMultiply(add(a11, a12), b22); const p6 = strassenMultiply(subtract(a21, a11), add(b11, b12)); const p7 = strassenMultiply(subtract(a12, a22), add(b21, b22)); // Compute the quadrants of the result const c11 = add(subtract(add(p1, p4), p5), p7); const c12 = add(p3, p5); const c21 = add(p2, p4); const c22 = add(subtract(add(p1, p3), p2), p6); // Combine the quadrants into a single matrix return combineMatrices(c11, c12, c21, c22);} // Helper functionsfunction submatrix(matrix, startRow, startCol, size) { const result = []; for (let i = 0; i < size; i++) { result[i] = []; for (let j = 0; j < size; j++) { result[i][j] = matrix[startRow + i][startCol + j]; } } return result;} function add(A, B) { const n = A.length; const result = []; for (let i = 0; i < n; i++) { result[i] = []; for (let j = 0; j < n; j++) { result[i][j] = A[i][j] + B[i][j]; } } return result;} function subtract(A, B) { const n = A.length; const result = []; for (let i = 0; i < n; i++) { result[i] = []; for (let j = 0; j < n; j++) { result[i][j] = A[i][j] - B[i][j]; } } return result;} function combineMatrices(c11, c12, c21, c22) { const n = c11.length; const result = []; for (let i = 0; i < n * 2; i++) { result[i] = []; for (let j = 0; j < n * 2; j++) { if (i < n && j < n) { result[i][j] = c11[i][j]; } else if (i < n && j >= n) { result[i][j] = c12[i][j - n]; } else if (i >= n && j < n) { result[i][j] = c21[i - n][j]; } else { result[i][j] = c22[i - n][j - n]; } } } return result;}
Note: This implementation assumes that the input matrices are square and have dimensions that are powers of 2. In practice, you would need to pad the matrices with zeros to meet this requirement.
O(n^log₂7) ≈ O(n^2.81), where n is the size of the matrices.
Explanation: The recurrence relation is T(n) = 7T(n/2) + O(n²), which resolves to O(n^log₂7) using the Master Theorem. Since log₂7 ≈ 2.81, the time complexity is approximately O(n^2.81).
O(n²) for storing the matrices and intermediate results.
Explanation: We need O(n²) space to store the input matrices and the result matrix. The recursion depth is O(log n), but the space used at each level decreases geometrically, so the total space complexity remains O(n²).
Used in high-performance computing libraries for large matrix operations.
Example: Scientific simulations, computer graphics, and machine learning with very large datasets.
Modern implementations often use Strassen's algorithm for large matrices and switch to the standard algorithm for smaller submatrices.
Example: Using Strassen's algorithm until the matrices are smaller than 128×128, then switching to the standard algorithm.
Strassen's algorithm inspired further research, leading to even faster algorithms like the Coppersmith-Winograd algorithm.
Example: The current theoretical best is approximately O(n^2.37), though it's not practical due to enormous constant factors.
The techniques used in Strassen's algorithm have been applied to other problems in linear algebra.
Example: Fast algorithms for computing determinants, matrix inversion, and solving systems of linear equations.
Strassen's algorithm is a divide and conquer approach for matrix multiplication that reduces the number of recursive calls from 8 to 7, resulting in a more efficient algorithm.
The standard matrix multiplication algorithm has a time complexity of O(n³), while Strassen's algorithm achieves O(n^2.81).
The key insight of Strassen's algorithm is that it's possible to multiply two 2×2 matrices using only 7 multiplications instead of the 8 required by the standard algorithm.
This reduction in the number of multiplications leads to a more efficient algorithm when applied recursively to larger matrices.
Strassen's algorithm computes the following seven products:
These products are then combined to form the four quadrants of the result matrix:
Here's an implementation of Strassen's matrix multiplication algorithm:
O(n^log₂7) ≈ O(n^2.81), where n is the size of the matrices.
O(n²) for storing the matrices and intermediate results.
Learn the matrix/numerical pattern of divide and conquer through Strassen's matrix multiplication algorithm.
Strassen's algorithm is a divide and conquer approach for matrix multiplication that reduces the number of recursive calls from 8 to 7, resulting in a more efficient algorithm.
The standard matrix multiplication algorithm has a time complexity of O(n³), while Strassen's algorithm achieves O(n^2.81).
Example:
Input: Two 2×2 matrices A and B
Standard Algorithm: 8 multiplications
Strassen's Algorithm: 7 multiplications
Explanation: By cleverly combining additions and subtractions with multiplications, Strassen reduced the number of required multiplications.
Developed by Volker Strassen in 1969, this algorithm was the first to break the O(n³) barrier for matrix multiplication, showing that the naive algorithm is not optimal.
Strassen's algorithm demonstrated that asymptotic complexity can be improved through clever algorithmic design, inspiring further research in fast matrix multiplication.
While theoretically faster, Strassen's algorithm is often not used in practice for small matrices due to its overhead and numerical stability issues. However, it's important for large matrices and theoretical computer science.
The standard approach to multiply two 2×2 matrices requires 8 scalar multiplications:
C11 = A11 × B11 + A12 × B21
C12 = A11 × B12 + A12 × B22
C21 = A21 × B11 + A22 × B21
C22 = A21 × B12 + A22 × B22
Strassen's key insight was that it's possible to compute the same result using only 7 multiplications by cleverly combining additions and subtractions:
P1 = (A11 + A22) × (B11 + B22)
P2 = (A21 + A22) × B11
P3 = A11 × (B12 - B22)
P4 = A22 × (B21 - B11)
P5 = (A11 + A12) × B22
P6 = (A21 - A11) × (B11 + B12)
P7 = (A12 - A22) × (B21 + B22)
C11 = P1 + P4 - P5 + P7
C12 = P3 + P5
C21 = P2 + P4
C22 = P1 + P3 - P2 + P6
Here's an implementation of Strassen's matrix multiplication algorithm:
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899function strassenMultiply(A, B) { const n = A.length; // Base case: 1x1 matrices if (n === 1) { return [[A[0][0] * B[0][0]]]; } // Ensure n is a power of 2 (for simplicity) // In a real implementation, you would pad the matrices with zeros // Divide matrices into quadrants const mid = n / 2; // Create submatrices const a11 = submatrix(A, 0, 0, mid); const a12 = submatrix(A, 0, mid, mid); const a21 = submatrix(A, mid, 0, mid); const a22 = submatrix(A, mid, mid, mid); const b11 = submatrix(B, 0, 0, mid); const b12 = submatrix(B, 0, mid, mid); const b21 = submatrix(B, mid, 0, mid); const b22 = submatrix(B, mid, mid, mid); // Compute the seven products (recursively) const p1 = strassenMultiply(add(a11, a22), add(b11, b22)); const p2 = strassenMultiply(add(a21, a22), b11); const p3 = strassenMultiply(a11, subtract(b12, b22)); const p4 = strassenMultiply(a22, subtract(b21, b11)); const p5 = strassenMultiply(add(a11, a12), b22); const p6 = strassenMultiply(subtract(a21, a11), add(b11, b12)); const p7 = strassenMultiply(subtract(a12, a22), add(b21, b22)); // Compute the quadrants of the result const c11 = add(subtract(add(p1, p4), p5), p7); const c12 = add(p3, p5); const c21 = add(p2, p4); const c22 = add(subtract(add(p1, p3), p2), p6); // Combine the quadrants into a single matrix return combineMatrices(c11, c12, c21, c22);} // Helper functionsfunction submatrix(matrix, startRow, startCol, size) { const result = []; for (let i = 0; i < size; i++) { result[i] = []; for (let j = 0; j < size; j++) { result[i][j] = matrix[startRow + i][startCol + j]; } } return result;} function add(A, B) { const n = A.length; const result = []; for (let i = 0; i < n; i++) { result[i] = []; for (let j = 0; j < n; j++) { result[i][j] = A[i][j] + B[i][j]; } } return result;} function subtract(A, B) { const n = A.length; const result = []; for (let i = 0; i < n; i++) { result[i] = []; for (let j = 0; j < n; j++) { result[i][j] = A[i][j] - B[i][j]; } } return result;} function combineMatrices(c11, c12, c21, c22) { const n = c11.length; const result = []; for (let i = 0; i < n * 2; i++) { result[i] = []; for (let j = 0; j < n * 2; j++) { if (i < n && j < n) { result[i][j] = c11[i][j]; } else if (i < n && j >= n) { result[i][j] = c12[i][j - n]; } else if (i >= n && j < n) { result[i][j] = c21[i - n][j]; } else { result[i][j] = c22[i - n][j - n]; } } } return result;}
Note: This implementation assumes that the input matrices are square and have dimensions that are powers of 2. In practice, you would need to pad the matrices with zeros to meet this requirement.
O(n^log₂7) ≈ O(n^2.81), where n is the size of the matrices.
Explanation: The recurrence relation is T(n) = 7T(n/2) + O(n²), which resolves to O(n^log₂7) using the Master Theorem. Since log₂7 ≈ 2.81, the time complexity is approximately O(n^2.81).
O(n²) for storing the matrices and intermediate results.
Explanation: We need O(n²) space to store the input matrices and the result matrix. The recursion depth is O(log n), but the space used at each level decreases geometrically, so the total space complexity remains O(n²).
Used in high-performance computing libraries for large matrix operations.
Example: Scientific simulations, computer graphics, and machine learning with very large datasets.
Modern implementations often use Strassen's algorithm for large matrices and switch to the standard algorithm for smaller submatrices.
Example: Using Strassen's algorithm until the matrices are smaller than 128×128, then switching to the standard algorithm.
Strassen's algorithm inspired further research, leading to even faster algorithms like the Coppersmith-Winograd algorithm.
Example: The current theoretical best is approximately O(n^2.37), though it's not practical due to enormous constant factors.
The techniques used in Strassen's algorithm have been applied to other problems in linear algebra.
Example: Fast algorithms for computing determinants, matrix inversion, and solving systems of linear equations.