Strassen matrix multiply

matrix/strassen_matrix_multiply.py

@@ -0,0 +1,197 @@
+"""
+Strassen's Matrix Multiplication Algorithm
+------------------------------------------
+An optimized divide-and-conquer algorithm for matrix multiplication that
+reduces the number of multiplications from 8 (in the naive approach)
+to 7 per recursion step.
+
+This results in a time complexity of approximately O(n^2.807),
+which is faster than the standard O(n^3) algorithm for large matrices.
+
+Reference:
+https://en.wikipedia.org/wiki/Strassen_algorithm
+"""
+
+Matrix = list[list[int]]
+
+
+def add(matrix_a: Matrix, matrix_b: Matrix) -> Matrix:
+    """
+    Add two square matrices of the same size.
+
+    >>> add([[1,2],[3,4]], [[5,6],[7,8]])
+    [[6, 8], [10, 12]]
+    """
+    n = len(matrix_a)
+    return [[matrix_a[i][j] + matrix_b[i][j] for j in range(n)] for i in range(n)]
+
+
+def sub(matrix_a: Matrix, matrix_b: Matrix) -> Matrix:
+    """
+    Subtract matrix_b from matrix_a.
+
+    >>> sub([[5,6],[7,8]], [[1,2],[3,4]])
+    [[4, 4], [4, 4]]
+    """
+    n = len(matrix_a)
+    return [[matrix_a[i][j] - matrix_b[i][j] for j in range(n)] for i in range(n)]
+
+
+def naive_mul(matrix_a: Matrix, matrix_b: Matrix) -> Matrix:
+    """
+    Multiply two square matrices using the naive O(n^3) method.
+
+    >>> naive_mul([[1,2],[3,4]], [[5,6],[7,8]])
+    [[19, 22], [43, 50]]
+    """
+    n = len(matrix_a)
+    result = [[0] * n for _ in range(n)]
+    for i in range(n):
+        row_a = matrix_a[i]
+        row_result = result[i]
+        for k in range(n):
+            a_ik = row_a[k]
+            col_b = matrix_b[k]
+            for j in range(n):
+                row_result[j] += a_ik * col_b[j]
+    return result
+
+
+def next_power_of_two(n: int) -> int:
+    """
+    Return the next power of two greater than or equal to n.
+
+    >>> next_power_of_two(5)
+    8
+    """
+    power = 1
+    while power < n:
+        power <<= 1
+    return power
+
+
+def pad_matrix(matrix: Matrix, size: int) -> Matrix:
+    """
+    Pad a matrix with zeros to reach the given size.
+
+    >>> pad_matrix([[1,2],[3,4]], 4)
+    [[1, 2, 0, 0], [3, 4, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]
+    """
+    rows = len(matrix)
+    cols = len(matrix[0])
+    padded = [[0] * size for _ in range(size)]
+    for i in range(rows):
+        for j in range(cols):
+            padded[i][j] = matrix[i][j]
+    return padded
+
+
+def unpad_matrix(matrix: Matrix, rows: int, cols: int) -> Matrix:
+    """
+    Remove padding from a matrix.
+
+    >>> unpad_matrix([[1,2,0],[3,4,0],[0,0,0]], 2, 2)
+    [[1, 2], [3, 4]]
+    """
+    return [row[:cols] for row in matrix[:rows]]
+
+
+def split(matrix: Matrix) -> tuple:
+    """
+    Split a matrix into four quadrants (top-left, top-right, bottom-left, bottom-right).
+
+    >>> split([[1,2],[3,4]])
+    ([[1]], [[2]], [[3]], [[4]])
+    """
+    n = len(matrix)
+    mid = n // 2
+    top_left = [[matrix[i][j] for j in range(mid)] for i in range(mid)]
+    top_right = [[matrix[i][j] for j in range(mid, n)] for i in range(mid)]
+    bottom_left = [[matrix[i][j] for j in range(mid)] for i in range(mid, n)]
+    bottom_right = [[matrix[i][j] for j in range(mid, n)] for i in range(mid, n)]
+    return top_left, top_right, bottom_left, bottom_right
+
+
+def join(c11: Matrix, c12: Matrix, c21: Matrix, c22: Matrix) -> Matrix:
+    """
+    Join four quadrants into a single matrix.
+
+    >>> join([[1]], [[2]], [[3]], [[4]])
+    [[1, 2], [3, 4]]
+    """
+    n2 = len(c11)
+    n = n2 * 2
+    result = [[0] * n for _ in range(n)]
+    for i in range(n2):
+        for j in range(n2):
+            result[i][j] = c11[i][j]
+            result[i][j + n2] = c12[i][j]
+            result[i + n2][j] = c21[i][j]
+            result[i + n2][j + n2] = c22[i][j]
+    return result
+
+
+def strassen(matrix_a: Matrix, matrix_b: Matrix, threshold: int = 64) -> Matrix:
+    """
+    Multiply two square matrices using Strassen's algorithm.
+    Uses naive multiplication for matrices smaller than threshold.
+
+    >>> strassen([[1,2],[3,4]], [[5,6],[7,8]])
+    [[19, 22], [43, 50]]
+    """
+    assert len(matrix_a) == len(matrix_a[0]) == len(matrix_b) == len(matrix_b[0]), (
+        "Only square matrices supported"
+    )
+
+    n_orig = len(matrix_a)
+    if n_orig == 0:
+        return []
+
+    if (m := next_power_of_two(n_orig)) != n_orig:
+        a_pad = pad_matrix(matrix_a, m)
+        b_pad = pad_matrix(matrix_b, m)
+    else:
+        a_pad, b_pad = matrix_a, matrix_b
+
+    c_pad = _strassen_recursive(a_pad, b_pad, threshold)
+    return unpad_matrix(c_pad, n_orig, n_orig)
+
+
+def _strassen_recursive(matrix_a: Matrix, matrix_b: Matrix, threshold: int) -> Matrix:
+    n = len(matrix_a)
+    if n <= threshold:
+        return naive_mul(matrix_a, matrix_b)
+    if n == 1:
+        return [[matrix_a[0][0] * matrix_b[0][0]]]
+
+    a11, a12, a21, a22 = split(matrix_a)
+    b11, b12, b21, b22 = split(matrix_b)
+
+    m1 = _strassen_recursive(add(a11, a22), add(b11, b22), threshold)
+    m2 = _strassen_recursive(add(a21, a22), b11, threshold)
+    m3 = _strassen_recursive(a11, sub(b12, b22), threshold)
+    m4 = _strassen_recursive(a22, sub(b21, b11), threshold)
+    m5 = _strassen_recursive(add(a11, a12), b22, threshold)
+    m6 = _strassen_recursive(sub(a21, a11), add(b11, b12), threshold)
+    m7 = _strassen_recursive(sub(a12, a22), add(b21, b22), threshold)
+
+    c11 = add(sub(add(m1, m4), m5), m7)
+    c12 = add(m3, m5)
+    c21 = add(m2, m4)
+    c22 = add(sub(add(m1, m3), m2), m6)
+
+    return join(c11, c12, c21, c22)
+
+
+if __name__ == "__main__":
+    A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
+    B = [[9, 8, 7], [6, 5, 4], [3, 2, 1]]
+
+    C = strassen(A, B, threshold=1)
+    print("A * B =")
+    for row in C:
+        print(row)
+
+    expected = naive_mul(A, B)
+    assert expected == C, "Strassen result differs from naive multiplication!"
+    print("Verified: result matches naive multiplication.")