Added Strassen divide and conquer algorithm to multiply matrices. (Th…

…eAlgorithms#1648)

* Added Strassen divide and conquer algorithm to multiply matrices

* Divide and conquer algorith to calculate pow(a,b) or a raised to the power of b

* Putting docstring inside the function.

* Added doctests

divide_and_conquer/power.py

@@ -0,0 +1,33 @@
+def actual_power(a: int, b: int):
+    """
+    Function using divide and conquer to calculate a^b.
+    It only works for integer a,b.
+    """
+    if b == 0:
+        return 1
+    if (b % 2) == 0:
+        return actual_power(a, int(b / 2)) * actual_power(a, int(b / 2))
+    else:
+        return a * actual_power(a, int(b / 2)) * actual_power(a, int(b / 2))
+
+
+def power(a: int, b: int) -> float:
+    """
+    >>> power(4,6)
+    4096
+    >>> power(2,3)
+    8
+    >>> power(-2,3)
+    -8
+    >>> power(2,-3)
+    0.125
+    >>> power(-2,-3)
+    -0.125
+    """
+    if b < 0:
+        return 1 / actual_power(a, b)
+    return actual_power(a, b)
+
+
+if __name__ == "__main__":
+    print(power(-2, -3))

divide_and_conquer/strassen_matrix_multiplication.py

@@ -0,0 +1,161 @@
+import math
+from typing import List, Tuple
+
+
+def default_matrix_multiplication(a: List, b: List) -> List:
+    """
+    Multiplication only for 2x2 matrices
+    """
+    if len(a) != 2 or len(a[0]) != 2 or len(b) != 2 or len(b[0]) != 2:
+        raise Exception("Matrices are not 2x2")
+    new_matrix = [
+        [a[0][0] * b[0][0] + a[0][1] * b[1][0], a[0][0] * b[0][1] + a[0][1] * b[1][1]],
+        [a[1][0] * b[0][0] + a[1][1] * b[1][0], a[1][0] * b[0][1] + a[1][1] * b[1][1]],
+    ]
+    return new_matrix
+
+
+def matrix_addition(matrix_a: List, matrix_b: List):
+    return [
+        [matrix_a[row][col] + matrix_b[row][col] for col in range(len(matrix_a[row]))]
+        for row in range(len(matrix_a))
+    ]
+
+
+def matrix_subtraction(matrix_a: List, matrix_b: List):
+    return [
+        [matrix_a[row][col] - matrix_b[row][col] for col in range(len(matrix_a[row]))]
+        for row in range(len(matrix_a))
+    ]
+
+
+def split_matrix(a: List,) -> Tuple[List, List, List, List]:
+    """
+    Given an even length matrix, returns the top_left, top_right, bot_left, bot_right quadrant.
+
+    >>> split_matrix([[4,3,2,4],[2,3,1,1],[6,5,4,3],[8,4,1,6]])
+    ([[4, 3], [2, 3]], [[2, 4], [1, 1]], [[6, 5], [8, 4]], [[4, 3], [1, 6]])
+    >>> split_matrix([[4,3,2,4,4,3,2,4],[2,3,1,1,2,3,1,1],[6,5,4,3,6,5,4,3],[8,4,1,6,8,4,1,6],[4,3,2,4,4,3,2,4],[2,3,1,1,2,3,1,1],[6,5,4,3,6,5,4,3],[8,4,1,6,8,4,1,6]])
+    ([[4, 3, 2, 4], [2, 3, 1, 1], [6, 5, 4, 3], [8, 4, 1, 6]], [[4, 3, 2, 4], [2, 3, 1, 1], [6, 5, 4, 3], [8, 4, 1, 6]], [[4, 3, 2, 4], [2, 3, 1, 1], [6, 5, 4, 3], [8, 4, 1, 6]], [[4, 3, 2, 4], [2, 3, 1, 1], [6, 5, 4, 3], [8, 4, 1, 6]])
+    """
+    if len(a) % 2 != 0 or len(a[0]) % 2 != 0:
+        raise Exception("Odd matrices are not supported!")
+
+    matrix_length = len(a)
+    mid = matrix_length // 2
+
+    top_right = [[a[i][j] for j in range(mid, matrix_length)] for i in range(mid)]
+    bot_right = [
+        [a[i][j] for j in range(mid, matrix_length)] for i in range(mid, matrix_length)
+    ]
+
+    top_left = [[a[i][j] for j in range(mid)] for i in range(mid)]
+    bot_left = [[a[i][j] for j in range(mid)] for i in range(mid, matrix_length)]
+
+    return top_left, top_right, bot_left, bot_right
+
+
+def matrix_dimensions(matrix: List) -> Tuple[int, int]:
+    return len(matrix), len(matrix[0])
+
+
+def print_matrix(matrix: List) -> None:
+    for i in range(len(matrix)):
+        print(matrix[i])
+
+
+def actual_strassen(matrix_a: List, matrix_b: List) -> List:
+    """
+    Recursive function to calculate the product of two matrices, using the Strassen Algorithm.
+    It only supports even length matrices.
+    """
+    if matrix_dimensions(matrix_a) == (2, 2):
+        return default_matrix_multiplication(matrix_a, matrix_b)
+
+    a, b, c, d = split_matrix(matrix_a)
+    e, f, g, h = split_matrix(matrix_b)
+
+    t1 = actual_strassen(a, matrix_subtraction(f, h))
+    t2 = actual_strassen(matrix_addition(a, b), h)
+    t3 = actual_strassen(matrix_addition(c, d), e)
+    t4 = actual_strassen(d, matrix_subtraction(g, e))
+    t5 = actual_strassen(matrix_addition(a, d), matrix_addition(e, h))
+    t6 = actual_strassen(matrix_subtraction(b, d), matrix_addition(g, h))
+    t7 = actual_strassen(matrix_subtraction(a, c), matrix_addition(e, f))
+
+    top_left = matrix_addition(matrix_subtraction(matrix_addition(t5, t4), t2), t6)
+    top_right = matrix_addition(t1, t2)
+    bot_left = matrix_addition(t3, t4)
+    bot_right = matrix_subtraction(matrix_subtraction(matrix_addition(t1, t5), t3), t7)
+
+    # construct the new matrix from our 4 quadrants
+    new_matrix = []
+    for i in range(len(top_right)):
+        new_matrix.append(top_left[i] + top_right[i])
+    for i in range(len(bot_right)):
+        new_matrix.append(bot_left[i] + bot_right[i])
+    return new_matrix
+
+
+def strassen(matrix1: List, matrix2: List) -> List:
+    """
+    >>> strassen([[2,1,3],[3,4,6],[1,4,2],[7,6,7]], [[4,2,3,4],[2,1,1,1],[8,6,4,2]])
+    [[34, 23, 19, 15], [68, 46, 37, 28], [28, 18, 15, 12], [96, 62, 55, 48]]
+    >>> strassen([[3,7,5,6,9],[1,5,3,7,8],[1,4,4,5,7]], [[2,4],[5,2],[1,7],[5,5],[7,8]])
+    [[139, 163], [121, 134], [100, 121]]
+    """
+    if matrix_dimensions(matrix1)[1] != matrix_dimensions(matrix2)[0]:
+        raise Exception(
+            f"Unable to multiply these matrices, please check the dimensions. \nMatrix A:{matrix1} \nMatrix B:{matrix2}"
+        )
+    dimension1 = matrix_dimensions(matrix1)
+    dimension2 = matrix_dimensions(matrix2)
+
+    if dimension1[0] == dimension1[1] and dimension2[0] == dimension2[1]:
+        return matrix1, matrix2
+
+    maximum = max(max(dimension1), max(dimension2))
+    maxim = int(math.pow(2, math.ceil(math.log2(maximum))))
+    new_matrix1 = matrix1
+    new_matrix2 = matrix2
+
+    # Adding zeros to the matrices so that the arrays dimensions are the same and also power of 2
+    for i in range(0, maxim):
+        if i < dimension1[0]:
+            for j in range(dimension1[1], maxim):
+                new_matrix1[i].append(0)
+        else:
+            new_matrix1.append([0] * maxim)
+        if i < dimension2[0]:
+            for j in range(dimension2[1], maxim):
+                new_matrix2[i].append(0)
+        else:
+            new_matrix2.append([0] * maxim)
+
+    final_matrix = actual_strassen(new_matrix1, new_matrix2)
+
+    # Removing the additional zeros
+    for i in range(0, maxim):
+        if i < dimension1[0]:
+            for j in range(dimension2[1], maxim):
+                final_matrix[i].pop()
+        else:
+            final_matrix.pop()
+    return final_matrix
+
+
+if __name__ == "__main__":
+    matrix1= [
+        [2, 3, 4, 5],
+        [6, 4, 3, 1],
+        [2, 3, 6, 7],
+        [3, 1, 2, 4],
+        [2, 3, 4, 5],
+        [6, 4, 3, 1],
+        [2, 3, 6, 7],
+        [3, 1, 2, 4],
+        [2, 3, 4, 5],
+        [6, 2, 3, 1],
+    ]
+    matrix2 = [[0, 2, 1, 1], [16, 2, 3, 3], [2, 2, 7, 7], [13, 11, 22, 4]]
+    print(strassen(matrix1, matrix2))