Task: Strassen's algorithm







Python: 

from __future__ import annotations
from itertools import chain
from typing import List
from typing import NamedTuple
from typing import Optional


class Shape(NamedTuple):
    rows: int
    cols: int


class Matrix(List):
    

    @classmethod
    def block(cls, blocks) -> Matrix:
        
        m = Matrix()
        for hblock in blocks:
            for row in zip(*hblock):
                m.append(list(chain.from_iterable(row)))

        return m

    def dot(self, b: Matrix) -> Matrix:
        
        assert self.shape.cols == b.shape.rows
        m = Matrix()
        for row in self:
            new_row = []
            for c in range(len(b[0])):
                col = [b[r][c] for r in range(len(b))]
                new_row.append(sum(x * y for x, y in zip(row, col)))
            m.append(new_row)
        return m

    def __matmul__(self, b: Matrix) -> Matrix:
        return self.dot(b)

    def __add__(self, b: Matrix) -> Matrix:
        
        assert self.shape == b.shape
        rows, cols = self.shape
        return Matrix(
            [[self[i][j] + b[i][j] for j in range(cols)] for i in range(rows)]
        )

    def __sub__(self, b: Matrix) -> Matrix:
        
        assert self.shape == b.shape
        rows, cols = self.shape
        return Matrix(
            [[self[i][j] - b[i][j] for j in range(cols)] for i in range(rows)]
        )

    def strassen(self, b: Matrix) -> Matrix:
        
        rows, cols = self.shape

        assert rows == cols, "matrices must be square"
        assert self.shape == b.shape, "matrices must be the same shape"
        assert rows and (rows & rows - 1) == 0, "shape must be a power of 2"

        if rows == 1:
            return self.dot(b)

        p = rows // 2  

        a11 = Matrix([n[:p] for n in self[:p]])
        a12 = Matrix([n[p:] for n in self[:p]])
        a21 = Matrix([n[:p] for n in self[p:]])
        a22 = Matrix([n[p:] for n in self[p:]])

        b11 = Matrix([n[:p] for n in b[:p]])
        b12 = Matrix([n[p:] for n in b[:p]])
        b21 = Matrix([n[:p] for n in b[p:]])
        b22 = Matrix([n[p:] for n in b[p:]])

        m1 = (a11 + a22).strassen(b11 + b22)
        m2 = (a21 + a22).strassen(b11)
        m3 = a11.strassen(b12 - b22)
        m4 = a22.strassen(b21 - b11)
        m5 = (a11 + a12).strassen(b22)
        m6 = (a21 - a11).strassen(b11 + b12)
        m7 = (a12 - a22).strassen(b21 + b22)

        c11 = m1 + m4 - m5 + m7
        c12 = m3 + m5
        c21 = m2 + m4
        c22 = m1 - m2 + m3 + m6

        return Matrix.block([[c11, c12], [c21, c22]])

    def round(self, ndigits: Optional[int] = None) -> Matrix:
        return Matrix([[round(i, ndigits) for i in row] for row in self])

    @property
    def shape(self) -> Shape:
        cols = len(self[0]) if self else 0
        return Shape(len(self), cols)


def examples():
    a = Matrix(
        [
            [1, 2],
            [3, 4],
        ]
    )
    b = Matrix(
        [
            [5, 6],
            [7, 8],
        ]
    )
    c = Matrix(
        [
            [1, 1, 1, 1],
            [2, 4, 8, 16],
            [3, 9, 27, 81],
            [4, 16, 64, 256],
        ]
    )
    d = Matrix(
        [
            [4, -3, 4 / 3, -1 / 4],
            [-13 / 3, 19 / 4, -7 / 3, 11 / 24],
            [3 / 2, -2, 7 / 6, -1 / 4],
            [-1 / 6, 1 / 4, -1 / 6, 1 / 24],
        ]
    )
    e = Matrix(
        [
            [1, 2, 3, 4],
            [5, 6, 7, 8],
            [9, 10, 11, 12],
            [13, 14, 15, 16],
        ]
    )
    f = Matrix(
        [
            [1, 0, 0, 0],
            [0, 1, 0, 0],
            [0, 0, 1, 0],
            [0, 0, 0, 1],
        ]
    )

    print("Naive matrix multiplication:")
    print(f"  a * b = {a @ b}")
    print(f"  c * d = {(c @ d).round()}")
    print(f"  e * f = {e @ f}")

    print("Strassen's matrix multiplication:")
    print(f"  a * b = {a.strassen(b)}")
    print(f"  c * d = {c.strassen(d).round()}")
    print(f"  e * f = {e.strassen(f)}")


if __name__ == "__main__":
    examples()