Strassen算法于1969年由德國數學家Strassen提出，該方法引入七個中間變量，每個中間變量都只需要進行一次乘法運算。而樸素算法卻需要進行8次乘法運算。

原理

Strassen算法的原理如下所示，使用sympy驗證Strassen算法的正確性

import sympy as s

A = s.Symbol("A")

B = s.Symbol("B")

C = s.Symbol("C")

D = s.Symbol("D")

E = s.Symbol("E")

F = s.Symbol("F")

G = s.Symbol("G")

H = s.Symbol("H")

p1 = A * (F - H)

p2 = (A + B) * H

p3 = (C + D) * E

p4 = D * (G - E)

p5 = (A + D) * (E + H)

p6 = (B - D) * (G + H)

p7 = (A - C) * (E + F)

print(A * E + B * G, (p5 + p4 - p2 + p6).simplify())

print(A * F + B * H, (p1 + p2).simplify())

print(C * E + D * G, (p3 + p4).simplify())

print(C * F + D * H, (p1 + p5 - p3 - p7).simplify())

復雜度分析

$$f(N)=7\times f(\frac{N}{2})=7^2\times f(\frac{N}{4})=...=7^k\times f(\frac{N}{2^k})$$

最終復雜度為$7^{log_2 N}=N^{log_2 7}$

java矩陣乘法(Strassen算法)

代碼如下，可以看看數據結構的定義，時間換空間。

public class Matrix {

private final Matrix[] _matrixArray;

private final int n;

private int element;

public Matrix(int n) {

this.n = n;

if (n != 1) {

this._matrixArray = new Matrix[4];

for (int i = 0; i < 4; i++) {

this._matrixArray[i] = new Matrix(n / 2);

}

} else {

this._matrixArray = null;

}

}

private Matrix(int n, boolean needInit) {

this.n = n;

if (n != 1) {

this._matrixArray = new Matrix[4];

} else {

this._matrixArray = null;

}

}

public void set(int i, int j, int a) {

if (n == 1) {

element = a;

} else {

int size = n / 2;

this._matrixArray[(i / size) * 2 + (j / size)].set(i % size, j % size, a);

}

}

public Matrix multi(Matrix m) {

Matrix result = null;

if (n == 1) {

result = new Matrix(1);

result.set(0, 0, (element * m.element));

} else {

result = new Matrix(n, false);

result._matrixArray[0] = P5(m).add(P4(m)).minus(P2(m)).add(P6(m));

result._matrixArray[1] = P1(m).add(P2(m));

result._matrixArray[2] = P3(m).add(P4(m));

result._matrixArray[3] = P5(m).add(P1(m)).minus(P3(m)).minus(P7(m));

}

return result;

}

public Matrix add(Matrix m) {

Matrix result = null;

if (n == 1) {

result = new Matrix(1);

result.set(0, 0, (element + m.element));

} else {

result = new Matrix(n, false);

result._matrixArray[0] = this._matrixArray[0].add(m._matrixArray[0]);

result._matrixArray[1] = this._matrixArray[1].add(m._matrixArray[1]);

result._matrixArray[2] = this._matrixArray[2].add(m._matrixArray[2]);

result._matrixArray[3] = this._matrixArray[3].add(m._matrixArray[3]);;

}

return result;

}

public Matrix minus(Matrix m) {

Matrix result = null;

if (n == 1) {

result = new Matrix(1);

result.set(0, 0, (element - m.element));

} else {

result = new Matrix(n, false);

result._matrixArray[0] = this._matrixArray[0].minus(m._matrixArray[0]);

result._matrixArray[1] = this._matrixArray[1].minus(m._matrixArray[1]);

result._matrixArray[2] = this._matrixArray[2].minus(m._matrixArray[2]);

result._matrixArray[3] = this._matrixArray[3].minus(m._matrixArray[3]);;

}

return result;

}

protected Matrix P1(Matrix m) {

return _matrixArray[0].multi(m._matrixArray[1]).minus(_matrixArray[0].multi(m._matrixArray[3]));

}

protected Matrix P2(Matrix m) {

return _matrixArray[0].multi(m._matrixArray[3]).add(_matrixArray[1].multi(m._matrixArray[3]));

}

protected Matrix P3(Matrix m) {

return _matrixArray[2].multi(m._matrixArray[0]).add(_matrixArray[3].multi(m._matrixArray[0]));

}

protected Matrix P4(Matrix m) {

return _matrixArray[3].multi(m._matrixArray[2]).minus(_matrixArray[3].multi(m._matrixArray[0]));

}

protected Matrix P5(Matrix m) {

return (_matrixArray[0].add(_matrixArray[3])).multi(m._matrixArray[0].add(m._matrixArray[3]));

}

protected Matrix P6(Matrix m) {

return (_matrixArray[1].minus(_matrixArray[3])).multi(m._matrixArray[2].add(m._matrixArray[3]));

}

protected Matrix P7(Matrix m) {

return (_matrixArray[0].minus(_matrixArray[2])).multi(m._matrixArray[0].add(m._matrixArray[1]));

}

public int get(int i, int j) {

if (n == 1) {

return element;

} else {

int size = n / 2;

return this._matrixArray[(i / size) * 2 + (j / size)].get(i % size, j % size);

}

}

public void display() {

for (int i = 0; i < n; i++) {

for (int j = 0; j < n; j++) {

System.out.print(get(i, j));

System.out.print(" ");

}

System.out.println();

}

}

public static void main(String[] args) {

Matrix m = new Matrix(2);

Matrix n = new Matrix(2);

m.set(0, 0, 1);

m.set(0, 1, 3);

m.set(1, 0, 5);

m.set(1, 1, 7);

n.set(0, 0, 8);

n.set(0, 1, 4);

n.set(1, 0, 6);

n.set(1, 1, 2);

Matrix res = m.multi(n);

res.display();

}

}

總結

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