矩阵连乘问题(递归+动态规划+备忘录法)

2024年6月9日 109次阅读
矩阵连乘问题的介绍网上很多，就不复述了，以下分别用递归算法、动态规划算法和备忘录法实现．
  • 递归算法实现
/******************** Divide-Conquer ********************/
int MatrixChain_Recursive(int i, int j, int *size, int **s)
{
    if (i == j)
        return 0;
    int result = INT_MAX;
    int tmp;
    for (int k = i; k < j; k++)
    {
        tmp = MatrixChain_Recursive(i, k, size, s) + MatrixChain_Recursive(k + 1, j, size, s) + size[i - 1] * size[k] * size[j];
        if (tmp < result)
        {
            result = tmp;
            s[i][j] = k;
        }
    }
    return result;
}
  • 动态规划算法实现
/******************** Dynamic Programming ********************/
void MatrixChain_Dynamic(int n, int *size, int **m, int **s)
{
    for (int i = 1; i <= n; i++)
        m[i][i] = 0;             // l = 1
    for (int l = 2; l <= n; l++) // l is the chain length, 自底向上！
    {
        int j;
        for (int i = 1; i <= n + 1 - l; i++)
        {
            j = l + i - 1;
            m[i][j] = m[i][i] + m[i + 1][j] + size[i - 1] * size[i] * size[j]; // k = i
            s[i][j] = i;                                                       // k = i
            int tmp;
            for (int k = i + 1; k < j; k++)
            {
                tmp = m[i][k] + m[k + 1][j] + size[i - 1] * size[k] * size[j];
                if (tmp < m[i][j])
                {
                    m[i][j] = tmp;
                    s[i][j] = k;
                }
            }
        }
    }
}
  • 备忘录法实现
/******************** 备忘录法 ********************/
int MatrixChain_LookUp(int i, int j, int *size, int **m, int **s)
{
    if (m[i][j] > 0)
        return m[i][j];
    if (i == j)
        return 0;
    int result = INT_MAX;
    int tmp;
    for (int k = i; k < j; k++)
    {
        tmp = MatrixChain_LookUp(i, k, size, m, s) + MatrixChain_LookUp(k + 1, j, size, m, s) + size[i - 1] * size[k] * size[j];
        if (tmp < result)
        {
            result = tmp;
            s[i][j] = k;
        }
    }
    m[i][j] = result;
    return result;
}

int MemorizedMatrixChain(int i, int j, int n, int *size, int **m, int **s)
{
    for (int p = 1; p <= n; p++)
        for (int q = 1; q <= n; q++)
            m[p][q] = 0;
    return MatrixChain_LookUp(i, j, size, m, s);
}
  • 完整程序
#include <iostream>
#include <climits>
using namespace std;

/******************** Dynamic Programming ********************/
void MatrixChain_Dynamic(int n, int *size, int **m, int **s)
{
    for (int i = 1; i <= n; i++)
        m[i][i] = 0;             // l = 1
    for (int l = 2; l <= n; l++) // l is the chain length, 自底向上！
    {
        int j;
        for (int i = 1; i <= n + 1 - l; i++)
        {
            j = l + i - 1;
            m[i][j] = m[i][i] + m[i + 1][j] + size[i - 1] * size[i] * size[j]; // k = i
            s[i][j] = i;                                                       // k = i
            int tmp;
            for (int k = i + 1; k < j; k++)
            {
                tmp = m[i][k] + m[k + 1][j] + size[i - 1] * size[k] * size[j];
                if (tmp < m[i][j])
                {
                    m[i][j] = tmp;
                    s[i][j] = k;
                }
            }
        }
    }
}

/******************** Divide-Conquer ********************/
int MatrixChain_Recursive(int i, int j, int *size, int **s)
{
    if (i == j)
        return 0;
    int result = INT_MAX;
    int tmp;
    for (int k = i; k < j; k++)
    {
        tmp = MatrixChain_Recursive(i, k, size, s) + MatrixChain_Recursive(k + 1, j, size, s) + size[i - 1] * size[k] * size[j];
        if (tmp < result)
        {
            result = tmp;
            s[i][j] = k;
        }
    }
    return result;
}

/******************** 备忘录法 ********************/
int MatrixChain_LookUp(int i, int j, int *size, int **m, int **s)
{
    if (m[i][j] > 0)
        return m[i][j];
    if (i == j)
        return 0;
    int result = INT_MAX;
    int tmp;
    for (int k = i; k < j; k++)
    {
        tmp = MatrixChain_LookUp(i, k, size, m, s) + MatrixChain_LookUp(k + 1, j, size, m, s) + size[i - 1] * size[k] * size[j];
        if (tmp < result)
        {
            result = tmp;
            s[i][j] = k;
        }
    }
    m[i][j] = result;
    return result;
}

int MemorizedMatrixChain(int i, int j, int n, int *size, int **m, int **s)
{
    for (int p = 1; p <= n; p++)
        for (int q = 1; q <= n; q++)
            m[p][q] = 0;
    return MatrixChain_LookUp(i, j, size, m, s);
}

void ConstructSolution(int i, int j, int **s)
{
    if (i == j)
        cout << 'A' << i;
    else
    {
        cout << '(';
        ConstructSolution(i, s[i][j], s);
        ConstructSolution(s[i][j] + 1, j, s);
        cout << ')';
    }
}

int main()
{
    int size[] = {30, 35, 15, 5, 10, 20, 25};
    int n = sizeof(size) / sizeof(*size) - 1;
    int **m = new int *[n + 1];
    int **s = new int *[n + 1];
    for (int i = 1; i <= n; i++)
    {
        m[i] = new int[n + 1];
        s[i] = new int[n + 1];
    }
    /******************** Dynamic Programming ********************/
    // MatrixChain_Dynamic(n, size, m, s);
    // cout << m[1][n] << endl;
    // cout << m[2][5] << endl;

    /******************** Divide-Conquer ********************/
    // cout << MatrixChain_Recursive(1, n, size, s) << endl;
    // cout << MatrixChain_Recursive(2, 5, size, s) << endl;

    /******************** 备忘录法 ********************/
    cout << MemorizedMatrixChain(1, n, n, size, m, s) << endl;
    cout << MemorizedMatrixChain(2, 5, n, size, m, s) << endl;

    ConstructSolution(1, n, s);
    cout << endl;
    for (int i = 1; i <= n; i++)
    {
        delete[] m[i];
        delete[] s[i];
    }
    delete[] m;
    delete[] s;

    return 0;
}
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