动态规划

一.钢条切割

#include<iostream>
using namespace std;



int Cut_Rod(int A[], int length, int S[])
{
    int *r = new int[length + 1];
    for (int i = 0; i < 11; i++)
    {
        r[i] = 0;
    }
    r[0] = 0;
    S[0] = 1;
    if (length <= 0 || A == NULL || S == NULL)
    {
        return r[0];
    }

    for (int i = 1; i <= length; i++)
    {
        for (int j = 1; j <= i; j++)
        {
            if (r[i] < A[j - 1] + r[i - j])
            {
                r[i] = A[j - 1] + r[i - j];
                S[i - 1] = j;
            }
        }
    }
    return r[10];
}

int main()
{
    int A[] = { 1, 5, 8, 9, 10, 17, 17, 20, 24, 30};
    int S[10];

    cout << Cut_Rod(A, 10, S) << endl;
    for (int i = 0; i < 10; i++)
    {
        cout << S[i] << " ";
    }
    cout << endl;
    return 0;
}

二.矩阵链乘法

#include<iostream>
using namespace std;

inline int  min(int a, int b)
{
    if (a > b)
    {
        return b;
    }
    else
    {
        return a;
    }
}

int Matrix_Chain_Order(int matrix[], int low, int high, int **m, int **s)
{
    for (int i = low; i <= high; i++)
    {
        m[i][i] = 0;
    }
    for (int len = 2; len <= high - low + 1; len++)
    {
        for (int i = low; i <= high - len + 1; i++)
        {
            int j = i + len - 1;
            for (int k = i; k < j; k++)
            {
                if (k == i)
                {
                    m[i][j] = m[i][k] + m[k + 1][j] + matrix[i * 2 - 2] * matrix[k * 2 - 1] * matrix[j * 2 - 1];
                    s[i][j] = k;
                }
                else if (m[i][j] > m[i][k] + m[k + 1][j] + matrix[i * 2 - 2] * matrix[k * 2 - 1] * matrix[j * 2 - 1])
                {
                    m[i][j] = m[i][k] + m[k + 1][j] + matrix[i * 2 - 2] * matrix[k * 2 - 1] * matrix[j * 2 - 1];
                    s[i][j] = k;
                }
            }
        }
    }
    return m[low][high];
}

void Print_Chain(int **s, int low, int high)
{
    if (low == high)
    {
        cout << "A" << low;
    }
    else
    {
        cout << "(";
        Print_Chain(s, low, s[low][high]);
        Print_Chain(s, s[low][high] + 1, high);
        cout << ")";
    }
}

int main()
{
    int A[] = { 30, 35, 35, 15, 15, 5, 5, 10, 10, 20, 20, 25};
    int **m = new int*[7];
    for (int i = 0; i < 7; i++)
    {
        m[i] = new int[7];
    }

    int **s = new int*[7];
    for (int i = 0; i < 7; i++)
    {
        s[i] = new int[7];
    }
    cout << Matrix_Chain_Order(A, 1, 6, m, s) << endl;
    Print_Chain(s, 1, 6);
    return 0;
}

三.最大子段和

#include<iostream>
using namespace std;

void MaxSubSum(int A[], int length, int sum[], int num[])
{
    sum[0] = A[0];
    num[0] = 0;
    for (int i = 1; i < length; i++)
    {
        if (sum[i - 1] > 0)
        {
            sum[i] = A[i] + sum[i - 1];
            num[i] = num[i - 1];
        }
        else
        {
            sum[i] = A[i];
            num[i] = i;
        }
    }
}

int main()
{
    int A[] = { -2, 11, -4, 13, -5, -2 };
    int sum[6], num[6];
    MaxSubSum(A, 6, sum, num);
    int max = 0;
    for (int i = 0; i < 6; i++)
    {
        if (sum[max] < sum[i])
        {
            max = i;
        }
    }
    cout << sum[max] << endl;
    for (int i = num[max]; i <= max; i++)
    {
        cout << A[i] << " ";
    }
    cout << endl;
    return 0;
}

四.LCS

#include<iostream>
#include<string>
#include<stack>
using namespace std;

int LCS(string x, string y, int **b)
{
    int i = x.length();
    int j = y.length();
    int *C = new int[i];
    for (int t = 0; t < i; t++)
    {
        C[t] = 0;
    }
    for (int m = 0; m < j; m++)
    {
        int num = 0;
        for (int n = 0; n < i; n++)
        {
            int preNum = C[n];
            if (y[m] == x[n])
            {
                C[n] = num + 1;
                b[m][n] = 0;
            }
            else
            {
                if (n != 0)
                {
                    if (C[n - 1] > C[n])
                    {
                        C[n] = C[n - 1];
                        b[m][n] = -1;
                    }
                    else
                    {
                        b[m][n] = 1;
                    }
                }
                else
                {
                    b[m][n] = 1;
                }
            }
            num = preNum;
        }
    }
    return C[i - 1];
}

void PrintLCS(string str, int i, int j, int **b)
{
    stack<char> s;
    while (j >= 0 && i >= 0)
    {
        if (b[j][i] == 0)
        {
            s.push(str[i]);
            i--; j--;
            continue;
        }
        if (b[j][i] == -1)
        {
            i--;
            continue;
        }
        if (b[j][i] == 1)
        {
            j--;
            continue;
        }
    }
    while (!s.empty())
    {
        cout << s.top() << " ";
        s.pop();
    }
    cout << endl;
}

int main()
{
    string x = "BDCABA";
    string y = "ABCBDAB";
    int **b = new int*[7];
    for (int i = 0; i < 7; i++)
    {
        b[i] = new int[6];
    }
    cout << LCS(x, y, b) << endl; //4
    PrintLCS(x, 5, 6, b); // BCBA
    return 0;
}

五.0-1背包

#include<iostream>
using namespace std;

inline int min(int a, int b)
{
    if (a > b)
    {
        return b;
    }
    else
    {
        return a;
    }
}

inline int max(int a, int b)
{
    if (a > b)
    {
        return a;
    }
    else
    {
        return b;
    }
}

int KnapSack(int w[], int v[], int c, int n, int**m)
{
    if (w == NULL || v == NULL || c <= 0 || n <= 0 || m == NULL)
    {
        return 0;
    }
    int JMax = min(w[n - 1] - 1, c);
    for (int j = 0; j <= JMax; j++)
    {
        m[n - 1][j] = 0;
    }
    for (int j = w[n - 1]; j <= c; j++)
    {
        m[n - 1][j] = v[n - 1];
    }
    for (int i = n - 2; i >= 1; i--)
    {
        JMax = min(w[i] - 1, c);
        for (int j = 0; j <= JMax; j++)
        {
            m[i][j] = m[i + 1][j];
        }
        for (int j = w[i]; j <= c; j++)
        {
            m[i][j] = max(m[i + 1][j], m[i + 1][j - w[i]] + v[i]);
        }
    }
    if (w[0] > c)
    {
        m[0][c] = m[1][c];
    }
    else
    {
        m[0][c] = max(m[1][c], m[1][c - w[0]] + v[0]);
    }
    return m[0][c];
}

void TraceBack(int w[], int c, int n, int **m)
{
    int *x = new int[n];
    for (int i = 0; i < n - 1; i++)
    {
        if (m[i][c] == m[i + 1][c])
        {
            x[i] = 0;
        }
        else
        {
            x[i] = 1;
            c -= w[i];
        }
    }
    if (m[n - 1][c] != 0)
    {
        x[n - 1] = 1;
    }
    else
    {
        x[n - 1] = 0;
    }
    for (int i = 0; i < n; i++)
    {
        if (x[i] == 1)
        {
            cout << i + 1 << " ";
        }
    }
    cout << endl;
}

int main()
{
    int w[] = { 4, 5, 6 };
    int v[] = { 3, 4, 5 };
    int c = 10;
    int **m = new int*[3];
    for (int i = 0; i < 3; i++)
    {
        m[i] = new int[11];
    }
    cout << KnapSack(w, v, c, 3, m) << endl; // 8
    TraceBack(w, c, 3, m); // 1 3

    return 0;
}