背包问题

01背包问题

给定物品的数量n（每一种物品只有一件）以及每件物品的价值，一个容量为v的背包，问怎样装物品使总价值最大。

朴素算法

#include<iostream>
#include<cstdio>
#include<algorithm>

using namespace std;

const int N = 1010;

int v[N], w[N]; // v 代表体积 w 代表权重 y总写的和其他人正好反着。 其他人 w 代表weight v 代表 value 
int f[N][N];
int n, m;

int main()
{
    scanf("%d%d", &n, &m);

    for (int i = 1; i <= n; i++) scanf("%d%d", &v[i], &w[i]);

    for (int i = 1; i <= n; i++)
        for (int j = 0; j <= m; j++)
        {
            f[i][j] = f[i - 1][j];
            if (v[i] <= j) f[i][j] = max(f[i][j], f[i - 1][j - v[i]] + w[i]);
        }
    
    cout << f[n][m] << endl;

    return 0;
}

优化

优化之前先看一下滚动数组。使用滚动数组可以大大减少空间复杂度。

二维变一维，最终优化结果：

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>

using namespace std;

const int N = 1010;

int v[N], w[N];
int n, m, f[N];

int main()
{
    cin >> n >> m;

    for (int i = 1; i <= n; i++)
        cin >> v[i] >> w[i];
    
    for (int i = 1; i <= n; i++)
        for (int j = m; j >= v[i]; j--)
            f[j] = max(f[j], f[j - v[i]] + w[i]);
    
    cout << f[m] << endl;
    
    return 0;
}

完全背包问题

给定n种物品（每一种物品有无限个）以及每一种物品的价值，一个容量为v的背包，问怎样装物品使价值最大。

朴素算法

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>

using namespace std;

const int N = 1010;

int v[N], w[N];
int n, m;
int f[N][N];

int main()
{
    cin >> n >> m;

    for (int i = 1; i <= n; i++)
        cin >> v[i] >> w[i];
    
    for (int i = 1; i <= n; i++)
        for (int j = 0; j <= m; j++)
            for (int k = 0; k * v[i] <= j; k++)
                f[i][j] = max(f[i][j], f[i - 1][j - k * v[i]] + k * w[i]);

    cout << f[n][m] << endl;

    return 0;
}

优化

/*
 * f[i][j] = max(f[i - 1][j], f[i - 1][j - v] + w, f[i - 1][j - 2v] + 2w, ....,f[i - 1][j - x * v] + x * w)
 * f[i][j - v] = max(         f[i - 1][j - v]    , f[i - 1][j - 2v] + w, .....,f[i - 1][j - x * v] + (x - 1) * w)
 * 
 * f[i][j] = max(f[i - 1][j], f[i][j - v] + w);
 */

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>

using namespace std;

const int N = 1010;

int v[N], w[N];
int n, m;
int f[N][N];

int main()
{
    cin >> n >> m;

    for (int i = 1; i <= n; i++)
        cin >> v[i] >> w[i];
    
    for (int i = 1; i <= n; i++)
        for (int j = 0; j <= m; j++)
        {
            f[i][j] = f[i - 1][j];
            if (j >= v[i]) f[i][j] = max(f[i][j], f[i][j - v[i]] + w[i]);
        }

    cout << f[n][m] << endl;

    return 0;
}

优化成一维

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>

using namespace std;

const int N = 1010;

int v[N], w[N];
int n, m;
int f[N];

int main()
{
    cin >> n >> m;

    for (int i = 1; i <= n; i++)
        cin >> v[i] >> w[i];
    
    for (int i = 1; i <= n; i++)
        for (int j = v[i]; j <= m; j++)
            f[j] = max(f[j], f[j - v[i]] + w[i]);

    cout << f[m] << endl;

    return 0;
}

多重背包问题

朴素算法