《算法基础》 动态规划-背包问题

• 《算法基础》 动态规划-背包问题
• • 1. 01背包问题
  • 2.完全背包问题
  • 3.多重背包问题
  • 4.分组背包问题

最核心最简洁的的代码：
1.01背包问题

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]);
        }

2.完全背包问题

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]);

3.多重背包问题

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

4.分组背包问题

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

#include 
#include 
#include 
using namespace std;

const int N = 1010;
int v[N], w[N];
int n, m;
int f[N][N];
// f[0][0 - m] 自动初始化为0

int main()
{
    cin >> 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 (j >= v[i]) {
                f[i][j] = max(f[i][j], f[i - 1][j - v[i]] + w[i]);
            }
        }
    }

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


    return 0;
}

//一维滚动数组

#include 
#include 
#include 
#include 
using namespace std;

typedef long long LL;
const int N = 110, mod = 1e9 + 7;
unordered_map primes;

int main()
{
	int n;
	cin >> n;
	while (n --) {
		int x;
		cin >> x;
		for (int i = 2; i <= x / i; i ++) {
			while (x % i == 0) {
				x /= i;
				primes[i] ++;
			}
		}
		if (x > 1)
			primes[x] ++;
	}
	LL res = 1;
	for (auto prime : primes) {
		res = res * (prime.second + 1) % mod;
	}
	cout << res << endl;

	return 0;
}

#include 
#include 
using namespace std;

const int N = 110;
int n, m;
int v[N], w[N], s[N];
int f[N][N];

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

    for (int i = 1; i <= n; i ++) cin >> v[i] >> w[i] >> s[i];

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

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

    return 0;
}

多重背包问题的二进制优化

#include 
#include 
using namespace std;

const int N = 12010, M = 2010;
int n, m;
int v[N], w[N];
int f[M];

int main()
{
    cin >> n >> m;
    int cnt  = 0;

    for (int i = 1; i <= n; i ++) {
        int a, b, s;
        cin >> a >> b >> s;
        int k = 1;
        while (k <=s) {
            cnt ++;
            v[cnt] = a * k;
            w[cnt] = b * k;
            s = s - k;
            k = k * 2;
        }
        if (s > 0) {
            cnt ++;
            v[cnt] = a * s;
            w[cnt] = b * s;
        }
    }

    n = cnt;

    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;
}

#include 
#include 
using namespace std;

const int N = 110;
int n , m;
int v[N][N], w[N][N], s[N];
int f[N];

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

    for (int i = 1; i <= n; i ++) {
        cin >> s[i];
        for (int j = 0; j < s[i]; j ++) {
            cin >> v[i][j] >> w[i][j];
        }
    }

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

    cout << f[m] << endl;

    return 0;
}