区间dp题汇

1.Luogu 4170 [CQOI2007]涂色

#include <bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for (int i = a; i <= b; ++i)
const int N = 57;

int n, f[N][N]; char a[N];

int main() {
    scanf("%s", a + 1); n = strlen(a + 1); memset(f, 0x3f, sizeof(f)); 
    rep(i, 1, n) {
        f[i][i] = 1;
    }
    
    rep(len, 2, n) rep(i, 1, n - len + 1) { int j = i + len - 1;
        rep(k, i, j - 1) {
            if (a[i] == a[j]) f[i][j] = min(f[i][j], f[i][k] + f[k + 1][j] - 1);
            else f[i][j] = min(f[i][j], f[i][k] + f[k + 1][j]);
        }
    }
    
    printf("%d\n", f[1][n]);
    return 0;
}

2.Luogu 1063 能量项链

#include <bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for (int i = a; i <= b; ++i)
const int N = 207;

int n, a[N], f[N][N], ans;

int main() {
    scanf("%d", &n);
    rep(i, 1, n) {
        scanf("%d", &a[i]);
        a[n + i] = a[i];
    }
    
    rep(len, 2, n + 1) rep(i, 1, n * 2 - len + 1) { int j = i + len - 1; 
        rep(k, i + 1, j - 1) {
            f[i][j] = max(f[i][j], f[i][k] + f[k][j] + a[i] * a[k] * a[j]);
        }
//        cout << i << ' ' << j << ' ' << f[i][j] << '\n';
        if (len == n + 1) ans = max(ans, f[i][j]);
    }
    
    printf("%d\n", ans);
    return 0;
}

3.Luogu 1005 矩阵取数游戏

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
#define rep(i, a, b) for (ll i = a; i <= b; ++i)
const ll N = 87;

ll n, m, a[N], f[N][N], ans;

int main() {
    scanf("%lld%lld", &n, &m);
    
    while (n--) {
        memset(f, 0, sizeof(f));
        rep(i, 1, m) {
            scanf("%lld", &a[i]);
            f[i][i] = a[i] * (1 << m);
        }
        rep(len, 2, m) rep(i, 1, m - len + 1) { ll j = i + len - 1;
            f[i][j] = max(f[i][j], max(f[i][j - 1] + a[j] * (1 << (m + 1 - len)), f[i + 1][j] + a[i] * (1 << (m + 1 - len))));
        }
        ans += f[1][m];
    }
    printf("%lld\n", ans);
    return 0;
}

4.Luogu 1220 关路灯

f[i][j][state] : [i, j]区间 state=0 当前选i state = 1 当前选j

注意枚举的顺序

转移的设计时 在同时刻不在[i, j]区间里的数也要考虑 不符合局部最优性

#include <bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for (int i = a; i <= b; ++i)

const int N = 57;

int n, c, p[N], v[N], s[N], f[N][N][2];

inline int cal(int x, int y, int l, int r) {
    return (p[y] - p[x]) * (s[n] - s[r] + s[l - 1]);
}

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

    memset(f, 0x3f, sizeof(f));
    f[c][c][0] = f[c][c][1] = 0;

    rep(i, 1, n) {
        scanf("%d%d", &p[i], &v[i]);
        s[i] = s[i - 1] + v[i];
    }

    rep(l, 2, n) rep(i, 1, n - l + 1) {
        int j = i + l - 1;
        f[i][j][0] = min(f[i + 1][j][0] + cal(i, i + 1, i + 1, j), f[i + 1][j][1] + cal(i, j, i + 1, j));
        f[i][j][1] = min(f[i][j - 1][0] + cal(i, j, i, j - 1), f[i][j - 1][1] + cal(j - 1, j, i, j - 1));
    }

    printf("%d\n", min(f[1][n][0], f[1][n][1]));

    return 0;
}