区间DP初探 P1880 [NOI1995]石子合并

石子合并

区间$dp$,顾名思义,是以区间为阶段的一种线性$dp$的拓展

状态常定义为$f[i][j]$,表示区间$[i,j]$的某种解;

通常先枚举区间长度,再枚举左端点,最后枚举断点$(k)$

石子合并便是一道经典的区间$dp$

#include <bits/stdc++.h>
#define read read()
#define up(i,l,r) for(int i = (l);i <= (r); i++)
#define inf 0x3f3f3f3f
using namespace std;
int read
{
    int x = 0;char ch = getchar();
    while(ch < 48 || ch > 57) ch = getchar();
    while(ch>= 48 && ch <= 57) {x = 10 * x + ch - 48; ch = getchar();}
    return x;
}
const int N = 205;
int n,cnt[N],sum[N],f1[N][N],f2[N][N];
int main()
{
    freopen("stone.in","r",stdin);
    n = read;
    //memset(f2,0x3f,sizeof(f2)); 
    up(i,1,n) cnt[i] = cnt[i + n] = read;//,f1[i][i] = 0,f2[i][i] = 0;  -> 
    up(i,1,((n<<1)-1)) sum[i] = sum[i - 1] + cnt[i];//前缀和            ->[1,2n-1] 处理环; 
    up(L,2,n)//[2,n] //枚举区间长度 
        up(i,1,( (n<<1) - L + 1) ) //枚举左端点 
            {
                int j = i + L - 1;//右端点; 
                f1[i][j] = 0; f2[i][j] = inf;//初始化; 
                up(k,i,(j - 1))//枚举断点 [i,j) 
                {
                    f1[i][j] = max(f1[i][j],f1[i][k] + f1[k + 1][j]);
                    f2[i][j] = min(f2[i][j],f2[i][k] + f2[k + 1][j]);
                }
                f1[i][j] += (sum[j] - sum[i - 1]);
                f2[i][j] += (sum[j] - sum[i - 1]);//!!加上这次合并[i,j]的分数; 
            }
    int max_ans = 0,min_ans = inf;
    up(i,1,n)//[1,n]
    {
        int j = i + n - 1;
        max_ans = max(max_ans,f1[i][j]);
        min_ans = min(min_ans,f2[i][j]);
    }
    printf("%d
",min_ans);
    printf("%d",max_ans);
    return 0;
}