E29 区间DP 环形石子合并

视频链接：https://www.bilibili.com/video/BV1oK4y1j7W3/

Luogu P1880 [NOI1995] 石子合并

#include <iostream>
#include <cstring>
#include <algorithm>
using namespace std;

const int N=210;
int n, a[N], s[N];
int f[N][N];  //f[i][j]表示把从i到j合并成一堆的得分最小值 
int g[N][N];  //g[i][j]表示把从i到j合并成一堆的得分最大值 

int main(){
  memset(f,0x3f,sizeof f); memset(g,-0x3f,sizeof g);
  scanf("%d",&n);
  for(int i=1; i<=n; i++)scanf("%d",&a[i]), a[i+n]=a[i];
  for(int i=1; i<=2*n; i++)s[i]=s[i-1]+a[i], g[i][i]=0, f[i][i]=0;
  
  int minv=1e9, maxv=-1e9;
  for(int len=2; len<=n; len++){            //区间长度 
    for(int i=1,j; (j=i+len-1)<=2*n; i++){  //区间起点
      for(int k=i; k<j; k++){               //区间分割点 
        f[i][j]=min(f[i][j],f[i][k]+f[k+1][j]+s[j]-s[i-1]);
        g[i][j]=max(g[i][j],g[i][k]+g[k+1][j]+s[j]-s[i-1]); 
      }
      minv=min(minv,f[i][i+n-1]); //f[1,n]...f[n,2n-1] 
      maxv=max(maxv,g[i][i+n-1]); //g[1,n]...g[n,2n-1]      
    }
  }
  printf("%d\n%d\n",minv,maxv);
}

 

// 四边形不等式优化
#include <iostream>
#include <cstring>
#include <algorithm>
using namespace std;

const int N=210;
int n,a[N],s[N];
int f[N][N]; //f[i][j]表示把从i到j合并成一堆的得分最小值
int g[N][N]; //g[i][j]表示把从i到j合并成一堆的得分最大值
int p[N][N];

int main(){
  scanf("%d",&n);
  for(int i=1;i<=n;i++) scanf("%d",&a[i]),a[i+n]=a[i];
  for(int i=1;i<=2*n;i++) s[i]=s[i-1]+a[i];
  memset(f,0x3f,sizeof f); memset(g,-0x3f,sizeof g);
  for(int i=1;i<=2*n;i++) g[i][i]=0,f[i][i]=0,p[i][i]=i;
  
  int minv=1e9, maxv=-1e9;
  for(int len=2; len<=n; len++){               //区间长度
    for(int i=1,j; (j=i+len-1)<=2*n; i++){     //区间端点
      for(int k=p[i][j-1]; k<=p[i+1][j]; k++){ //区间分割点
        if(f[i][k]+f[k+1][j]+s[j]-s[i-1]<f[i][j])
          f[i][j]=f[i][k]+f[k+1][j]+s[j]-s[i-1], p[i][j]=k;
      }
      g[i][j]=max(g[i][j-1],g[i+1][j])+s[j]-s[i-1];
      minv=min(minv,f[i][i+n-1]); //f[1,n]...f[n,2n-1]
      maxv=max(maxv,g[i][i+n-1]); //g[1,n]...g[n,2n-1]
    }
  }
  printf("%d\n%d\n",minv,maxv);
}