【CodeForces】【321E】Ciel and Gondolas

DP优化/四边形不等式

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　　这题……跟邮局那题简直一模一样吧……好水的E题……

　　设dp[i][j]表示前 i 艘“gondola”坐了前 j 个人，那么方程即为$dp(i,j)=min\{ dp[i-1][k]+w[k][j] \} (i\leq k\leq j)$

　　很明显$w(l,r)=\sum_{i=l}^r \sum_{j=l}^r u(i,j) /2$是满足四边形不等式的……那么根据决策单调性直接搞就行了……

//CF 321E
#include<vector>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<iostream>
#include<algorithm>
#define rep(i,n) for(int i=0;i<n;++i)
#define F(i,j,n) for(int i=j;i<=n;++i)
#define D(i,j,n) for(int i=j;i>=n;--i)
using namespace std;
//#define debug
int getint(){
    int v=0,sign=1; char ch=getchar();
    while(ch<'0'||ch>'9') {if (ch=='-') sign=-1; ch=getchar();}
    while(ch>='0'&&ch<='9') {v=v*10+ch-'0'; ch=getchar();}
    return v*sign;
}
typedef long long LL;
const int N=100010,INF=~0u>>2;
/*******************tamplate********************/
LL u[4001][4001],n,m,w[4001][4001],dp[801][4001];
int s[801][4001];
int main(){
#ifndef ONLINE_JUDGE
    freopen("input.txt","r",stdin);
//    freopen("output.txt","w",stdout);
#endif
    n=getint(); m=getint();
    F(i,1,n) F(j,1,n) u[i][j]=getint();
    F(i,1,n) F(j,1,n) u[i][j]+=u[i-1][j]+u[i][j-1]-u[i-1][j-1];
    
    F(i,1,n) F(j,i+1,n) w[i][j]=u[j][j]-u[i-1][j]-u[j][i-1]+u[i-1][i-1];
    
    #ifdef debug
    F(i,1,n) {F(j,1,n) printf("%3d",w[i][j]);puts("");}
    #endif
    F(i,1,m) F(j,1,n) dp[i][j]=INF;
    F(i,1,n){
        dp[1][i]=w[1][i];
        s[1][i]=0;
    }
    F(i,2,m){
        s[i][n+1]=n;
        D(j,n,i)
            F(k,s[i-1][j],s[i][j+1])
                if (dp[i-1][k]+w[k+1][j]<dp[i][j]){
                    s[i][j]=k;
                    dp[i][j]=dp[i-1][k]+w[k+1][j];
                }
    }
    printf("%I64d\n",dp[m][n]/2);
    return 0;
}