DP优化/四边形不等式

　　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……

 1 //CF 321E
 2 #include<vector>
 3 #include<cstdio>
 4 #include<cstdlib>
 5 #include<cstring>
 6 #include<iostream>
 7 #include<algorithm>
 8 #define rep(i,n) for(int i=0;i<n;++i)
 9 #define F(i,j,n) for(int i=j;i<=n;++i)
10 #define D(i,j,n) for(int i=j;i>=n;--i)
11 using namespace std;
12 //#define debug
13 int getint(){
14     int v=0,sign=1; char ch=getchar();
15     while(ch<‘0‘||ch>‘9‘) {if (ch==‘-‘) sign=-1; ch=getchar();}
16     while(ch>=‘0‘&&ch<=‘9‘) {v=v*10+ch-‘0‘; ch=getchar();}
17     return v*sign;
18 }
19 typedef long long LL;
20 const int N=100010,INF=~0u>>2;
21 /*******************tamplate********************/
22 LL u[4001][4001],n,m,w[4001][4001],dp[801][4001];
23 int s[801][4001];
24 int main(){
25 #ifndef ONLINE_JUDGE
26     freopen("input.txt","r",stdin);
27 //    freopen("output.txt","w",stdout);
28 #endif
29     n=getint(); m=getint();
30     F(i,1,n) F(j,1,n) u[i][j]=getint();
31     F(i,1,n) F(j,1,n) u[i][j]+=u[i-1][j]+u[i][j-1]-u[i-1][j-1];
32
33     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];
34
35     #ifdef debug
36     F(i,1,n) {F(j,1,n) printf("%3d",w[i][j]);puts("");}
37     #endif
38     F(i,1,m) F(j,1,n) dp[i][j]=INF;
39     F(i,1,n){
40         dp[1][i]=w[1][i];
41         s[1][i]=0;
42     }
43     F(i,2,m){
44         s[i][n+1]=n;
45         D(j,n,i)
46             F(k,s[i-1][j],s[i][j+1])
47                 if (dp[i-1][k]+w[k+1][j]<dp[i][j]){
48                     s[i][j]=k;
49                     dp[i][j]=dp[i-1][k]+w[k+1][j];
50                 }
51     }
52     printf("%I64d\n",dp[m][n]/2);
53     return 0;
54 }