BZOJ1096 [ZJOI2007]仓库建设——斜率优化

方程：

$Large f(i)=min(f(j)+sumlimits_{k=j+1}^{i}(x_i-x_k)*p_k)+c_i$

显然这样的方程复杂度为$O(n^3)$极限爆炸，所以我们要换一个方程

设$S(i)=sumlimits_{k=1}^i(x_n-x_k)*p_k$且$A(i)=sumlimits_{k=1}^ip_k$

则$S(i)-S(j)=sumlimits_{k=j+1}^i(x_n-x_k)*p_k$,这和原方程很像

最终方程就可以化成

$Large f(i)=min(f(j)+S(i)-S(j)-(A(i)-A(j))*(x_n-x_i))+c_i$

若对于某个$i$,$j$比$k$优，则

$f(j)+S(i)-S(j)-(A(i)-A(j))*(x_n-x_i)le f(k)+S(i)-S(k)-(A(i)-A(k))*(x_n-x_i)$

化简得

$frac{f(j)-S(j)-f(k)+S(k)}{A(j)-A(k)}le x_i-x_n$

维护一个下凸壳即可

代码

#include<cstdio>
#define LL long long
#define maxn 1000005
#define inf 0x3fffffffffffffff
int x[maxn],p[maxn],c[maxn],que[maxn],s,t=1;
LL S[maxn],A[maxn],f[maxn];
LL calc(int i,int j){
    if(A[i]==A[j])return inf;
    return (f[i]-S[i]-f[j]+S[j])/(A[i]-A[j]);
}
void insert(int i){
    while(s<t-1&&calc(i,que[t-1])<=calc(que[t-1],que[t-2]))t--;
    que[t++]=i;
}
int main(){
    int n;scanf("%d",&n);
    for(int i=1;i<=n;i++)scanf("%d%d%d",x+i,p+i,c+i),A[i]=A[i-1]+p[i];
    for(int i=1;i<=n;i++){
        S[i]=S[i-1]+1ll*(x[n]-x[i])*p[i];
        while(s<t-1&&calc(que[s+1],que[s])<=x[i]-x[n])s++;
        int w=que[s];
        f[i]=f[w]+S[i]-S[w]-(A[i]-A[w])*(x[n]-x[i])+c[i];
        insert(i);
    }
    printf("%lld",f[n]);
    return 0;
}