bzoj3437 小p的牧场

https://www.lydsy.com/JudgeOnline/problem.php?id=3437

题干：略。

暴力做法：O（n^2），强行无脑dp。

整解：s1为一维前缀和，s2为二维前缀和（存 b[i]*i）

f[i] = min( f[j] + i*(s1[i]-s1[j]) - (s2[i] - s2[j]));

若 j 优于 k

则有 ( (f[ j ]+s2[ j ]) - ( f[ k ]+s2[ k ] ) ) / ( s1[ j ] - s1[ k ] ) < i

然后想起来 △y / △x = k

发现这道题是一道斜率优化的dp题。（但是并不需要维护凸包）

代码：

#include<cstdio>
#include<algorithm>
using namespace std;
#define ll long long
#define N 1000050
int n;
ll a[N],b[N],s1[N],s2[N],f[N];
ll x(int i)
{
    return s1[i];
}
ll y(int i)
{
    return f[i]+s2[i];
}
ll q[N],hd,tl;
int main()
{
    scanf("%d",&n);
    for(int i=1;i<=n;i++)scanf("%I64d",&a[i]);
    for(int i=1;i<=n;i++)
    {
        scanf("%I64d",&b[i]);
        s1[i] = s1[i-1]+b[i];
        s2[i] = s2[i-1]+i*b[i];
    }
    for(ll i=1;i<=n;i++)
    {
        while(hd<tl&&y(q[hd+1])-y(q[hd])<=i*(x(q[hd+1])-x(q[hd])))hd++;
        f[i] = a[i]+f[q[hd]]+i*s1[i]-i*s1[q[hd]]-s2[i]+s2[q[hd]];
        while(hd<tl&&((y(i)-y(q[tl]))*(x(q[tl])-x(q[tl-1]))<=(y(q[tl])-y(q[tl-1]))*(x(i)-x(q[tl]))))tl--;
        q[++tl] = i;
    }
    printf("%I64d
",f[n]);
    return 0;
}