[省选前题目整理][BZOJ 1911][APIO 2010]特别行动队(斜率优化DP)

题目链接

http://www.lydsy.com/JudgeOnline/problem.php?id=1911

思路

注：单调递减的队列里，相邻两个点的连线的斜率是单调递增的，即维护一个下凸壳

代码

#include <iostream>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <algorithm>

#define MAXN 1200000

using namespace std;

typedef long long int LL;

LL f[MAXN],sum[MAXN],q[MAXN],a,b,c,n;

double getSlope(int x,int y) //求(x,f[x])和(y,f[y])之间的斜率
{
    return (double)((f[x]+a*sum[x]*sum[x]-b*sum[x])-(f[y]+a*sum[y]*sum[y]-b*sum[y]))/(2*a*(sum[x]-sum[y]));
}

int main()
{
    scanf("%lld%lld%lld%lld",&n,&a,&b,&c);
    for(int i=1;i<=n;i++)
    {
        LL x;
        scanf("%lld",&x);
        sum[i]=sum[i-1]+x;
    }
    f[0]=0;
    int h=1,t=1;
    q[h]=0;
    for(int i=1;i<=n;i++)
    {
        while(h<t&&getSlope(q[h],q[h+1])<=sum[i]) h++;
        f[i]=f[q[h]]+a*(sum[i]-sum[q[h]])*(sum[i]-sum[q[h]])+b*(sum[i]-sum[q[h]])+c;
        while(h<t&&getSlope(q[t-1],q[t])>getSlope(q[t],i)) t--;
        q[++t]=i; //!!!!
    }
    printf("%lld\n",f[n]);
    return 0;
}