斜率优化DP入门_HDU3507_Print Article

一次开火车遇到的知识盲区，只能从基础入门了，找了一个hdu的题学习了一发。

个人比较喜欢纸质笔记，字丑见谅。

#include <bits/stdc++.h>
const long long mod = 1e9+7;
const double ex = 1e-10;
#define inf 0x3f3f3f3f
#define iinf 0x3f3f3f3f3f3f3f3f
using namespace std;
long long dp[500011];
long long sum[500011];
int Q[500011];
double getk(int a,int b) // get the slope
{
    if (dp[a]+sum[a]*sum[a] - dp[b]-sum[b]*sum[b] ==0 ) return 0;
    if (sum[a]==sum[b]) return 1e233;
    return (1.0*dp[a]+1.0*sum[a]*sum[a] - 1.0*dp[b]-1.0*sum[b]*sum[b])/(2.0*(sum[a]-sum[b]));
}
int main()
{
    int N,M;
    while (cin >> N >> M)
    {
        memset(dp,iinf,sizeof(dp));
        memset(sum,0,sizeof(sum));
        dp[0] = 0;
        int a;
        for (int i = 1;i<=N; i++){
            scanf("%d",&a);
            sum[i]+=sum[i-1]+a;
        }
        int head=2 , tail = 1;
        Q[1] = 0;
        for (int i = 1;i<=N;i++){
            while (head>tail+1 && sum[i]>getk(Q[tail+1],Q[tail])) tail++; // Find the best shift && Drop some shift because of the increase of sum
            dp[i] = dp[Q[tail]] + M + ( sum[i] - sum[Q[tail]] ) * ( sum[i] - sum[Q[tail]] );//Shift
            while (head>tail+1 && getk(i,Q[head-1])<getk(Q[head-1],Q[head-2])) head--;//Insert the shift i && Maintain the increase of the slope
            Q[head++] = i;//Insert i
        }
        cout << dp[N] <<endl;
    }
    return 0;
}