斜率优化DP

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Time Limit: 9000/3000 MS (Java/Others)    Memory Limit: 131072/65536 K (Java/Others)

Problem Description

Zero has an old printer that doesn‘t work well sometimes. As it is antique, he still like to use it to print articles. But it is too old to work for a long time and it will certainly wear and tear, so Zero use a cost to evaluate this degree.

One day Zero want to print an article which has N words, and each word i has a cost Ci to be printed. Also, Zero know that print k words in one line will cost

M is a const number.

Now Zero want to know the minimum cost in order to arrange the article perfectly.

Input

There are many test cases. For each test case, There are two numbers N and M in the first line (0 ≤ n ≤ 500000, 0 ≤ M ≤ 1000). Then, there are N numbers in the next 2 to N + 1 lines. Input are terminated by EOF.

Output

A single number, meaning the mininum cost to print the article.

Sample Input

5 5
5
9
5
7
5

Sample Output

230

Author

Xnozero

Source

2010
ACM-ICPC Multi-University Training Contest（7）——Host by HIT

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>

using namespace std;

const int maxn=500500;

int n;
int q[maxn],head,tail;
int dp[maxn],sum[maxn],M;

int main()
{
    while(scanf("%d%d",&n,&M)!=EOF)
    {
        head=0,tail=0,dp[0]=0,sum[0]=0;
        q[tail++]=0;

        for(int i=1;i<=n;i++)
        {
            scanf("%d",sum+i);
            sum[i]+=sum[i-1];
        }

        for(int i=1;i<=n;i++)
        {
            while(head+1<tail)
            {
                int p1=q[head];
                int p2=q[head+1];
                int x1=dp[p1]+sum[p1]*sum[p1];
                int x2=dp[p2]+sum[p2]*sum[p2];
                int y1=sum[p1];
                int y2=sum[p2];
                if( (x2-x1) <= (y2-y1) * 2 * sum[i] ) head++;
                else break;
            }
            int k=q[head];
            dp[i]=dp[k]+(sum[i]-sum[k])*(sum[i]-sum[k])+M;
            while(head+1<tail)
            {
                int p1=q[tail-2];
                int p2=q[tail-1];
                int p3=i;
                int x1=dp[p1]+sum[p1]*sum[p1];
                int x2=dp[p2]+sum[p2]*sum[p2];
                int x3=dp[p3]+sum[p3]*sum[p3];
                int y1=sum[p1],y2=sum[p2],y3=sum[p3];
                if((x3-x2)*(y2-y1)<=(x2-x1)*(y3-y2))
                    tail--;
                else break;
            }
            q[tail++]=i;
        }
       printf("%d\n",dp[n]);
    }
    return 0;
}

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