HDU 4923 Room and Moor (数学＋单调栈)

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本文介绍了一种算法，用于解决给定01序列时如何构造一个实数序列，使得两者之间的平方差之和最小。该算法通过忽略两端连续相同的数字、处理中间变化部分，并使用单调栈来实现。

Room and Moor

Time Limit: 12000/6000 MS (Java/Others) Memory Limit: 262144/262144 K (Java/Others)

Total Submission(s): 1292 Accepted Submission(s): 418

Problem Description

PM Room defines a sequence A = {A₁, A₂,..., A_N}, each of which is either 0 or 1. In order to beat him, programmer Moor has to construct another sequence B = {B₁, B₂,... , B_N} of the same length, which satisfies that:

Input

The input consists of multiple test cases. The number of test cases T(T<=100) occurs in the first line of input.

For each test case:
The first line contains a single integer N (1<=N<=100000), which denotes the length of A and B.
The second line consists of N integers, where the ith denotes A_i.

Output

Output the minimal f (A, B) when B is optimal and round it to 6 decimals.

Sample Input

  
   4
9
1 1 1 1 1 0 0 1 1
9
1 1 0 0 1 1 1 1 1
4
0 0 1 1
4
0 1 1 1

Sample Output

Author

BUPT

Source

2014 Multi-University Training Contest 6

题目链接：http://acm.hdu.edu.cn/showproblem.php?pid=4923

题目大意：给一个0 1序列，要求构造一个实数序列，(要求bi属于[0, 1]且bi单调递增(非严格递增)) 使得(ai - bi)^2的和最小，输出这个结果

题目分析：首先原来0 1序列的最前面的连续0和最后面的连续1可以忽略，因为序列b中可以构造对应的0 1，使其差为0，接着我们取剩下序列中形如10，1100。。。的子序列，即连续1加上连续0的子序列，尽可能让每一个这样的子序列构造出的结果最小，策略显然是尽量取平均！比如最简单的1100，我们构造的bi对应的值应该是0.5 0.5 0.5 0.5，假设子序列有c0个0，c1个1，平均值为val，则val = (c1 * 1 + c0 * 0) / (c0 + c1)，因此我们可以得到val为c1 / (c0 + c1)，然后维护一个单调栈即可，当当前的val比栈顶的val小时，取出栈顶元素，将这段序列与栈顶序列中和，再入栈，中和的方式就是重新计算val值，最后将栈中的元素纷纷弹出并累计c1 * (1 - val)^2 + c0 * val^2
比如样例1：1 1 1 1 1 0 0，val = 5 / 7，则ans = 5 * (1 - 5/7)^2 + 2 * (5/7)^2

#include <cstdio>
#include <cstring>
#include <stack>
using namespace std;
int const MAX = 1e5 + 5;

struct DIVIDE
{
    int c0, c1;
    double val;
}d[MAX];

int a[MAX];
stack <DIVIDE> stk;

int main()
{
    int T, n;
    scanf("%d", &T);
    while(T--)
    {
        double ans = 0;
        memset(a, 0, sizeof(a));
        scanf("%d", &n);
        int st = 1, ed = n;
        for(int i = 1; i <= n; i++)
            scanf("%d", &a[i]);
        while(ed >= 1 && a[ed] == 1)
            ed --;
        while(st <= n && a[st] == 0)
            st ++;
        int cnt = 0;
        for(int i = st; i <= ed; i++)
        {
            if(a[i] == 1)
            {
                if(a[i - 1] == 0)
                {
                    cnt ++;
                    d[cnt].c0 = d[cnt].c1 = 0;
                }
                d[cnt].c1 ++;
            }
            else
                d[cnt].c0 ++;
        }
        for(int i = 1; i <= cnt; i++)
            d[i].val = (d[i].c1 * 1.0) / ((d[i].c1 + d[i].c0) * 1.0);
        for(int i = 1; i <= cnt; i++)
        {
            if(stk.empty() || d[i].val >= stk.top().val)
                stk.push(d[i]);
            else
            {
                while(!stk.empty() && d[i].val < stk.top().val)
                {
                    d[i].c0 += stk.top().c0;
                    d[i].c1 += stk.top().c1;
                    d[i].val = (d[i].c1 * 1.0) / ((d[i].c1 + d[i].c0) * 1.0);
                    stk.pop();
                }
                i --;
            }
        }
        while(!stk.empty())
        {
            int c1 = stk.top().c1;
            int c0 = stk.top().c0;
            double val = stk.top().val;
            ans += c1 * (1.0 - val) * (1.0 - val) + c0 * val * val;
            stk.pop();
        }
        printf("%.6f\n", ans);
    }
}