题目链接：hdu 4923 Room and Moor

题目大意：给定一个序列a，元素由0，1组成，求一个序列b，元素在0~1之间，并且保证递增。输出最小的∑(ai?bi)2，

解题思路：首先剔除为首的0，和末尾的1，然后将中间部分成若干段由连续1开头，连续0结尾的各个段落。对于每一段有一个最优的值x=aa+b(a为1的个数，b为0的个数)，用栈维护各个段的x值，如果当前x值小于前面一个段的x值，那么就要将两个段合并，a=ai?1+ai,b=bi?1+bi.

#include <cstdio>
#include <cstring>
#include <cmath>
#include <queue>
#include <stack>
#include <algorithm>

using namespace std;
const int maxn = 1e5+5;

int n, a[maxn], l, r;

struct state {
    int p, q;
    double x;
    state (int p, int q) {
        this->p = p;
        this->q = q;
        this->x = p * 1.0 / (p + q);
    }
};

double solve () {
    l = 0; r = n-1;
    while (l < n && a[l] == 0)
        l++;
    while (r >= 0 && a[r] == 1)
        r--;

    if (r <= l)
        return 0;

    bool flag = true;
    int cntp = 0, cntq = 0;
    queue<state> que;

    for (int i = l; i <= r; i++) {
        if (flag && a[i] == 1)
            cntp++;
        else if (flag == false && a[i] == 0)
            cntq++;
        else if (flag == false && a[i] == 1) {
            que.push(state(cntp, cntq));
            cntp = 1;
            flag = true;
        } else {
            cntq = 1;
            flag = false;
        }
    }

    que.push(state(cntp, cntq));

    stack<state> sta;
    while (!que.empty()) {
        state u = que.front();
        que.pop();

        while (!sta.empty() && sta.top().x > u.x) {
            u.p += sta.top().p;
            u.q += sta.top().q;
            u.x = u.p * 1.0 / (u.p + u.q);
            sta.pop();
        }

        sta.push(u);
    }

    double ans = 0;
    while (!sta.empty()) {
        state u = sta.top();    
        sta.pop();
        ans += (pow(1-u.x, 2) * u.p + pow(u.x, 2) * u.q);
    }
    return ans;;
}

int main () {
    int cas;
    scanf("%d", &cas);
    while (cas--) {
        scanf("%d", &n);
        for (int i = 0; i < n; i++)
            scanf("%d", &a[i]);
        printf("%.6lf\n", solve());
    }
    return 0;
}