leetcode1515 服务中心的最佳位置

思路：

是一道凸优化的问题，可以使用爬山法。

实现：

class Solution
{
public:

    double getDis(double x, double y, vector<vector<int>>& p)
    {
        double sum = 0;
        for (int i = 0; i < p.size(); i++)
        {
            sum += sqrt(pow(x - p[i][0], 2) + pow(y - p[i][1], 2));
        }
        return sum;
    }
    double getMinDistSum(vector<vector<int>>& positions)
    {
        int n = positions.size();
        double x = 0, y = 0;
        for (int i = 0; i < n; i++)
        {
            x += positions[i][0]; y += positions[i][1];
        }
        x /= n; y /= n;
        double cur = getDis(x, y, positions), eps = 1e-7, step = 1, decay = 0.5;
        int dx[4] = {1, 0, -1, 0}, dy[4] = {0, 1, 0, -1};
        while (step > eps)
        {
            bool modified = false;
            for (int i = 0; i < 4; i++)
            {
                double nx = x + dx[i] * step, ny = y + dy[i] * step;
                double tmp = getDis(nx, ny, positions);
                if (tmp < cur)
                {
                    cur = tmp;
                    modified = true;
                    x = nx; y = ny;
                    break;
                }
            }
            if (!modified) step = step * (1 - decay);
        }
        return cur;
    }
}