<p>The <strong>median</strong> is the middle value in an ordered integer list. If the size of the list is even, there is no middle value. So the median is the mean of the two middle values.</p>

<ul>
	<li>For examples, if <code>arr = [2,<u>3</u>,4]</code>, the median is <code>3</code>.</li>
	<li>For examples, if <code>arr = [1,<u>2,3</u>,4]</code>, the median is <code>(2 + 3) / 2 = 2.5</code>.</li>
</ul>

<p>You are given an integer array <code>nums</code> and an integer <code>k</code>. There is a sliding window of size <code>k</code> which is moving from the very left of the array to the very right. You can only see the <code>k</code> numbers in the window. Each time the sliding window moves right by one position.</p>

<p>Return <em>the median array for each window in the original array</em>. Answers within <code>10<sup>-5</sup></code> of the actual value will be accepted.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,3,-1,-3,5,3,6,7], k = 3
<strong>Output:</strong> [1.00000,-1.00000,-1.00000,3.00000,5.00000,6.00000]
<strong>Explanation:</strong> 
Window position                Median
---------------                -----
[<strong>1  3  -1</strong>] -3  5  3  6  7        1
 1 [<strong>3  -1  -3</strong>] 5  3  6  7       -1
 1  3 [<strong>-1  -3  5</strong>] 3  6  7       -1
 1  3  -1 [<strong>-3  5  3</strong>] 6  7        3
 1  3  -1  -3 [<strong>5  3  6</strong>] 7        5
 1  3  -1  -3  5 [<strong>3  6  7</strong>]       6
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,3,4,2,3,1,4,2], k = 3
<strong>Output:</strong> [2.00000,3.00000,3.00000,3.00000,2.00000,3.00000,2.00000]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= k &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>-2<sup>31</sup> &lt;= nums[i] &lt;= 2<sup>31</sup> - 1</code></li>
</ul>

class Solution {
public:
    vector<double> medianSlidingWindow(vector<int>& nums, int k) {
        set<pair<int, int>> maxh, minh;
        
        // Initialize maxh with the first k elements
        for (int i = 0; i < k; i++) {
            maxh.insert({nums[i], i});
        }
        
        // Balance maxh and minh
        for (int i = 0; i < k / 2; i++) {
            minh.insert(*maxh.rbegin());
            maxh.erase(*maxh.rbegin());
        }
        
        vector<double> res;
        int n = nums.size();
        
        for (int i = k; i < n; i++) {
            // Calculate the median
            res.push_back(k % 2 == 0 ? 
                ((double)(maxh.rbegin())->first + (double)(minh.begin())->first) / 2.0 : 
                (double)(maxh.rbegin())->first);
            
            // Remove the element that is out of the window
            removeElement(maxh, nums[i - k], i - k);
            removeElement(minh, nums[i - k], i - k);
            
            // Insert the new element
            insertElement(maxh, minh, nums[i], i);
            
            // Balance the sets
            balanceSets(maxh, minh);
        }
        
        // Calculate the final median
        res.push_back(k % 2 == 0 ? 
            ((double)(maxh.rbegin())->first + (double)(minh.begin())->first) / 2.0 : 
            (double)(maxh.rbegin())->first);
        
        return res;
    }
    
private:
    void removeElement(set<pair<int, int>>& s, int num, int idx) {
        if (s.count({num, idx})) {
            s.erase({num, idx});
        }
    }
    
    void insertElement(set<pair<int, int>>& maxh, set<pair<int, int>>& minh, int num, int idx) {
        if (maxh.empty() || num <= maxh.rbegin()->first) {
            maxh.insert({num, idx});
        } else {
            minh.insert({num, idx});
        }
    }
    
    void balanceSets(set<pair<int, int>>& maxh, set<pair<int, int>>& minh) {
        if (minh.size() > maxh.size()) {
            pair<int, int> x = *minh.begin();
            minh.erase(x);
            maxh.insert(x);
        }
        if (maxh.size() > minh.size() + 1) {
            pair<int, int> x = *maxh.rbegin();
            maxh.erase(x);
            minh.insert(x);
        }
    }
};
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