<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>

import heapq
from typing import List

class TwoHeaps:
    def __init__(self):
        self.min_heap = []  # min heap (high) for the larger half of the numbers
        self.max_heap = []  # max heap (low) for the smaller half of the numbers
    
    def add(self, num):
        # Add to max_heap (low) as negative to simulate max heap behavior
        heapq.heappush(self.max_heap, -num)

        # Move the largest from max_heap to min_heap if necessary
        if self.min_heap and (-self.max_heap[0]) > self.min_heap[0]:
            heapq.heappush(self.min_heap, -heapq.heappop(self.max_heap))

        # Balance heaps
        if len(self.max_heap) > len(self.min_heap) + 1:
            heapq.heappush(self.min_heap, -heapq.heappop(self.max_heap))
        elif len(self.min_heap) > len(self.max_heap):
            heapq.heappush(self.max_heap, -heapq.heappop(self.min_heap))
    
    def remove(self, num):
        if self.min_heap and num >= self.min_heap[0]:
            self._remove(self.min_heap, num)
        else:
            self._remove(self.max_heap, -num)
    
    def _remove(self, heap, num):
        # Remove the element from the heap
        index = heap.index(num)
        heap[index] = heap[-1]
        heap.pop()
        if index < len(heap):
            heapq._siftup(heap, index)
            heapq._siftdown(heap, 0, index)

    def getMedian(self):
        if len(self.max_heap) == len(self.min_heap):
            return (-self.max_heap[0] + self.min_heap[0]) / 2
        return -self.max_heap[0]


class Solution:
    def medianSlidingWindow(self, nums: List[int], k: int) -> List[float]:
        twoHeaps = TwoHeaps()
        result = []

        L = 0
        for R in range(len(nums)):
            twoHeaps.add(nums[R])

            if R - L + 1 == k:
                result.append(twoHeaps.getMedian())
                twoHeaps.remove(nums[L])
                L += 1

        return result

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