<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>

from sortedcontainers import SortedDict
from typing import List

class MedianFinder:
    def __init__(self, k: int):
        self.k = k
        self.small = SortedDict()  # Elements less than or equal to the median
        self.large = SortedDict()  # Elements greater than the median
        self.small_count = 0
        self.large_count = 0

    def add_number(self, number):
        if not self.small or number <= next(reversed(self.small)):
            if number in self.small:
                self.small[number] += 1
            else:
                self.small[number] = 1
            self.small_count += 1
        else:
            if number in self.large:
                self.large[number] += 1
            else:
                self.large[number] = 1
            self.large_count += 1

        self.rebalance()

    def remove_number(self, number):
        if number in self.small:
            if self.small[number] == 1:
                del self.small[number]
            else:
                self.small[number] -= 1
            self.small_count -= 1
        else:
            if self.large[number] == 1:
                del self.large[number]
            else:
                self.large[number] -= 1
            self.large_count -= 1

        self.rebalance()

    def rebalance(self):
        if self.small_count > self.large_count + 1:
            move_number = next(reversed(self.small))
            if self.small[move_number] == 1:
                del self.small[move_number]
            else:
                self.small[move_number] -= 1
            self.small_count -= 1

            if move_number in self.large:
                self.large[move_number] += 1
            else:
                self.large[move_number] = 1
            self.large_count += 1

        elif self.large_count > self.small_count:
            move_number = next(iter(self.large))
            if self.large[move_number] == 1:
                del self.large[move_number]
            else:
                self.large[move_number] -= 1
            self.large_count -= 1

            if move_number in self.small:
                self.small[move_number] += 1
            else:
                self.small[move_number] = 1
            self.small_count += 1

    def find_median(self):
        if self.small_count > self.large_count:
            return float(next(reversed(self.small)))
        else:
            return (next(reversed(self.small)) + next(iter(self.large))) / 2.0

class Solution:
    def medianSlidingWindow(self, nums: List[int], k: int) -> List[float]:
        mf = MedianFinder(k)
        ans = []
        for i in range(k):
            mf.add_number(nums[i])
        ans.append(mf.find_median())
        N = len(nums)
        for i in range(k, N):
            mf.add_number(nums[i])
            mf.remove_number(nums[i - k])
            ans.append(mf.find_median())
        return ans

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