<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 HashHeap:
    def __init__(self):
        self.pq = [] # min heap: [val, nums_idx]
        self.idx_to_idx = {} # nums idx to pq idx
    
    def push(self, nums_idx, num):
        n = len(self.pq)
        self.pq.append([num, nums_idx])
        self.idx_to_idx[nums_idx] = n
        self.siftup(n)
    
    def pop(self):
        return self.remove(0)
    
    def remove(self, idx):
        # remove by pq index
        n = len(self.pq)
        self.swap(idx, n - 1)
        res = self.pq.pop()
        del self.idx_to_idx[res[1]]
        if idx != n - 1:
            self.siftup(idx)
            self.siftdown(idx)
        return res

    def siftup(self, idx):
        while idx > 0:
            parent = (idx - 1) // 2
            if self.pq[idx] < self.pq[parent]:
                self.swap(idx, parent)
                idx = parent
            else:
                break
    
    def siftdown(self, idx):
        while idx < len(self.pq):
            left, right, min_idx = 2 * idx + 1, 2 * idx + 2, idx
            if left < len(self.pq) and self.pq[left] < self.pq[min_idx]:
                min_idx = left
            if right < len(self.pq) and self.pq[right] < self.pq[min_idx]:
                min_idx = right
            if min_idx != idx:
                self.swap(idx, min_idx)
                idx = min_idx
            else:
                break
    
    def swap(self, idx1, idx2):
        self.pq[idx1], self.pq[idx2] = self.pq[idx2], self.pq[idx1]
        self.idx_to_idx[self.pq[idx1][1]] = idx1
        self.idx_to_idx[self.pq[idx2][1]] = idx2   


class Solution:
    def medianSlidingWindow(self, nums: List[int], k: int) -> List[float]:
        if not nums or len(nums) < k:
            return []
        
        min_heap = HashHeap() # larger half
        max_heap = HashHeap() # smaller half
        ans = []
        for nums_idx, num in enumerate(nums):
            if not max_heap.pq or num < -max_heap.pq[0][0]:
                max_heap.push(nums_idx, -num)
            else:
                min_heap.push(nums_idx, num)
            #self.balance(min_heap, max_heap)
            if nums_idx >= k:
                if nums_idx - k in min_heap.idx_to_idx:
                    pq_idx = min_heap.idx_to_idx[nums_idx - k]
                    min_heap.remove(pq_idx)
                else:
                    pq_idx = max_heap.idx_to_idx[nums_idx - k]
                    max_heap.remove(pq_idx)
            self.balance(min_heap, max_heap)
            if nums_idx >= k - 1:
                ans.append(self.get_median(min_heap, max_heap))
        return ans


    def balance(self, min_heap, max_heap):
        while len(min_heap.pq) > len(max_heap.pq):
            num, idx = min_heap.pop()
            max_heap.push(idx, -num)
        while len(max_heap.pq) > len(min_heap.pq) + 1:
            n_num, idx = max_heap.pop()
            min_heap.push(idx, -n_num)
    
    def get_median(self, min_heap, max_heap):
        if not max_heap.pq and not min_heap.pq:
            return None
        if len(max_heap.pq) > len(min_heap.pq):
            return -max_heap.pq[0][0]
        else:
            return 0.5 * (-max_heap.pq[0][0] + min_heap.pq[0][0])
        
        
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