<p>You are given an <code>m x n</code> matrix <code>mat</code> that has its rows sorted in non-decreasing order and an integer <code>k</code>.</p>

<p>You are allowed to choose <strong>exactly one element</strong> from each row to form an array.</p>

<p>Return <em>the </em><code>k<sup>th</sup></code><em> smallest array sum among all possible arrays</em>.</p>

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

<pre>
<strong>Input:</strong> mat = [[1,3,11],[2,4,6]], k = 5
<strong>Output:</strong> 7
<strong>Explanation:</strong> Choosing one element from each row, the first k smallest sum are:
[1,2], [1,4], [3,2], [3,4], [1,6]. Where the 5th sum is 7.
</pre>

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

<pre>
<strong>Input:</strong> mat = [[1,3,11],[2,4,6]], k = 9
<strong>Output:</strong> 17
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> mat = [[1,10,10],[1,4,5],[2,3,6]], k = 7
<strong>Output:</strong> 9
<strong>Explanation:</strong> Choosing one element from each row, the first k smallest sum are:
[1,1,2], [1,1,3], [1,4,2], [1,4,3], [1,1,6], [1,5,2], [1,5,3]. Where the 7th sum is 9.  
</pre>

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

<ul>
	<li><code>m == mat.length</code></li>
	<li><code>n == mat.length[i]</code></li>
	<li><code>1 &lt;= m, n &lt;= 40</code></li>
	<li><code>1 &lt;= mat[i][j] &lt;= 5000</code></li>
	<li><code>1 &lt;= k &lt;= min(200, n<sup>m</sup>)</code></li>
	<li><code>mat[i]</code> is a non-decreasing array.</li>
</ul>

import heapq


class PriorityQueue:
    def __init__(self):
        self._queue = []
        
    def push(self, matrixSum, indexes):
        heapq.heappush(self._queue, (matrixSum, indexes))
        
    def pop(self) -> tuple:
        return heapq.heappop(self._queue)

    
class Solution:
    def kthSmallest(self, mat: List[List[int]], k: int) -> int:
        queue = PriorityQueue()
        
        m = len(mat)  # row
        n = len(mat[0]) # col
        
        smallestSum = sum(mat[i][0] for i in range(0, m))
        firstColIndexes = tuple([0 for _ in range(0, m)])
        queue.push(smallestSum, firstColIndexes)
        
        
        usedIndexes = set()
        usedIndexes.add(firstColIndexes)
        
        kCount = 0
        
        while 1:
            matSum, colIndexes = queue.pop()
            kCount += 1
            
            if kCount == k:
                return matSum

            for rowIndex, colIndex in enumerate(colIndexes):
                if colIndex < n - 1:
                    colIndexesCopy = list(colIndexes)
                    colIndexesCopy[rowIndex] += 1
                    colIndexesCopy = tuple(colIndexesCopy)
                    
                    if colIndexesCopy not in usedIndexes:
                        matSum = sum(mat[row][colIndexesCopy[row]] for row in range(0, m))
                        queue.push(matSum, colIndexesCopy)
                        usedIndexes.add(colIndexesCopy)
189