<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>

class Solution:
    def getNextCombs(self, currCombination, mat):
        result = []
        currCombination[1]
        for i, index in enumerate(currCombination[1]):
            if index+1 < len(mat[0]):
                newValue = list(currCombination[1]).copy()
                newValue[i] += 1
                result.append(tuple(newValue))
        return result

    def addCombsToHeap(self, minHeap, visited, nextCombs, mat):
        for comb in nextCombs:
            if comb not in visited:
                next_val_sum = 0
                for row, col in enumerate(comb):
                    next_val_sum += mat[row][col]
                heapq.heappush(minHeap, (next_val_sum, comb))
                visited.add(comb)


    def kthSmallest(self, mat: List[List[int]], k: int) -> int:
        minHeap, visited = [], set()
        initSum, initComb = 0, []
        for row in mat:
            initSum += row[0]
            initComb.append(0)
        minHeap.append((initSum, tuple(initComb)))
        visited.add(tuple(initComb))
        heapq.heapify(minHeap)
        while k:
            result = heapq.heappop(minHeap)
            if k == 0:
                return result[0]
            nextCombs = self.getNextCombs(result, mat)
            self.addCombsToHeap(minHeap, visited, nextCombs, mat)
            k -= 1
        return result[0]
        
177