<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 find_next_possible_combinations(self, crnt_combination, rows, cols):
        ans = []
        for i in range(rows):
            crnt_combination_copy = crnt_combination[:]
            if crnt_combination_copy[i] + 1 < cols:
                crnt_combination_copy[i] += 1                    
                ans.append(crnt_combination_copy)
        return ans

    def add_all_combinations_in_heap(self, next_possible_combinations, min_heap, mat, combination_set):
        for comb in next_possible_combinations:
            i = 0
            crnt_comb_sum = 0
            if tuple(comb) not in combination_set:
                for j in comb:
                    crnt_comb_sum += mat[i][j]
                    i += 1

                heapq.heappush(min_heap, (crnt_comb_sum, comb))
                combination_set.add(tuple(comb))

        return min_heap

    def kthSmallest(self, mat: List[List[int]], k: int) -> int:
        min_heap = []
        combination_set = set()
        rows = len(mat)
        cols = len(mat[0])
        crnt_subarr_sum = 0
        crnt_combination = []
        count = 0
        for i in range(rows):
            crnt_subarr_sum += mat[i][0]
            crnt_combination.append(0)

        if k==1:
            return crnt_subarr_sum

        heapq.heappush(min_heap, (crnt_subarr_sum, crnt_combination))

        while(min_heap):
            crnt_sum, crnt_combination = heapq.heappop(min_heap)
            next_possible_combinations = self.find_next_possible_combinations(crnt_combination, rows, cols)
            min_heap = self.add_all_combinations_in_heap(next_possible_combinations, min_heap, mat, combination_set)
            count += 1
            if count == k:
                return crnt_sum
        
173