<p>Given an <code>n x n</code> <code>matrix</code> where each of the rows and columns is sorted in ascending order, return <em>the</em> <code>k<sup>th</sup></code> <em>smallest element in the matrix</em>.</p>

<p>Note that it is the <code>k<sup>th</sup></code> smallest element <strong>in the sorted order</strong>, not the <code>k<sup>th</sup></code> <strong>distinct</strong> element.</p>

<p>You must find a solution with a memory complexity better than <code>O(n<sup>2</sup>)</code>.</p>

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

<pre>
<strong>Input:</strong> matrix = [[1,5,9],[10,11,13],[12,13,15]], k = 8
<strong>Output:</strong> 13
<strong>Explanation:</strong> The elements in the matrix are [1,5,9,10,11,12,13,<u><strong>13</strong></u>,15], and the 8<sup>th</sup> smallest number is 13
</pre>

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

<pre>
<strong>Input:</strong> matrix = [[-5]], k = 1
<strong>Output:</strong> -5
</pre>

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

<ul>
	<li><code>n == matrix.length == matrix[i].length</code></li>
	<li><code>1 &lt;= n &lt;= 300</code></li>
	<li><code>-10<sup>9</sup> &lt;= matrix[i][j] &lt;= 10<sup>9</sup></code></li>
	<li>All the rows and columns of <code>matrix</code> are <strong>guaranteed</strong> to be sorted in <strong>non-decreasing order</strong>.</li>
	<li><code>1 &lt;= k &lt;= n<sup>2</sup></code></li>
</ul>

<p>&nbsp;</p>
<p><strong>Follow up:</strong></p>

<ul>
	<li>Could you solve the problem with a constant memory (i.e., <code>O(1)</code> memory complexity)?</li>
	<li>Could you solve the problem in <code>O(n)</code> time complexity? The solution may be too advanced for an interview but you may find reading <a href="http://www.cse.yorku.ca/~andy/pubs/X+Y.pdf" target="_blank">this paper</a> fun.</li>
</ul>

#include <tuple>
class Solution {
public:
    bool is_valid(vector<vector<int>> &matrix, int y, int x){
        if (y<0 || y >= matrix.size())
            return false;
        if (x<0 || x >= matrix.size())
            return false;
        return true;
    }
    int kthSmallest(vector<vector<int>>& matrix, int k) {
        if( k == 1) return matrix[0][0];
        set<tuple<int,int,int>> grey; // min heap {value, y, x}
            int cur_rank = 2;
            grey.insert(make_tuple(matrix[0][1], 0,1));
            grey.insert(make_tuple(matrix[1][0], 1,0));
            while( !grey.empty() ){
                auto [value, y, x] = *grey.begin();
                grey.erase(grey.begin());
                if(cur_rank==k)
                    return value;
                if( is_valid(matrix, y+1, x) && grey.find(make_tuple(matrix[y+1][x], y+1, x))==grey.end()){
                    grey.insert(make_tuple(matrix[y+1][x], y+1, x));
                }
                if( is_valid(matrix, y, x+1) && grey.find(make_tuple(matrix[y][x+1], y, x+1))==grey.end()){
                    grey.insert(make_tuple(matrix[y][x+1], y, x+1));
                }
                cur_rank += 1;
            }
        return -1;          

    }
};
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