Kth Smallest Element in Sorted Matrix

/**
Given a n x n matrix where each of the rows and columns are sorted in ascending order, find the kth smallest element in the matrix.

Note that it is the kth smallest element in the sorted order, not the kth distinct element.

Example:

matrix = [
   [ 1,  5,  9],
   [10, 11, 13],
   [12, 13, 15]
],
k = 8,

return 13.
Note:
You may assume k is always valid, 1 ≤ k ≤ n2.
*/

/**
Build a minHeap of elements from the first row.
Do the following operations k-1 times :
Every time when you poll out the root(Top Element in Heap), you need to know the row number and column number of that element(so we can create a tuple class here), replace that root with the next element from the same column.
After you finish this problem, thinks more :

For this question, you can also build a min Heap from the first column, and do the similar operations as above.(Replace the root with the next element from the same row)
What is more, this problem is exact the same with Leetcode373 Find K Pairs with Smallest Sums, I use the same code which beats 96.42%, after you solve this problem, you can check with this link:
https://discuss.leetcode.com/topic/52953/share-my-solution-which-beat-96-42
ref: https://leetcode.com/problems/kth-smallest-element-in-a-sorted-matrix/discuss/85173/Share-my-thoughts-and-Clean-Java-Code
*/
public class Solution {
    public int kthSmallest(int[][] matrix, int k) {
        int n = matrix.length;
        PriorityQueue<Tuple> pq = new PriorityQueue<Tuple>();
//insert the first row into a minheap
        for(int j = 0; j <= n-1; j++) pq.offer(new Tuple(0, j, matrix[0][j]));
//pop the first k-1 elements from the heap and insert the other elements from the matrix meanwhile, one at a time
        for(int i = 0; i < k-1; i++) {
            Tuple t = pq.poll();
            if(t.x == n-1) continue;//matrix row exhausted
            pq.offer(new Tuple(t.x+1, t.y, matrix[t.x+1][t.y]));
        }
        return pq.poll().val;
    }
}

class Tuple implements Comparable<Tuple> {
    int x, y, val;
    public Tuple (int x, int y, int val) {
        this.x = x;
        this.y = y;
        this.val = val;
    }
   
    @Override
    public int compareTo (Tuple that) {
        return this.val - that.val;
    }
}


public class KthSmallestElementInSortedMatrix{

public int kthSmallest(int[][] mat, int k) {
        int low = mat[0][0];
int high = mat[mat.length-1][mat[0].length-1];// + 1;
while(low<high){
int mid = low +(high-low)/2;
int count = 0;
for(int i=0;i<mat.length; i++){
int j = mat[0].length-1;
               while(j>=0 && mat[i][j]>mid){
j--;
}
count+=j+1;
}
if(count<k){
low = mid+1;
}else{
high = mid;
}
}
return low; 
    }
}