Top left is the smallest one, obviously. Insert that, along with (0, 0), into a min heap. while less than k elements have been popped: pop the smallest element. Say it's coordinates are (i, j) Insert (i+1, j), (i, j+1) into the min heap. These are the possible next smallest numbers. The element at the root of the min-heap is the k-th smallest element now.

The key feature of such a matrix is: The element at (m,n) should be the largest one in the sub matrix (0,0,m,n). So, the search should start from (0,N), then do zig-zag search based on the above feature.

Walk through the matrix starting (0, 0). The comparisons happen between bottom and right values. If the right value is smaller than bottom, go to the bottom. If bottom is smaller than right, go right. Count every time you make a move. Do this until counter == K or you reach end of matrix.