How to sort a m x n matrix which has all its m rows sorted and n columns sorted?

Question

Given a matrix with m rows and n columns, each of which are sorted. How to efficiently sort the entire matrix?

I know a solution which runs in O(m n log(min(m,n)). I

Answer 1

By creating a Binary Search Tree, we can achieve this in O(mn) time. Take the last element from the first column (element 3 in the example mentioned above), make it as a root. Right nodes will be the n greater elements of that last row and left node will be the one above element ie. the (m-1)th or the 1st element from the second last row. Similarly for this element, the right nodes will be the n elements of that row. Again m-2 will be the left element and all the n elements in it's row will be the right elements. Similarly moving forward we'll have a binary search tree created in O(mn) time. This is O(mn) because we are not searching while inserting, it's a simple insert while traversing by shifting the root node pointer. Then inorder traversal of this BST will be done which will also be O(mn) time.