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Question

Two of the nodes of a Binary Search Tree are swapped.

Input Tree:

     10
    /  \
   5    8
  / \
 2   20

In the above tree, nodes 20 and 8 must be swapped to fix the tree.

Output tree:

     10
    /  \
   5    20
  / \
 2   8

I followed the solution given in here . But I feel the solution is incorrect because:

As per the site:

1. The swapped nodes are not adjacent in the inorder traversal of the BST.

   For example, Nodes 5 and 25 are swapped in {3 5 7 8 10 15 20 25} .
   The inorder traversal of the given tree is 3 25 7 8 10 15 20 5 If we observe carefully, during inorder traversal, we find node 7 is smaller than the previous visited node 25. Here save the context of node 25 (previous node). Again, we find that node 5 is smaller than the previous node 20. This time, we save the context of node 5 ( current node ). Finally swap the two node's values.

So my point is if it is considering 25 because it is greater than 7 than it should consider 20 as well because it is also greater than 5. So is this correct solution or I am missing something?

Answer 1

Yes. It is considering 25 because it is greater than 7. But, it should not consider 20 as well because it is also greater than 5. Instead, it should consider 5 because it is less than 20.

This example is not very good, because the position of 5 in the original array is the last one. Let's consider a sorted array {1, 2, 3, 4, 5} . Swap 2 and 4, then we get {1, 4, 3, 2, 5} . If two elements (not adjacent) in a sorted array is swapped, for all pairs like (A[i], A[i+1]) , there will be exactly two pairs that is in wrong order, namely descending order. In the case of {1, 4, 3, 2, 5} , we have pair (4, 3) , and pair (3, 2) . Suppose we have pair (A[p], A[p+1]) and pair (A[q], A[q+1]) , such that A[p] > A[p+1] and A[q] > A[q+1] , we can claim that it is A[p] and A[q+1] being swapped. In the case of {1, 4, 3, 2, 5} , it is 4 and 2 being swapped.

Now come back to the example 3 25 7 8 10 15 20 5 , in which 25, 7 and 20 5 are the only two pairs in wrong order. Then 25 and 5 are the two elements being swapped.

Answer 2

Following @jeffreys' notation,

if we have pair (A[p], A[p+1]) and pair (A[q], A[q+1]), such that A[p] > A[p+1] and A[q] > A[q+1], we can claim that it is A[p] and A[q+1] being swapped

You know that there's only a single swap, that would create either 2 discrepancies in the sorted order, or only one if they're adjacent. Let's say p < q, so the A[p],A[p+1] is the first descending pair, and the q's are the second.

• If there's no second couple, than swapping the first couple would fix the tree, that's the easy part. Otherwise we know there are two non-adjacent nodes.

• Out of the A[p] and A[p+1] let's say that A[p+1] was the one out of place. Since this is the first couple we would have to move A[p+1] forward towards the second couple, but that means that it's still going to be smaller than the earlier A[p] that stayed in place, so we would not create a sorted array. We must therefore chose A[p].

• Same goes for the A[q] and A[q+1], let's say that A[q] was out of place, that means we'll have to move it backwards, and it would still be larger than A[q+1] appearing later, again breaking sort.