Cartesian tree from inorder traversal | Segment Tree

Given an in-order traversal of a cartesian tree, the task is to build the entire tree from it.

Examples: 

Input: arr[] = {1, 5, 3}
Output: 1 5 3
 5
 / \
1 3

Input: arr[] = {3, 7, 4, 8}
Output: 3 7 4 8
 8
 /
 7
 / \
 3 4

Approach: We have already seen an algorithm here that takes O(NlogN) time on an average but can get to O(N²) in the worst case.
In this article, we will see how to build the cartesian in worst case running time of O(Nlog(N)). For this, we will use segment-tree to answer range-max queries.

Below will be our recursive algorithm on range {L, R}:  

1. Find the maximum in this range {L, R} using range-max query on the segment-tree. Let's say 'M' is index the maximum in the range.
2. Pick up 'arr[M]' as the value for current node and create a node with this value.
3. Solve for range {L, M-1} and {M+1, R}.
4. Set the node returned by {L, M-1} as the left child of current node and {M+1, R} as the right child.

Below is the implementation of the above approach:  

8 11 21 100 5 70 55

Time complexity: O(N+logN)
Auxiliary Space: O(N).  

Related Topic: Segment Tree