Length of Longest Increasing Subsequences (LIS) using Segment Tree

Given an array arr[] of size N, the task is to count the number of longest increasing subsequences present in the given array.

Example:

Input: arr[] = {2, 2, 2, 2, 2}
Output: 5
Explanation: The length of the longest increasing subsequence is 1, i.e. {2}. Therefore, count of longest increasing subsequences of length 1 is 5. 
 

Input: arr[] = {1, 3, 5, 4, 7}
Output: 2
Explanation: The length of the longest increasing subsequence is 4, and there are 2 longest increasing subsequences of length 4, i.e. {1, 3, 4, 7} and {1, 3, 5, 7}.

Approach: An approach to the given problem has been already discussed using dynamic programming in this article. 
This article suggests a different approach using segment trees. Follow the below steps to solve the given problem:

• Initialise the segment tree as an array of pairs initially containing pairs of (0, 0), where the 1^st element represents the length of LIS and 2^nd element represents the count of LIS of current length.
• The 1^st element of the segment tree can be calculated similarly to the approach discussed in this article.
• The 2^nd element of the segment tree can be calculated using the following steps:
  • If cases where the length of left child > length of right child, the parent node becomes equal to the left child as LIS will that be of the left child.
  • If cases where the length of left child < length of right child, the parent node becomes equal to the right child as LIS will that be of the right child.
  • If cases where the length of left child = length of right child, the parent node becomes equal to the sum of the count of LIS of the left child and the right child.
• The required answer is the 2^nd element of the root of the segment tree.

Below is the implementation of the above approach:

2

Time Complexity: O(N*log N)
Auxiliary Space: O(N)

Related Topic: Segment Tree