Longest Decreasing Subsequence

Posted: 26 Oct, 2020
Difficulty: Moderate

PROBLEM STATEMENT

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You are given an array/list ARR consisting of N integers. Your task is to find the length of the longest decreasing subsequence.

A subsequence is a sequence of numbers obtained by deleting zero or more elements from the array/list, keeping the relative positions of elements same as it was in the initial sequence. A decreasing subsequence is a subsequence in which every element is strictly less than the previous number.

Note:

There can be more than one subsequences with the longest length.
For example:-
For the given array [5, 0, 3, 2, 9], the longest decreasing subsequence is of length 3, i.e. [5, 3, 2]
Note:
Try to solve the problem in O(N log N) time complexity.
Input Format :
The first line of input contains an integer 'T' representing the number of the test case. Then the test case follows.

The first line of each test case contains an integer ‘N’ representing the size of the array/list.

The second line of each test case contains N single space-separated integers representing the array/list elements.
Output Format :
For each test case, print the integer denoting the length of the longest decreasing subsequence.

Print the output of each test case in a separate line.
Note:
You do not need to print anything; it has already been taken care of. Just implement the given function.
Constraints:
1 <= T <= 50 
1 <= N <= 5000
1 <= ARR[i] <= 10^9

Time Limit: 1 sec
Approach 1

We can use recursion to solve this problem.

  1. For each element, there are two possibilities.
    1. We include the current item in LDS if it is smaller than the previous element and recurse for remaining items.
    2. We exclude the current item from LDS and recurse for remaining items.
  2. Finally, we return the length of LDS we get by including or excluding the current item.
  3. The base case of the recursion would be to return 0 when no items are left.
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