Length of longest increasing absolute even subsequence

Given an array arr[] consisting of N integers, the task is to find the length of the longest increasing absolute even subsequence.

An increasing absolute even subsequence is an increasing subsequence of array elements having absolute difference between adjacent pairs as even.

Examples:

Input: arr[] = {10, 22, 9, 33, 21, 50, 41, 60}
Output: 4
Explanation: The longest increasing absolute even subsequence is {10, 22, 50, 60}. Therefore, the required length is 4.

Input: arr[] = {11, -22, 43, -54, 66, 5}
Output: 3
Explanation:
The longest increasing absolute even subsequence is 3 i.e. {-22, -54, 66}. Therefore, the required length is 4.

Naive Approach: The simplest approach is to generate all possible subsequence of the given array and for each subsequence, check if the subsequence is increasing and absolute difference between adjacent elements is even or not. Print the length of the longest such subsequence.

Time Complexity: O(2N)
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach, the idea is similar to finding longest increasing subsequence. But the only condition to be changed is to check if the absolute difference between two adjacent elements of the subsequence is even or not. Follow the steps below to solve the problem:

1. Initialize an auxiliary array dp[] where all are initially 1.
2. Traverse the given array arr[] using variable i over the range [0, N) and for each index do the following:
• Iterate using variable j over the range [0, i) and check for the following three conditions:
1. If absolute value of arr[i] > arr[j].
2. If arr[i] and arr[j] both are even or not.
3. If dp[i] < dp[j] + 1.
• If the above three conditions are satisfied for any index j, then update dp[i] = dp[j] + 1.
3. Print the maximum element of the array dp[] as the required result.

Below is the implementation of the above approach:

 // C++14 program for the above approach #include using namespace std;   // Function to find the longest // increasing absolute even subsequence void EvenLIS(int arr[], int n) {           // Stores length of     // required subsequence     int lis[n];     for(int i = 0; i < n; i++)         lis[i] = 1;            // Traverse the array     for(int i = 1; i < n; i++)     {                   // Traverse prefix of current         // array element         for(int j = 0; j < i; j++)         {                           // Check if the subsequence is             // LIS and have even absolute             // difference of adjacent pairs             if (abs(arr[i]) > abs(arr[j]) &&                 abs(arr[i]) % 2 == 0 &&                 abs(arr[j]) % 2 == 0 &&                     lis[i] < lis[j] + 1)                    // Update lis[]                 lis[i] = lis[j] + 1;         }     }            // Stores maximum length     int maxlen = 0;            // Find the length of longest     // abolute even subsequence     for(int i = 0; i < n; i++)         maxlen = max(maxlen, lis[i]);        // Return the maximum length of     // absolute even subsequence     cout << maxlen << endl; }    // Driver code int main() {           // Given array arr[] and brr[]     int arr[] = { 11, -22, 43, -54, 66, 5 };        int N = sizeof(arr) / sizeof(arr[0]);            // Function call     EvenLIS(arr, N); }   // This code is contributed by code_hunt

 // Java program for the above approach import java.util.*; import java.io.*;   class GFG{       // Function to find the longest // increasing absolute even subsequence static void EvenLIS(int arr[]) {           // Length of arr     int n = arr.length;           // Stores length of     // required subsequence     int lis[] = new int[n];     Arrays.fill(lis, 1);           // Traverse the array     for(int i = 1; i < n; i++)     {               // Traverse prefix of current         // array element         for(int j = 0; j < i; j++)         {               // Check if the subsequence is             // LIS and have even absolute             // difference of adjacent pairs             if (Math.abs(arr[i]) > Math.abs(arr[j]) &&                 Math.abs(arr[i]) % 2 == 0 &&                 Math.abs(arr[j]) % 2 == 0 &&                           lis[i] < lis[j] + 1)                   // Update lis[]                 lis[i] = lis[j] + 1;         }     }           // Stores maximum length     int maxlen = 0;           // Find the length of longest     // abolute even subsequence     for(int i = 0; i < n; i++)         maxlen = Math.max(maxlen, lis[i]);       // Return the maximum length of     // absolute even subsequence     System.out.println(maxlen); }   // Driver code public static void main(String args[]) {           // Given array arr[] and brr[]     int arr[] = { 11, -22, 43, -54, 66, 5 };       int N = arr.length;           // Function call     EvenLIS(arr); } }   // This code is contributed by bikram2001jha

 # Python3 program for the above approach     # Function to find the longest # increasing absolute even subsequence def EvenLIS(arr):       # Length of arr     n = len(arr)       # Stores length of     # required subsequence     lis = [1]*n       # Traverse the array     for i in range(1, n):               # Traverse prefix of current         # array element         for j in range(0, i):               # Check if the subsequence is             # LIS and have even absolute             # difference of adjacent pairs               if abs(arr[i]) > abs(arr[j]) \             and abs(arr[i] % 2) == 0 \             and abs(arr[j] % 2) == 0 \             and lis[i] < lis[j]+1:                   # Update lis[]                 lis[i] = lis[j]+1       # Stores maximum length     maxlen = 0       # Find the length of longest     # abolute even subsequence     for i in range(n):         maxlen = max(maxlen, lis[i])       # Return the maximum length of     # absolute even subsequence     print(maxlen)   # Driver Code   # Given arr[] arr = [11, -22, 43, -54, 66, 5]   # Function Call EvenLIS(arr)

 // C# program for the above approach using System;   class GFG{       // Function to find the longest // increasing absolute even subsequence static void EvenLIS(int []arr) {           // Length of arr     int n = arr.Length;           // Stores length of     // required subsequence     int []lis = new int[n];     for(int i = 0; i < n; i++)         lis[i] = 1;          // Traverse the array     for(int i = 1; i < n; i++)     {               // Traverse prefix of current         // array element         for(int j = 0; j < i; j++)         {               // Check if the subsequence is             // LIS and have even absolute             // difference of adjacent pairs             if (Math.Abs(arr[i]) > Math.Abs(arr[j]) &&                 Math.Abs(arr[i]) % 2 == 0 &&                 Math.Abs(arr[j]) % 2 == 0 &&                          lis[i] < lis[j] + 1)                   // Update lis[]                 lis[i] = lis[j] + 1;         }     }           // Stores maximum length     int maxlen = 0;           // Find the length of longest     // abolute even subsequence     for(int i = 0; i < n; i++)         maxlen = Math.Max(maxlen, lis[i]);       // Return the maximum length of     // absolute even subsequence     Console.WriteLine(maxlen); }   // Driver code public static void Main(String []args) {     // Given array []arr and brr[]     int []arr = { 11, -22, 43, -54, 66, 5 };       int N = arr.Length;           // Function call     EvenLIS(arr); } }   // This code is contributed by Amit Katiyar

Output:
3

Time Complexity: O(N2)
Auxiliary Space: O(N)

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