Longest subsequence having difference between the maximum and minimum element equal to K

Given an array arr[] consisting of N integers and an integer K, the task is to find the longest subsequence of the given array such that the difference between the maximum and the minimum element in the subsequence is exactly K.

Examples:

Input: arr[] = {1, 3, 2, 2, 5, 2, 3, 7}, K = 1
Output: 5
Explanation:
The longest subsequence whose difference between the maximum and minimum element is K(= 1) is {3, 2, 2, 2, 3}.
Therefore, the length is 5.

Input: arr [] = {4, 3, 3, 4}, K = 4
Output: 0

Naive Approach: The simplest approach is to generate all possible subsequences of the given array and for every subsequence, find the difference between the maximum and minimum values in the subsequence. If it is equal to K, update the resultant longest subsequence length. After checking for all subsequences, print the maximum length obtained.

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

Efficient Approach: To optimize the above approach, the idea is based on the observation that in the required subsequence, only two unique elements can be present, and their difference should be K. The problem can be solved by Hashing, to store the frequency of each array element. Follow the steps below to solve the problem:

• Initialize a variable, say ans, to store the length of the longest subsequence.
• Initialize a hashmap, say M, that stores the frequency of the array elements.
• Traverse the array arr[] using the variable i and for each array element arr[i], increment the frequency of arr[] in M by 1.
• Now traverse the hashmap M and for each key(say X) in M, if (X + K) is also present in the M, then update the value of ans to the maximum of ans and the sum of the value of both the keys.
• After completing the above steps, print the value of ans as the result.

Below is the implementation of the above approach:

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find longest subsequence
// having absolute difference between
// maximum and minimum element K
void longestSubsequenceLength(int arr[],
                              int N, int K)
{
    // Stores the frequency of each
    // array element
    unordered_map<int, int> um;

    // Traverse the array arr[]
    for (int i = 0; i < N; i++)

        // Increment um[arr[i]] by 1
        um[arr[i]]++;

    // Store the required answer
    int ans = 0;

    // Traverse the hashmap
    for (auto it : um) {

        // Check if it.first + K
        // exists in the hashmap
        if (um.find(it.first + K)
            != um.end()) {

            // Update the answer
            ans = max(ans,
                      it.second
                          + um[it.first + K]);
        }
    }

    // Print the result
    cout << ans;
}

// Driver Code
int main()
{
    int arr[] = { 1, 3, 2, 2, 5, 2, 3, 7 };
    int N = sizeof(arr) / sizeof(arr[0]);
    int K = 1;

    longestSubsequenceLength(arr, N, K);

    return 0;
}

// Java program for the above approach
import java.util.*;

class GFG{
 
// Function to find longest subsequence
// having absolute difference between
// maximum and minimum element K
static void longestSubsequenceLength(int []arr,
                                     int N, int K)
{
    
    // Stores the frequency of each
    // array element
    Map<Integer, 
        Integer> um = new HashMap<Integer, 
                                  Integer>();

    // Traverse the array arr[]
    for(int i = 0; i < N; i++)
    {
        if (um.containsKey(arr[i]))
            um.put(arr[i], um.get(arr[i]) + 1);
        else
            um.put(arr[i], 1);
    }
    
    // Store the required answer
    int ans = 0;

    // Traverse the hashmap
    for(Map.Entry<Integer, Integer> e : um.entrySet())
    {
        
        // Check if it.first + K
        // exists in the hashmap
        if (um.containsKey(e.getKey() + K)) 
        {
            
            // Update the answer
            ans = Math.max(ans, e.getValue() + 
                           um.get(e.getKey() + K));
        }
    }

    // Print the result
    System.out.println(ans);
}

// Driver Code
public static void main(String args[])
{
    int []arr = { 1, 3, 2, 2, 5, 2, 3, 7 };
    int N = arr.length;
    int K = 1;
    
    longestSubsequenceLength(arr, N, K);
}
}

// This code is contributed by bgangwar59

# Python3 program for the above approach
from collections import defaultdict

# Function to find longest subsequence
# having absolute difference between
# maximum and minimum element K
def longestSubsequenceLength(arr, N, K):

    # Stores the frequency of each
    # array element
    um = defaultdict(int)

    # Traverse the array arr[]
    for i in range(N):

        # Increment um[arr[i]] by 1
        um[arr[i]] += 1

    # Store the required answer
    ans = 0

    # Traverse the hashmap
    for it in um.keys():

        # Check if it.first + K
        # exists in the hashmap
        if (it + K) in um:

            # Update the answer
            ans = max(ans,
                      um[it]
                      + um[it + K])

    # Print the result
    print(ans)

# Driver Code
if __name__ == "__main__":

    arr = [1, 3, 2, 2, 5, 2, 3, 7]
    N = len(arr)
    K = 1

    longestSubsequenceLength(arr, N, K)

    # This code is contributed by chitranayal.

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG
{

// Function to find longest subsequence
// having absolute difference between
// maximum and minimum element K
static void longestSubsequenceLength(int[] arr,
                              int N, int K)
{
  
    // Stores the frequency of each
    // array element
    Dictionary<int, int> um = new Dictionary<int, int>();

    // Traverse the array
    for(int i = 0; i < N; i++) 
    {
         
        // Increase the counter of
        // the array element by 1
        int count = um.ContainsKey(arr[i]) ? um[arr[i]] : 0; 
        if (count == 0)
        {
            um.Add(arr[i], 1);
        }
        else
        {
            um[arr[i]] = count + 1;
        }
    }

    // Store the required answer
    int ans = 0;

    // Traverse the hashmap
    foreach(KeyValuePair<int, int> it in um) 
    {

        // Check if it.first + K
        // exists in the hashmap
        if (um.ContainsKey(it.Key + K))
        {

            // Update the answer
            ans = Math.Max(ans, (it.Value + um[it.Key + K]));
        }
    }

    // Print the result
    Console.Write(ans);
}

// Driver Code
public static void Main()
{
    int[] arr = { 1, 3, 2, 2, 5, 2, 3, 7 };
    int N = arr.Length;
    int K = 1;

    longestSubsequenceLength(arr, N, K);
}
}

// This code is contributed by splevel62.

<script>

// Javascript program for the above approach

// Function to find longest subsequence
// having absolute difference between
// maximum and minimum element K
function longestSubsequenceLength(arr, N, K)
{
    // Stores the frequency of each
    // array element
    var um = new Map();

    // Traverse the array arr[]
    for (var i = 0; i < N; i++)

        // Increment um[arr[i]] by 1
        if(um.has(arr[i]))
        {
            um.set(arr[i], um.get(arr[i])+1);
        }
        else
        {
            um.set(arr[i], 1);
        }

    // Store the required answer
    var ans = 0;

    // Traverse the hashmap
    um.forEach((value, key) => {
         
        // Check if it.first + K
        // exists in the hashmap
        if (um.has(key+K)) {

            // Update the answer
            ans = Math.max(ans,
                      value
                          + um.get(key+K));
        }
    });

    // Print the result
    document.write( ans);
}

// Driver Code
var arr = [ 1, 3, 2, 2, 5, 2, 3, 7 ];
var N = arr.length;
var K = 1;
longestSubsequenceLength(arr, N, K);

</script>    

5

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