Maximum length subsequence with difference between adjacent elements as either 0 or 1

Given an array of n integers. The problem is to find maximum length of the subsequence with difference between adjacent elements in the subsequence as either 0 or 1. Time Complexity of O(n) is required.

Examples:

Input : arr[] = {2, 5, 6, 3, 7, 6, 5, 8}
Output : 5
The subsequence is {5, 6, 7, 6, 5}.

Input : arr[] = {-2, -1, 5, -1, 4, 0, 3}
Output : 4
The subsequence is {-2, -1, -1, 0}.

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Method 1: Previously an approach having time complexity of O(n²) have been discussed in this post.

Method 2 (Efficient Approach): The idea is to create a hash map having tuples in the form (ele, len), where len denotes the length of the longest subsequence ending with the element ele. Now, for each element arr[i] we can find the length of the values arr[i]-1, arr[i] and arr[i]+1 in the hash table and consider the maximum among them. Let this maximum value be max. Now, the length of longest subsequence ending with arr[i] would be max+1. Update this length along with the element arr[i] in the hash table. Finally, the element having the maximum length in the hash table gives the maximum length subsequence.

C++

// C++ implementation to find maximum length 
// subsequence with difference between adjacent  
// elements as either 0 or 1 
#include <bits/stdc++.h> 
using namespace std; 
   
// function to find maximum length subsequence  
// with difference between adjacent elements as 
// either 0 or 1 
int maxLenSub(int arr[], int n) 
{ 
    // hash table to map the array element with the 
    // length of the longest subsequence of which 
    // it is a part of and is the last element of 
    // that subsequence 
    unordered_map<int, int> um; 
      
    // to store the maximum length subsequence 
    int maxLen = 0; 
      
    // traverse the array elements 
    for (int i=0; i<n; i++) 
    { 
        // initialize current length  
        // for element arr[i] as 0 
        int len = 0; 
          
        // if 'arr[i]-1' is in 'um' and its length of  
        // subsequence is greater than 'len' 
        if (um.find(arr[i]-1) != um.end() && len < um[arr[i]-1]) 
            len = um[arr[i]-1]; 
          
        // if 'arr[i]' is in 'um' and its length of  
        // subsequence is greater than 'len'     
        if (um.find(arr[i]) != um.end() && len < um[arr[i]]) 
            len = um[arr[i]]; 
              
        // if 'arr[i]+1' is in 'um' and its length of  
        // subsequence is greater than 'len'         
        if (um.find(arr[i]+1) != um.end() && len < um[arr[i]+1]) 
            len = um[arr[i]+1];     
          
        // update arr[i] subsequence length in 'um'     
        um[arr[i]] = len + 1; 
          
        // update maximum length 
        if (maxLen < um[arr[i]])     
            maxLen = um[arr[i]]; 
    } 
       
    // required maximum length subsequence 
    return maxLen;         
} 
   
// Driver program to test above 
int main() 
{ 
    int arr[] = {2, 5, 6, 3, 7, 6, 5, 8}; 
    int n = sizeof(arr) / sizeof(arr[0]); 
    cout << "Maximum length subsequence = "
         << maxLenSub(arr, n); 
    return 0; 
}  

Python3

# Python3 implementation to find maximum  
# length subsequence with difference between  
# adjacent elements as either 0 or 1  
from collections import defaultdict 
  
# Function to find maximum length subsequence with  
# difference between adjacent elements as either 0 or 1  
def maxLenSub(arr, n):  
  
    # hash table to map the array element with the  
    # length of the longest subsequence of which it is a  
    # part of and is the last element of that subsequence  
    um = defaultdict(lambda:0) 
      
    # to store the maximum length subsequence  
    maxLen = 0
      
    # traverse the array elements  
    for i in range(0, n):  
      
        # initialize current length  
        # for element arr[i] as 0  
        length = 0
          
        # if 'arr[i]-1' is in 'um' and its length of  
        # subsequence is greater than 'len'  
        if (arr[i]-1) in um and length < um[arr[i]-1]: 
            length = um[arr[i]-1]  
          
        # if 'arr[i]' is in 'um' and its length of  
        # subsequence is greater than 'len'  
        if arr[i] in um and length < um[arr[i]]:  
            length = um[arr[i]]  
              
        # if 'arr[i]+1' is in 'um' and its length of  
        # subsequence is greater than 'len'   
        if (arr[i]+1) in um and length < um[arr[i]+1]:  
            length = um[arr[i]+1]  
          
        # update arr[i] subsequence length in 'um'  
        um[arr[i]] = length + 1
          
        # update maximum length  
        if maxLen < um[arr[i]]:  
            maxLen = um[arr[i]]  
      
    # required maximum length subsequence  
    return maxLen 
  
# Driver program to test above  
if __name__ == "__main__":  
  
    arr = [2, 5, 6, 3, 7, 6, 5, 8]  
    n = len(arr)  
    print("Maximum length subsequence =", maxLenSub(arr, n)) 
      
# This code is contributed by Rituraj Jain 

Output:

Maximum length subsequence = 5

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

Thanks to Neeraj for suggesting the above solution in the comments of this post.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

Maximum length subsequence with difference between adjacent elements as either 0 or 1 | Set 2

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Python3

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