Length of the longest subsequence such that xor of adjacent elements is non-decreasing

Given a sequence arr of N positive integers, the task is to find the length of the longest subsequence such that xor of adjacent integers in the subsequence must be non-decreasing.

Examples:

Input: N = 8, arr = {1, 100, 3, 64, 0, 5, 2, 15}
Output: 6
The subsequence of maximum length is {1, 3, 0, 5, 2, 15}
with XOR of adjacent elements as {2, 3, 5, 7, 13}



Input: N = 3, arr = {1, 7, 10}
Output: 3
The subsequence of maximum length is {1, 3, 7}
with XOR of adjacent elements as {2, 4}.

Approach:

  • This problem can be solved using dynamic programming where dp[i] will store the length of the longest valid subsequence that ends at index i.
  • First, store the xor of all the pairs of elements i.e. arr[i] ^ arr[j] and the pair (i, j) also and then sort them according to the value of xor as they need to be non-decreasing.
  • Now if the pair (i, j) is considered then the length of the longest subsequence that ends at j will be max(dp[j], 1 + dp[i]). In this way, calculate the maximum possible value of dp[] array for each position and then take the maximum of them.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the length of the longest
// subsequence such that the XOR of adjacent
// elements in the subsequence must
// be non-decreasing
int LongestXorSubsequence(int arr[], int n)
{
  
    vector<pair<int, pair<int, int> > > v;
  
    for (int i = 0; i < n; i++) {
        for (int j = i + 1; j < n; j++) {
  
            // Computing xor of all the pairs
            // of elements and store them
            // along with the pair (i, j)
            v.push_back(make_pair(arr[i] ^ arr[j],
                                  make_pair(i, j)));
        }
    }
  
    // Sort all possible xor values
    sort(v.begin(), v.end());
  
    int dp[n];
  
    // Initialize the dp array
    for (int i = 0; i < n; i++) {
        dp[i] = 1;
    }
  
    // Calculating the dp array
    // for each possible position
    // and calculating the max length
    // that ends at a particular index
    for (auto i : v) {
        dp[i.second.second]
            = max(dp[i.second.second],
                  1 + dp[i.second.first]);
    }
  
    int ans = 1;
  
    // Taking maximum of all position
    for (int i = 0; i < n; i++)
        ans = max(ans, dp[i]);
  
    return ans;
}
  
// Driver code
int main()
{
  
    int arr[] = { 2, 12, 6, 7, 13, 14, 8, 6 };
    int n = sizeof(arr) / sizeof(arr[0]);
  
    cout << LongestXorSubsequence(arr, n);
  
    return 0;
}

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Python3

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# Python3 implementation of the approach
  
# Function to find the length of the longest
# subsequence such that the XOR of adjacent
# elements in the subsequence must
# be non-decreasing
def LongestXorSubsequence(arr, n):
  
    v = []
  
    for i in range(0, n): 
        for j in range(i + 1, n): 
  
             # Computing xor of all the pairs
            # of elements and store them
            # along with the pair (i, j)
            v.append([(arr[i] ^ arr[j]), (i, j)])
  
        # v.push_back(make_pair(arr[i] ^ arr[j], make_pair(i, j)))
          
    # Sort all possible xor values
    v.sort()
      
    # Initialize the dp array
    dp = [1 for x in range(88)]
  
    # Calculating the dp array
    # for each possible position
    # and calculating the max length
    # that ends at a particular index
    for a, b in v:
        dp[b[1]] = max(dp[b[1]], 1 + dp[b[0]])
      
    ans = 1
  
    # Taking maximum of all position
    for i in range(0, n):
        ans = max(ans, dp[i])
  
    return ans
  
# Driver code
arr = [ 2, 12, 6, 7, 13, 14, 8, 6 ]
n = len(arr)
print(LongestXorSubsequence(arr, n))
  
# This code is contributed by Sanjit Prasad

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Output:

5


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Improved By : Sanjit_Prasad