Given a sequence of positive integers of length N. The only operation allowed is to insert a single integer of any value at any position in the sequence. The task is to find the sub-sequence of maximum length that contains consecutive values in increasing order.
Examples:
Input: arr[] = {2, 1, 4, 5}
Output: 4
Insert element with value 3 at the 3rd position.(1 based indexing)
The new sequence becomes {2, 1, 3, 4, 5}
Longest consecutive sub-sequence would be {2, 3, 4, 5}Input: arr[] = {2, 1, 2, 3, 5, 7}
Output: 5
Approach: The idea is to use Dynamic Programming.
Let dp[val][0] be the length of required subsequence that ends in an element equal to val and the element is not inserted yet. Let dp[val][1] be the length of required subsequence that ends in an element equal to val and some element has been inserted already.
Now break the problem into its subproblems as follows:
To calculate dp[val][0], as no element in inserted, the length of the subsequence will increase by 1 from its previous value
dp[val][0] = 1 + dp[val – 1][0].
To calculate dp[val][1], consider these two cases:
- When the element is already inserted for (val-1), then there would be an increment of length 1 from dp[ val-1 ][ 1 ]
- When the element has not been inserted yet, then the element with value (val-1) can be inserted . Hence there would be an increment of length 2 from dp[ val-2 ][ 0 ].
Take maximum of both the above cases.
dp[val][1] = max(1 + dp[val – 1][1], 2 + dp[val – 2][0]).
Below is the implementation of the above approach:
C++
// C++ implementation of above approach #include <bits/stdc++.h> using namespace std; // Function to return the length of longest // consecuetive subsequence after inserting an element int LongestConsSeq(int arr[], int N) { // Variable to find maximum value of the array int maxval = 1; // Calculating maximum value of the array for (int i = 0; i < N; i += 1) { maxval = max(maxval, arr[i]); } // Declaring the DP table int dp[maxval + 1][2] = { 0 }; // Variable to store the maximum length int ans = 1; // Iterating for every value present in the array for (int i = 0; i < N; i += 1) { // Recurrence for dp[val][0] dp[arr[i]][0] = (1 + dp[arr[i] - 1][0]); // No value can be inserted before 1, // hence the element value should be // greater than 1 for this recurrance relation if (arr[i] >= 2) // Recurrence for dp[val][1] dp[arr[i]][1] = max(1 + dp[arr[i] - 1][1], 2 + dp[arr[i] - 2][0]); else // Maximum length of consecutive sequence // ending at 1 is equal to 1 dp[arr[i]][1] = 1; // Update the ans variable with // the new maximum length possible ans = max(ans, dp[arr[i]][1]); } // Return the ans return ans; } // Driver code int main() { // Input array int arr[] = { 2, 1, 4, 5 }; int N = sizeof(arr) / sizeof(arr[0]); cout << LongestConsSeq(arr, N); return 0; } |
Java
// Java implementation of above approach class GFG { // Function to return the length of longest // consecuetive subsequence after inserting an element static int LongestConsSeq(int [] arr, int N) { // Variable to find maximum value of the array int maxval = 1; // Calculating maximum value of the array for (int i = 0; i < N; i += 1) { maxval = Math. max(maxval, arr[i]); } // Declaring the DP table int [][] dp = new int[maxval + 1][2]; // Variable to store the maximum length int ans = 1; // Iterating for every value present in the array for (int i = 0; i < N; i += 1) { // Recurrence for dp[val][0] dp[arr[i]][0] = (1 + dp[arr[i] - 1][0]); // No value can be inserted before 1, // hence the element value should be // greater than 1 for this recurrance relation if (arr[i] >= 2) // Recurrence for dp[val][1] dp[arr[i]][1] = Math.max(1 + dp[arr[i] - 1][1], 2 + dp[arr[i] - 2][0]); else // Maximum length of consecutive sequence // ending at 1 is equal to 1 dp[arr[i]][1] = 1; // Update the ans variable with // the new maximum length possible ans = Math.max(ans, dp[arr[i]][1]); } // Return the ans return ans; } // Driver code public static void main (String[] args) { // Input array int [] arr = { 2, 1, 4, 5 }; int N = arr.length; System.out.println(LongestConsSeq(arr, N)); } } // This code is contributed by ihritik |
Python3
# Python3 implementation of above approach # Function to return the length of longest # consecuetive subsequence after inserting an element def LongestConsSeq(arr, N): # Variable to find maximum value of the array maxval = 1 # Calculating maximum value of the array for i in range(N): maxval = max(maxval, arr[i]) # Declaring the DP table dp=[[ 0 for i in range(2)] for i in range(maxval + 1)] # Variable to store the maximum length ans = 1 # Iterating for every value present in the array for i in range(N): # Recurrence for dp[val][0] dp[arr[i]][0] = 1 + dp[arr[i] - 1][0] # No value can be inserted before 1, # hence the element value should be # greater than 1 for this recurrance relation if (arr[i] >= 2): # Recurrence for dp[val][1] dp[arr[i]][1] = max(1 + dp[arr[i] - 1][1], 2 + dp[arr[i] - 2][0]) else: # Maximum length of consecutive sequence # ending at 1 is equal to 1 dp[arr[i]][1] = 1 # Update the ans variable with # the new maximum length possible ans = max(ans, dp[arr[i]][1]) # Return the ans return ans # Driver code arr=[2, 1, 4, 5] N = len(arr) print(LongestConsSeq(arr, N)) # This code is contributed by mohit kumar 29 |
C#
// C# implementation of above approach using System; class GFG { // Function to return the length of longest // consecuetive subsequence after inserting an element static int LongestConsSeq(int [] arr, int N) { // Variable to find maximum value of the array int maxval = 1; // Calculating maximum value of the array for (int i = 0; i < N; i += 1) { maxval =Math.Max(maxval, arr[i]); } // Declaring the DP table int [ , ] dp = new int[maxval + 1, 2]; // Variable to store the maximum length int ans = 1; // Iterating for every value present in the array for (int i = 0; i < N; i += 1) { // Recurrence for dp[val][0] dp[arr[i], 0] = (1 + dp[arr[i] - 1, 0]); // No value can be inserted before 1, // hence the element value should be // greater than 1 for this recurrance relation if (arr[i] >= 2) // Recurrence for dp[val][1] dp[arr[i], 1] = Math.Max(1 + dp[arr[i] - 1, 1], 2 + dp[arr[i] - 2, 0]); else // Maximum length of consecutive sequence // ending at 1 is equal to 1 dp[arr[i], 1] = 1; // Update the ans variable with // the new maximum length possible ans = Math.Max(ans, dp[arr[i], 1]); } // Return the ans return ans; } // Driver code public static void Main () { // Input array int [] arr = new int [] { 2, 1, 4, 5 }; int N = arr.Length; Console.WriteLine(LongestConsSeq(arr, N)); } } // This code is contributed by ihritik |
4
Time Complexity: O(N)
Space Complexity: O(MaxValue) where MaxValue is the maximum value present in the array.
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