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Count of all subsequences having adjacent elements with different parity

  • Last Updated : 11 May, 2021

Given an array arr[] of size N, the task is to find the number of non-empty subsequences from the given array such that no two adjacent elements of the subsequence have the same parity.

Examples:  

Input: arr[] = [5, 6, 9, 7] 
Output:
Explanation: 
All such subsequences of given array will be {5}, {6}, {9}, {7}, {5, 6}, {6, 7}, {6, 9}, {5, 6, 9}, {5, 6, 7}.
Input: arr[] = [2, 3, 4, 8] 
Output: 9  

Naive Approach: Generate all non-empty subsequences and select the ones with alternate odd-even or even-odd numbers and count all such subsequences to obtain the answer. 
Time Complexity: O(2N)
Efficient Approach: 
The above approach can be optimized using Dynamic Programming. Follow the steps below to solve the problem: 
 

  • Consider a dp[] matrix of dimensions (N+1)*(2).
  • dp[i][0] stores the count of subsequences till ith index ending with an even element.
  • dp[i][1] stores the count of subsequences till ith index ending with an odd element.
  • Hence, for every ith element, check if the element is even or odd and proceed by including as well as excluding the ith element.
  • Hence, the recurrence relation if the ith element is odd: 

 dp[i][1] = dp[i – 1][0] (Including the ith element by considering all subsequences ending with even element till (i – 1)th index) + 1 + dp[i – 1][1] (Excluding the ith element)



  • Similarly, if the ith element is even:

 dp[i][0] = dp[i – 1][1] (Including the ith element by considering all subsequences ending with odd element till (i – 1)th index) + 1 + dp[i – 1][0] (Excluding the ith element) 

  • Finally, the sum of dp[n][0], which contains all such subsequences ending with an even element, and dp[n][1], which contains all such subsequences ending with an odd element, is the required answer.

Below is the implementation of the above approach: 

C++




// C++ Program to implement the
// above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to find required subsequences
int validsubsequences(int arr[], int n)
{
    // dp[i][0]: Stores the number of
    // subsequences till i-th index
    // ending with even element
    // dp[i][1]: Stores the number of
    // subsequences till i-th index
    // ending with odd element
    long long int dp[n + 1][2];
 
    // Initialise the dp[][] with 0.
    for (int i = 0; i < n + 1; i++) {
        dp[i][0] = 0;
        dp[i][1] = 0;
    }
 
    for (int i = 1; i <= n; i++) {
 
        // If odd element is
        // encountered
        if (arr[i - 1] % 2) {
 
            // Considering i-th element
            // will be present in
            // the subsequence
            dp[i][1] += 1;
 
            // Appending i-th element to all
            // non-empty subsequences
            // ending with even element
            // till (i-1)th indexes
            dp[i][1] += dp[i - 1][0];
 
            // Considering ith element will
            // not be present in
            // the subsequence
            dp[i][1] += dp[i - 1][1];
 
            dp[i][0] += dp[i - 1][0];
        }
        else {
 
            // Considering i-th element
            // will be present in
            // the subsequence
            dp[i][0] += 1;
 
            // Appending i-th element to all
            // non-empty subsequences
            // ending with odd element
            // till (i-1)th indexes
            dp[i][0] += dp[i - 1][1];
 
            // Considering ith element will
            // not be present in
            // the subsequence
            dp[i][0] += dp[i - 1][0];
            dp[i][1] += dp[i - 1][1];
        }
    }
 
    // Count of all valid subsequences
    return dp[n][0] + dp[n][1];
}
 
// Driver Code
int main()
{
    int arr[] = { 5, 6, 9, 7 };
    int n = sizeof(arr) / sizeof(arr[0]);
 
    cout << validsubsequences(arr, n);
 
    return 0;
}

Java




// Java Program implementation
// of the approach
import java.util.*;
import java.io.*;
 
class GFG{
 
// Function to find required subsequences    
static int validsubsequences(int arr[], int n)
{
     
    // dp[i][0]: Stores the number of
    // subsequences till i-th index
    // ending with even element
    // dp[i][1]: Stores the number of
    // subsequences till i-th index
    // ending with odd element
    long dp[][] = new long [n + 1][2];
 
    // Initialise the dp[][] with 0.
    for(int i = 0; i < n + 1; i++)
    {
        dp[i][0] = 0;
        dp[i][1] = 0;
    }
     
    for(int i = 1; i <= n; i++)
    {
         
        // If odd element is
        // encountered
        if (arr[i - 1] % 2 != 0)
        {
 
            // Considering i-th element
            // will be present in
            // the subsequence
            dp[i][1] += 1;
 
            // Appending i-th element to all
            // non-empty subsequences
            // ending with even element
            // till (i-1)th indexes
            dp[i][1] += dp[i - 1][0];
 
            // Considering ith element will
            // not be present in
            // the subsequence
            dp[i][1] += dp[i - 1][1];
            dp[i][0] += dp[i - 1][0];
        }
        else
        {
 
            // Considering i-th element
            // will be present in
            // the subsequence
            dp[i][0] += 1;
 
            // Appending i-th element to all
            // non-empty subsequences
            // ending with odd element
            // till (i-1)th indexes
            dp[i][0] += dp[i - 1][1];
 
            // Considering ith element will
            // not be present in
            // the subsequence
            dp[i][0] += dp[i - 1][0];
            dp[i][1] += dp[i - 1][1];
        }
    }
 
    // Count of all valid subsequences
    return (int)(dp[n][0] + dp[n][1]);
}
 
// Driver code
public static void main(String[] args)
{
    int arr[] = { 5, 6, 9, 7 };
    int n = arr.length;
         
    System.out.print(validsubsequences(arr, n));
}
}
 
// This code is contributed by code_hunt

Python3




# Python3 program to implement the
# above approach
 
# Function to find required subsequences
def validsubsequences(arr, n):
 
    # dp[i][0]: Stores the number of
    # subsequences till i-th index
    # ending with even element
    # dp[i][1]: Stores the number of
    # subsequences till i-th index
    # ending with odd element
 
    # Initialise the dp[][] with 0.
    dp = [[0 for i in range(2)]
             for j in range(n + 1)]
              
    for i in range(1, n + 1):
 
        # If odd element is
        # encountered
        if(arr[i - 1] % 2):
 
            # Considering i-th element
            # will be present in
            # the subsequence
            dp[i][1] += 1
 
            # Appending i-th element to all
            # non-empty subsequences
            # ending with even element
            # till (i-1)th indexes
            dp[i][1] += dp[i - 1][0]
 
            # Considering ith element will
            # not be present in
            # the subsequence
            dp[i][1] += dp[i - 1][1]
            dp[i][0] += dp[i - 1][0]
 
        else:
 
            # Considering i-th element
            # will be present in
            # the subsequence
            dp[i][0] += 1
 
            # Appending i-th element to all
            # non-empty subsequences
            # ending with odd element
            # till (i-1)th indexes
            dp[i][0] += dp[i - 1][1]
 
            # Considering ith element will
            # not be present in
            # the subsequence
            dp[i][0] += dp[i - 1][0]
            dp[i][1] += dp[i - 1][1]
 
    # Count of all valid subsequences
    return dp[n][0] + dp[n][1]
 
# Driver code
if __name__ == '__main__':
 
    arr = [ 5, 6, 9, 7 ]
    n = len(arr)
 
    print(validsubsequences(arr, n))
 
# This code is contributed by Shivam Singh

C#




// C# program implementation
// of the approach
using System;
 
class GFG{
 
// Function to find required subsequences    
static int validsubsequences(int[] arr, int n)
{
     
    // dp[i][0]: Stores the number of
    // subsequences till i-th index
    // ending with even element
    // dp[i][1]: Stores the number of
    // subsequences till i-th index
    // ending with odd element
    long[,] dp = new long [n + 1, 2];
 
    // Initialise the dp[][] with 0.
    for(int i = 0; i < n + 1; i++)
    {
        dp[i, 0] = 0;
        dp[i, 1] = 0;
    }
     
    for(int i = 1; i <= n; i++)
    {
         
        // If odd element is
        // encountered
        if (arr[i - 1] % 2 != 0)
        {
 
            // Considering i-th element
            // will be present in
            // the subsequence
            dp[i, 1] += 1;
 
            // Appending i-th element to all
            // non-empty subsequences
            // ending with even element
            // till (i-1)th indexes
            dp[i, 1] += dp[i - 1, 0];
 
            // Considering ith element will
            // not be present in
            // the subsequence
            dp[i, 1] += dp[i - 1, 1];
            dp[i, 0] += dp[i - 1, 0];
        }
        else
        {
 
            // Considering i-th element
            // will be present in
            // the subsequence
            dp[i, 0] += 1;
 
            // Appending i-th element to all
            // non-empty subsequences
            // ending with odd element
            // till (i-1)th indexes
            dp[i, 0] += dp[i - 1, 1];
 
            // Considering ith element will
            // not be present in
            // the subsequence
            dp[i, 0] += dp[i - 1, 0];
            dp[i, 1] += dp[i - 1, 1];
        }
    }
 
    // Count of all valid subsequences
    return (int)(dp[n, 0] + dp[n, 1]);
}
 
// Driver code
public static void Main()
{
    int[] arr = { 5, 6, 9, 7 };
    int n = arr.Length;
         
    Console.Write(validsubsequences(arr, n));
}
}
 
// This code is contributed by chitranayal

Javascript




<script>
 
// Javascript program for the above approach
  
// Function to find required subsequences    
function validsubsequences(arr, n)
{
       
    // dp[i][0]: Stores the number of
    // subsequences till i-th index
    // ending with even element
    // dp[i][1]: Stores the number of
    // subsequences till i-th index
    // ending with odd element
    let dp = new Array(n + 1);
    for (var i = 0; i < dp.length; i++) {
    dp[i] = new Array(2);
    }
   
    // Initialise the dp[][] with 0.
    for(let i = 0; i < n + 1; i++)
    {
        dp[i][0] = 0;
        dp[i][1] = 0;
    }
       
    for(let i = 1; i <= n; i++)
    {
           
        // If odd element is
        // encountered
        if (arr[i - 1] % 2 != 0)
        {
   
            // Considering i-th element
            // will be present in
            // the subsequence
            dp[i][1] += 1;
   
            // Appending i-th element to all
            // non-empty subsequences
            // ending with even element
            // till (i-1)th indexes
            dp[i][1] += dp[i - 1][0];
   
            // Considering ith element will
            // not be present in
            // the subsequence
            dp[i][1] += dp[i - 1][1];
            dp[i][0] += dp[i - 1][0];
        }
        else
        {
   
            // Considering i-th element
            // will be present in
            // the subsequence
            dp[i][0] += 1;
   
            // Appending i-th element to all
            // non-empty subsequences
            // ending with odd element
            // till (i-1)th indexes
            dp[i][0] += dp[i - 1][1];
   
            // Considering ith element will
            // not be present in
            // the subsequence
            dp[i][0] += dp[i - 1][0];
            dp[i][1] += dp[i - 1][1];
        }
    }
   
    // Count of all valid subsequences
    return (dp[n][0] + dp[n][1]);
}
 
// Driver Code
 
    let arr = [ 5, 6, 9, 7 ];
    let n = arr.length;
           
    document.write(validsubsequences(arr, n));
 
</script>
Output: 
9

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

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