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Find number of subarrays with even sum

  • Difficulty Level : Hard
  • Last Updated : 20 Apr, 2021

Given an array, find the number of subarrays whose sum is even.

Example : 

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Input : arr[] = {1, 2, 2, 3, 4, 1} 
Output : 9

There are possible subarrays with even
sum. The subarrays are 
1) {1, 2, 2, 3}  Sum = 8
2) {1, 2, 2, 3, 4}  Sum = 12
3) {2}  Sum = 2 (At index 1)
4) {2, 2}  Sum = 4
5) {2, 2, 3, 4, 1}  Sum = 12
6) {2}  Sum = 2 (At index 2)
7) {2, 3, 4, 1} Sum = 10
8) {3, 4, 1}  Sum = 8
9) {4}  Sum = 4 

O(n2) time and O(1) space method [Brute Force] 
We can simply generate all the possible sub-arrays and find whether the sum of all the elements in them is an even or not. If it is even then we will count that sub-array otherwise neglect it.
 



C++




/* C++ program to count number of sub-arrays
  whose sum is even using brute force
 Time Complexity - O(N^2)
 Space Complexity - O(1) */
#include<iostream>
using namespace std;
 
int countEvenSum(int arr[], int n)
{
    int result = 0;
 
    // Find sum of all subarrays and increment
    // result if sum is even
    for (int i=0; i<=n-1; i++)
    {
        int sum = 0;
        for (int j=i; j<=n-1; j++)
        {
            sum = sum + arr[j];
            if (sum % 2 == 0)
                    result++;
        }
    }
 
    return (result);
}
 
// Driver code
int main()
{
    int arr[] = {1, 2, 2, 3, 4, 1};
    int n = sizeof (arr) / sizeof (arr[0]);
 
    cout << "The Number of Subarrays with even"
            " sum is " << countEvenSum (arr, n);
 
    return (0);
}

Java




// Java program to count number
// of sub-arrays whose sum is
// even using brute force
// Time Complexity - O(N^2)
// Space Complexity - O(1)
import java.io.*;
 
class GFG
{
static int countEvenSum(int arr[],
                        int n)
{
    int result = 0;
 
    // Find sum of all subarrays
    // and increment result if
    // sum is even
    for (int i = 0; i <= n - 1; i++)
    {
        int sum = 0;
        for (int j = i; j <= n - 1; j++)
        {
            sum = sum + arr[j];
            if (sum % 2 == 0)
                    result++;
        }
    }
 
    return (result);
}
 
// Driver code
public static void main (String[] args)
{
int arr[] = {1, 2, 2,
             3, 4, 1};
int n = arr.length;
 
System.out.print("The Number of Subarrays"+
                     " with even sum is ");
                      
System.out.println(countEvenSum(arr, n));
}
}
 
// This code is contributed by ajit

Python3




# Python 3 program to count number
# of sub-arrays whose sum is even
# using brute force
# Time Complexity - O(N^2)
# Space Complexity - O(1)
 
def countEvenSum(arr, n):
    result = 0
 
    # Find sum of all subarrays and
    # increment result if sum is even
    for i in range(0, n, 1):
        sum = 0
        for j in range(i, n, 1):
            sum = sum + arr[j]
            if (sum % 2 == 0):
                    result = result + 1
 
    return (result)
 
# Driver code
if __name__ == '__main__':
    arr = [1, 2, 2, 3, 4, 1]
    n = len(arr)
    print("The Number of Subarrays" ,
                  "with even sum is",
               countEvenSum (arr, n))
 
# This code is contributed by
# Surendra_Gangwar

C#




// C# program to count number
// of sub-arrays whose sum is
// even using brute force
// Time Complexity - O(N^2)
// Space Complexity - O(1)
using System;
 
class GFG
{
static int countEvenSum(int []arr,
                        int n)
{
    int result = 0;
 
    // Find sum of all subarrays
    // and increment result if
    // sum is even
    for (int i = 0; i <= n - 1; i++)
    {
        int sum = 0;
        for (int j = i; j <= n - 1; j++)
        {
            sum = sum + arr[j];
            if (sum % 2 == 0)
                    result++;
        }
    }
 
    return (result);
}
 
// Driver code
static public void Main ()
{
    int []arr = {1, 2, 2,
                 3, 4, 1};
    int n = arr.Length;
 
    Console.Write("The Number of Subarrays"+
                      " with even sum is ");
                     
    Console.WriteLine(countEvenSum(arr, n));
    }
}
 
// This code is contributed by m_kit

PHP




<?php
// PHP program to count number
// of sub-arrays whose sum is
// even using brute force
// Time Complexity - O(N^2)
// Space Complexity - O(1)
function countEvenSum($arr, $n)
{
    $result = 0;
 
    // Find sum of all subarrays
    // and increment result if
    // sum is even
    for ($i = 0; $i <= $n - 1; $i++)
    {
        $sum = 0;
        for ($j = $i; $j <= $n - 1; $j++)
        {
            $sum = $sum + $arr[$j];
            if ($sum % 2 == 0)
                    $result++;
        }
    }
 
    return ($result);
}
 
// Driver code
$arr = array(1, 2, 2, 3, 4, 1);
$n = sizeof ($arr);
 
echo "The Number of Subarrays ",
           "with even sum is " ,
        countEvenSum ($arr, $n);
 
// This code is contributed by ajit
?>

Javascript




<script>
 
// Javascript program to count number
// of sub-arrays whose sum is
// even using brute force
// Time Complexity - O(N^2)
// Space Complexity - O(1)
 
function countEvenSum(arr,
                        n)
{
    let result = 0;
  
    // Find sum of all subarrays
    // and increment result if
    // sum is even
    for (let i = 0; i <= n - 1; i++)
    {
        let sum = 0;
        for (let j = i; j <= n - 1; j++)
        {
            sum = sum + arr[j];
            if (sum % 2 == 0)
                    result++;
        }
    }
  
    return (result);
}
 
// Driver Code
 
let arr = [1, 2, 2,
             3, 4, 1];
let n = arr.length;
  
document.write("The Number of Subarrays"+
                     " with even sum is ");
                       
document.write(countEvenSum(arr, n));
 
</script>
Output
The Number of Subarrays with even sum is 9

  
O(n) Time and O(1) Space Method [Efficient] 
If we do compute the cumulative sum array in temp[] of our input array, then we can see that the sub-array starting from i and ending at j, has an even sum if temp[] if (temp[j] – temp[i]) % 2 = 0. So, instead of building a cumulative sum array we build a cumulative sum modulo 2 array, and find how many times 0 and 1 appears in temp[] array using handshake formula. [n * (n-1) /2]

C++




/* C++ program to count number of sub-arrays
with even sum using an efficient algorithm
Time Complexity - O(N)
Space Complexity - O(1)*/
#include<iostream>
using namespace std;
 
int countEvenSum(int arr[], int n)
{
    // A temporary array of size 2. temp[0] is
    // going to store count of even subarrays
    // and temp[1] count of odd.
    // temp[0] is initialized as 1 because there
    // a single even element is also counted as
    // a subarray
    int temp[2] = {1, 0};
 
    // Initialize count.  sum is sum of elements
    // under modulo 2 and ending with arr[i].
    int result = 0, sum = 0;
 
    // i'th iteration computes sum of arr[0..i]
    // under modulo 2 and increments even/odd count
    // according to sum's value
    for (int i=0; i<=n-1; i++)
    {
        // 2 is added to handle negative numbers
        sum = ( (sum + arr[i]) % 2 + 2) % 2;
 
        // Increment even/odd count
        temp[sum]++;
    }
 
    // Use handshake lemma to count even subarrays
    // (Note that an even cam be formed by two even
    // or two odd)
    result = result + (temp[0]*(temp[0]-1)/2);
    result = result + (temp[1]*(temp[1]-1)/2);
 
    return (result);
}
 
// Driver code
int main()
{
    int arr[] = {1, 2, 2, 3, 4, 1};
    int n = sizeof (arr) / sizeof (arr[0]);
 
    cout << "The Number of Subarrays with even"
            " sum is " << countEvenSum (arr, n);
 
    return (0);
}

Java




// Java program to count
// number of sub-arrays
// with even sum using an
// efficient algorithm
// Time Complexity - O(N)
// Space Complexity - O(1)
import java.io.*;
 
class GFG
{
static int countEvenSum(int arr[], 
                        int n)
{
    // A temporary array of size 2.
    // temp[0] is going to store
    // count of even subarrays
    // and temp[1] count of odd.
    // temp[0] is initialized as
    // 1 because there a single even
    // element is also counted as
    // a subarray
    int temp[] = {1, 0};
 
    // Initialize count. sum is
    // sum of elements under modulo
    // 2 and ending with arr[i].
    int result = 0, sum = 0;
 
    // i'th iteration computes sum
    // of arr[0..i] under modulo 2
    // and increments even/odd count
    // according to sum's value
    for (int i = 0; i <= n - 1; i++)
    {
        // 2 is added to handle
        // negative numbers
        sum = ((sum + arr[i]) %
                    2 + 2) % 2;
 
        // Increment even/odd count
        temp[sum]++;
    }
 
    // Use handshake lemma to
    // count even subarrays
    // (Note that an even cam
    // be formed by two even
    // or two odd)
    result = result + (temp[0] *
                      (temp[0] - 1) / 2);
    result = result + (temp[1] *
                      (temp[1] - 1) / 2);
 
    return (result);
}
 
// Driver code
public static void main (String[] args)
{
 
int arr[] = {1, 2, 2, 3, 4, 1};
int n = arr.length;
 
System.out.println("The Number of Subarrays"+
                       " with even sum is " +
                      countEvenSum (arr, n));
}
}
 
// This code is contributed by ajit

Python 3




# Python 3 program to count number of sub-arrays
# with even sum using an efficient algorithm
# Time Complexity - O(N)
# Space Complexity - O(1)
def countEvenSum(arr, n):
 
    # A temporary array of size 2. temp[0] is
    # going to store count of even subarrays
    # and temp[1] count of odd.
    # temp[0] is initialized as 1 because there
    # a single even element is also counted as
    # a subarray
    temp = [1, 0]
 
    # Initialize count. sum is sum of elements
    # under modulo 2 and ending with arr[i].
    result = 0
    sum = 0
 
    # i'th iteration computes sum of arr[0..i]
    # under modulo 2 and increments even/odd
    # count according to sum's value
    for i in range( n):
         
        # 2 is added to handle negative numbers
        sum = ( (sum + arr[i]) % 2 + 2) % 2
 
        # Increment even/odd count
        temp[sum]+= 1
 
    # Use handshake lemma to count even subarrays
    # (Note that an even cam be formed by two even
    # or two odd)
    result = result + (temp[0] * (temp[0] - 1) // 2)
    result = result + (temp[1] * (temp[1] - 1) // 2)
 
    return (result)
 
# Driver code
if __name__ == "__main__":
     
    arr = [1, 2, 2, 3, 4, 1]
    n = len(arr)
    print( "The Number of Subarrays with even"
            " sum is" , countEvenSum (arr, n))
 
# This code is contributed by ita_c

C#




// C# program to count
// number of sub-arrays
// with even sum using an
// efficient algorithm
// Time Complexity - O(N)
// Space Complexity - O(1)
using System;
 
class GFG
{
static int countEvenSum(int []arr,
                        int n)
{
    // A temporary array of size 2.
    // temp[0] is going to store
    // count of even subarrays
    // and temp[1] count of odd.
    // temp[0] is initialized as
    // 1 because there a single even
    // element is also counted as
    // a subarray
    int []temp = {1, 0};
 
    // Initialize count. sum is
    // sum of elements under modulo
    // 2 and ending with arr[i].
    int result = 0, sum = 0;
 
    // i'th iteration computes sum
    // of arr[0..i] under modulo 2
    // and increments even/odd count
    // according to sum's value
    for (int i = 0; i <= n - 1; i++)
    {
        // 2 is added to handle
        // negative numbers
        sum = ((sum + arr[i]) %
                    2 + 2) % 2;
 
        // Increment even
        // or odd count
        temp[sum]++;
    }
 
    // Use handshake lemma to
    // count even subarrays
    // (Note that an even cam
    // be formed by two even
    // or two odd)
    result = result + (temp[0] *
                      (temp[0] - 1) / 2);
    result = result + (temp[1] *
                      (temp[1] - 1) / 2);
 
    return (result);
}
 
// Driver code
static public void Main ()
{
    int []arr = {1, 2, 2, 3, 4, 1};
    int n = arr.Length;
 
    Console.WriteLine("The Number of Subarrays"+
                          " with even sum is " +
                         countEvenSum (arr, n));
}
}
 
// This code is contributed
// by akt_mit

PHP




<?php
// PHP program to count number
// of sub-arrays with even sum
// using an efficient algorithm
// Time Complexity - O(N)
// Space Complexity - O(1)*/
function countEvenSum($arr, $n)
{
    // A temporary array of size 2.
    // temp[0] is going to store
    // count of even subarrays and
    // temp[1] count of odd. temp[0]
    // is initialized as 1 because
    // there a single even element
    // is also counted as a subarray
    $temp = array(1, 0);
 
    // Initialize count. sum is
    // sum of elements under
    // modulo 2 and ending with arr[i].
    $result = 0; $sum = 0;
 
    // i'th iteration computes
    // sum of arr[0..i] under
    // modulo 2 and increments
    // even/odd count according
    // to sum's value
    for ($i = 0; $i <= $n - 1; $i++)
    {
        // 2 is added to handle
        // negative numbers
        $sum = (($sum + $arr[$i]) %
                        2 + 2) % 2;
 
        // Increment even/odd
        // count
        $temp[$sum]++;
    }
 
    // Use handshake lemma to
    // count even subarrays
    // (Note that an even can
    // be formed by two even
    // or two odd)
    $result = $result + (int)($temp[0] *
                             ($temp[0] - 1) / 2);
    $result = $result + (int)($temp[1] *
                             ($temp[1] - 1) / 2);
 
    return ($result);
}
 
// Driver code
$arr = array (1, 2, 2,
              3, 4, 1);
$n = sizeof ($arr);
 
echo "The Number of Subarrays " .
        "with even", " sum is " ,
         countEvenSum ($arr, $n);
 
// This code is contributed by ajit
?>

Javascript




<script>
// Javascript program to count
// number of sub-arrays
// with even sum using an
// efficient algorithm
// Time Complexity - O(N)
// Space Complexity - O(1)
    function countEvenSum(arr,n)
    {
     
        // A temporary array of size 2.
        // temp[0] is going to store
        // count of even subarrays
        // and temp[1] count of odd.
        // temp[0] is initialized as
        // 1 because there a single even
        // element is also counted as
        // a subarray
        let temp = [1, 0];
      
        // Initialize count. sum is
        // sum of elements under modulo
        // 2 and ending with arr[i].
        let result = 0, sum = 0;
      
        // i'th iteration computes sum
        // of arr[0..i] under modulo 2
        // and increments even/odd count
        // according to sum's value
        for (let i = 0; i <= n - 1; i++)
        {
         
            // 2 is added to handle
            // negative numbers
            sum = ((sum + arr[i]) %
                        2 + 2) % 2;
      
            // Increment even/odd count
            temp[sum]++;
        }
      
        // Use handshake lemma to
        // count even subarrays
        // (Note that an even cam
        // be formed by two even
        // or two odd)
        result = result + (temp[0] *
                          (temp[0] - 1) / 2);
        result = result + (temp[1] *
                          (temp[1] - 1) / 2);
      
        return (result);
    }
     
    // Driver code
    let arr=[1, 2, 2, 3, 4, 1];
    let n = arr.length;
    document.write("The Number of Subarrays"+
                       " with even sum is " +
                      countEvenSum (arr, n));
     
    // This code is contributed by rag2127
</script>

Output : 

The Number of Subarrays with even sum is 9

O(n) Time and O(1) Space Method (bottom-up-approach)

If we start counting from last index and keep track of number of subarrays with even sum so far starting from present index then we can calculate number of subarrays with even sum starting from previous index

C++




/* C++ program to count number of sub-arrays
with even sum using an efficient algorithm
Time Complexity - O(N)
Space Complexity - O(1)*/
#include <iostream>
using namespace std;
 
long long countEvenSum(int a[], int n)
{
    // Result may be large enough not to
    // fit in int;
    long long res = 0;
   
    // To keep track of subarrays with even sum
    // starting from index i;
    int s = 0;
    for (int i = n - 1; i >= 0; i--)
    {
        if (a[i] % 2 == 1)
        {
            /* s is the count of subarrays starting from
             * index i+1 whose sum was even*/
            /*
            If a[i] is odd then all subarrays starting from
            index i+1 which was odd becomes even when a[i]
            gets added to it.
            */
            s = n - i - 1 - s;
        }
        else
        {
            /*
            If a[i] is even then all subarrays starting from
            index i+1 which was even remains even and one
            extra a[i] even subarray gets added to it.
            */
            s = s + 1;
        }
        res = res + s;
    }
    return res;
}
 
// Driver Code
int main()
{
 
    int arr[] = { 1, 2, 2, 3, 4, 1 };
    int n = sizeof(arr) / sizeof(arr[0]);
 
    cout << "The Number of Subarrays with even"
            " sum is "
         << countEvenSum(arr, n);
    return 0;
}
 
// This code is contributed by Aditya Anand

Java




// Java program to count
// number of sub-arrays
// with even sum using an
// efficient algorithm
// Time Complexity - O(N)
// Space Complexity - O(1)
import java.io.*;
 
class GFG {
 
    public static long countEvenSum(int a[], int n)
    {
        // result may be large enough not to
        // fit in int;
        long res = 0;
         
        // to keep track of subarrays with even
        // sum starting from index i
        int s = 0;
        for (int i = n - 1; i >= 0; i--)
        {
            if (a[i] % 2 == 1)
            {
                // s is the count of subarrays starting from
                // index i+1 whose sum was even
                /*if a[i] is odd then all subarrays starting
                from index i+1 which was odd becomeseven
                when a[i] gets added to it.*/
                s = n - i - 1 - s;
            }
            else
            {
                /*if a[i] is even then all subarrays
        starting from index i+1 which was even remainseven
        and one extra a[i] even subarray gets added to it.*/
                s = s + 1;
            }
            res = res + s;
        }
        return res;
    }
     
    // Driver Code
    public static void main(String[] args)
    {
        int arr[] = { 1, 2, 2, 3, 4, 1 };
        int n = arr.length;
 
        System.out.println("The Number of Subarrays"
                           + " with even sum is "
                           + countEvenSum(arr, n));
    }
}
 
// This code is contributed by Aditya Anand

Python3




# Python 3 program to count number of sub-arrays
# with even sum using an efficient algorithm
# Time Complexity - O(N)
# Space Complexity - O(1)
 
 
def countEvenSum(arr, n):
   
    # result may be large
    # enough not to fit in int;
    res = 0
     
    # to keep track of subarrays
    # with even sum starting from index i
    s = 0 
    for i in reversed(range(n)):
        if arr[i] % 2 == 1:
            # s is the count of subarrays
            # starting from index i+1
            # whose sum was even
            """
            if a[i] is odd then all subarrays
            starting from index i+1 which was
            odd becomes even when a[i] gets
            added to it.
            """
            s = n-i-1-s
        else:
           
            """
            if a[i] is even then all subarrays
            starting from index i+1 which was
            even remains even and one extra a[i]
            even subarray gets added to it.
            """
            s = s+1
        res = res + s
    return res
 
 
# Driver code
if __name__ == "__main__":
 
    arr = [1, 2, 2, 3, 4, 1]
    n = len(arr)
    print("The Number of Subarrays with even"
          " sum is", countEvenSum(arr, n))
 
# This code is contributed by Aditya Anand

C#




// C# program to count
// number of sub-arrays
// with even sum using an
// efficient algorithm
// Time Complexity - O(N)
// Space Complexity - O(1)
using System;
public class GFG
{
  public static long countEvenSum(int[] a, int n)
  {
 
    // result may be large enough not to
    // fit in int;
    long res = 0;
 
    // to keep track of subarrays with even
    // sum starting from index i
    int s = 0;
    for (int i = n - 1; i >= 0; i--)
    {
      if (a[i] % 2 == 1)
      {
 
        // s is the count of subarrays starting from
        // index i+1 whose sum was even
        /*if a[i] is odd then all subarrays starting
                from index i+1 which was odd becomeseven
                when a[i] gets added to it.*/
        s = n - i - 1 - s;
      }
      else
      {
 
        /*if a[i] is even then all subarrays
        starting from index i+1 which was even remainseven
        and one extra a[i] even subarray gets added to it.*/
        s = s + 1;
      }
      res = res + s;
    }
    return res;
  }
 
  // Driver Code
  static public void Main ()
  {
    int[] arr = { 1, 2, 2, 3, 4, 1 };
    int n = arr.Length;
 
    Console.WriteLine("The Number of Subarrays"
                      + " with even sum is "
                      + countEvenSum(arr, n));
  }
}
 
// This code is contributed by avanitrachhadiya2155

Javascript




<script>
 
    // Javascript program to count
    // number of sub-arrays
    // with even sum using an
    // efficient algorithm
    // Time Complexity - O(N)
    // Space Complexity - O(1)
     
    function countEvenSum(a, n)
    {
 
      // result may be large enough not to
      // fit in int;
      let res = 0;
 
      // to keep track of subarrays with even
      // sum starting from index i
      let s = 0;
      for (let i = n - 1; i >= 0; i--)
      {
        if (a[i] % 2 == 1)
        {
 
          // s is the count of subarrays starting from
          // index i+1 whose sum was even
          /*if a[i] is odd then all subarrays starting
            from index i+1 which was odd becomeseven
            when a[i] gets added to it.*/
          s = n - i - 1 - s;
        }
        else
        {
 
          /*if a[i] is even then all subarrays
          starting from index i+1 which was even remainseven
          and one extra a[i] even subarray gets added to it.*/
          s = s + 1;
        }
        res = res + s;
      }
      return res;
    }
       
    let arr = [ 1, 2, 2, 3, 4, 1 ];
    let n = arr.length;
  
    document.write("The Number of Subarrays" +
    " with even sum is " + countEvenSum(arr, n));
     
</script>
Output
The Number of Subarrays with even sum is 9

This article is contributed by Rachit Belwariar. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
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