Sum of all Subarrays | Set 1

Given an integer array ‘arr[]’ of size n, find sum of all sub-arrays of given array.

Examples :

Input   : arr[] = {1, 2, 3}
Output  : 20
Explanation : {1} + {2} + {3} + {2 + 3} + 
              {1 + 2} + {1 + 2 + 3} = 20

Input  : arr[] = {1, 2, 3, 4}
Output : 50

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

Method 1 (Simple Solution)
A simple solution is to generate all sub-array and compute their sum.

Below is the implementation of above idea.

C++

// Simple C++ program to compute sum of 
// subarray elements 
#include<bits/stdc++.h> 
using namespace std; 
  
// Computes sum all sub-array 
long int SubArraySum(int arr[], int n) 
{ 
    long int result = 0,temp=0; 
  
    // Pick starting point 
    for (int i=0; i <n; i++) 
    { 
        // Pick ending point 
        temp=0; 
        for (int j=i; j<n; j++) 
        { 
            // sum subarray between current 
            // starting and ending points 
            temp+=arr[j]; 
            result += temp ; 
        } 
    } 
    return result ; 
} 
  
// driver program to test above function 
int main() 
{ 
    int arr[] = {1, 2, 3} ; 
    int n = sizeof(arr)/sizeof(arr[0]); 
    cout << "Sum of SubArray : "
          << SubArraySum(arr, n) << endl; 
    return 0; 
} 

Java

// Simple Java program to compute sum of 
// subarray elements 
class GFG { 
      
    // Computes sum all sub-array 
    public static long SubArraySum(int arr[], int n) 
    { 
        long result = 0,temp=0; 
       
        // Pick starting point 
        for (int i = 0; i < n; i ++) 
        { 
            // Pick ending point 
            temp=0; 
            for (int j = i; j < n; j ++) 
            { 
                // sum subarray between current 
                // starting and ending points 
                temp+=arr[j]; 
                result += temp ; 
            } 
        } 
        return result ; 
    } 
      
    /* Driver program to test above function */
    public static void main(String[] args)  
    { 
        int arr[] = {1, 2, 3} ; 
        int n = arr.length; 
        System.out.println("Sum of SubArray : " +  
                          SubArraySum(arr, n)); 
    } 
} 
// This code is contributed by Arnav Kr. Mandal. 

Python 3

# Python3 program to compute  
# sum of subarray elements 
  
# Computes sum all sub-array 
def SubArraySum(arr, n): 
    temp,result = 0,0
  
    # Pick starting point 
    for i in range(0, n): 
  
        # Pick ending point 
        temp=0; 
        for j in range(i, n): 
  
            # sum subarray between  
            # current starting and  
            # ending points 
            temp+=arr[j] 
            result += temp  
    return result  
  
# driver program  
arr = [1, 2, 3] 
n = len(arr) 
print ("Sum of SubArray :" 
       ,SubArraySum(arr, n)) 
  
# This code is contributed by  
# TheGameOf256. 

C#

// Simple C# program to compute sum of 
// subarray elements 
using System; 
  
class GFG { 
      
    // Computes sum all sub-array 
    public static long SubArraySum(int []arr, 
                                      int n) 
    { 
        long result = 0,temp=0; 
      
        // Pick starting point 
        for (int i = 0; i < n; i ++) 
        { 
            // Pick ending point 
            temp=0; 
            for (int j = i; j < n; j ++) 
            { 
                  
                // sum subarray between current 
                // starting and ending points 
                temp+=arr[j]; 
                result += temp ; 
            } 
        } 
        return result ; 
    } 
      
    // Driver Code 
    public static void Main()  
    { 
        int []arr = {1, 2, 3} ; 
        int n = arr.Length; 
        Console.Write("Sum of SubArray : " +  
                        SubArraySum(arr, n)); 
    } 
} 
  
// This code is contributed by nitin mittal 

PHP

<?php 
// Simple PHP program to  
// compute sum of subarray  
// elements Computes sum 
// all sub-array 
  
function SubArraySum($arr, $n) 
{ 
    $result = 0; 
    $temp=0; 
    // Pick starting point 
    for ($i = 0; $i < $n; $i++) 
    { 
        // Pick ending point 
        $temp=0; 
        for ($j = $i; $j < $n; $j++) 
        { 
            // sum subarray between  
            // current starting and 
            // ending points 
            $temp+=$arr[$j] 
            $result += $temp ; 
        } 
    } 
    return $result ; 
} 
  
// Driver Code 
$arr = array(1, 2, 3) ; 
$n = sizeof($arr); 
echo "Sum of SubArray : ", 
    SubArraySum($arr, $n),"\n"; 
      
// This code is contributed by ajit 
?> 

Output:

Sum of SubArray : 20

Time Complexity : O(n²)

Method 2 (efficient solution)
If we take a close look then we observe a pattern. Let take an example

arr[] = [1, 2, 3], n = 3
All subarrays :  [1], [1, 2], [1, 2, 3], 
                 [2], [2, 3], [3]
here first element 'arr[0]' appears 3 times    
     second element 'arr[1]' appears 4 times  
     third element 'arr[2]' appears 3 times

Every element arr[i] appears in two types of subsets:
i)  In subarrays beginning with arr[i]. There are 
    (n-i) such subsets. For example [2] appears
    in [2] and [2, 3].
ii) In (n-i)*i subarrays where this element is not
    first element. For example [2] appears in 
    [1, 2] and [1, 2, 3].

Total of above (i) and (ii) = (n-i) + (n-i)*i 
                            = (n-i)(i+1)
                                  
For arr[] = {1, 2, 3}, sum of subarrays is:
  arr[0] * ( 0 + 1 ) * ( 3 - 0 ) + 
  arr[1] * ( 1 + 1 ) * ( 3 - 1 ) +
  arr[2] * ( 2 + 1 ) * ( 3 - 2 ) 

= 1*3 + 2*4 + 3*3 
= 20

Below is the implementation of above idea.

C++

// Efficient C++ program to compute sum of 
// subarray elements 
#include<bits/stdc++.h> 
using namespace std; 
  
//function compute sum all sub-array 
long int SubArraySum( int arr[] , int n ) 
{ 
    long int result = 0; 
  
    // computing sum of subarray using formula 
    for (int i=0; i<n; i++) 
        result += (arr[i] * (i+1) * (n-i)); 
  
    // return all subarray sum 
    return result ; 
} 
  
// driver program to test above function 
int main() 
{ 
    int arr[] = {1, 2, 3} ; 
    int n = sizeof(arr)/sizeof(arr[0]); 
    cout << "Sum of SubArray : "
         << SubArraySum(arr, n) << endl; 
    return 0; 
} 

Java

// Efficient Java program to compute sum of 
// subarray elements 
class GFG { 
      
    //function compute sum all sub-array 
    public static long SubArraySum( int arr[] , int n ) 
    { 
        long result = 0; 
       
        // computing sum of subarray using formula 
        for (int i=0; i<n; i++) 
            result += (arr[i] * (i+1) * (n-i)); 
       
        // return all subarray sum 
        return result ; 
    } 
      
    /* Driver program to test above function */
    public static void main(String[] args)  
    { 
        int arr[] = {1, 2, 3} ; 
        int n = arr.length; 
        System.out.println("Sum of SubArray " +  
                           SubArraySum(arr, n)); 
    } 
} 
// This code is contributed by Arnav Kr. Mandal. 

Python 3

#Python3 code to compute  
# sum of subarray elements 
  
#function compute sum  
# all sub-array 
def SubArraySum(arr, n ): 
    result = 0
  
    # computing sum of subarray  
    # using formula 
    for i in range(0, n): 
        result += (arr[i] * (i+1) * (n-i)) 
  
    # return all subarray sum 
    return result  
  
# driver program  
arr = [1, 2, 3] 
n = len(arr) 
print ("Sum of SubArray : ",  
      SubArraySum(arr, n)) 
  
# This code is contributed by  
# TheGameOf256. 

C#

// Efficient C# program 
// to compute sum of 
// subarray elements 
using System; 
  
class GFG 
{ 
          
    // function compute  
    // sum all sub-array 
    public static long SubArraySum(int []arr ,  
                                   int n ) 
    { 
        long result = 0; 
      
        // computing sum of  
        // subarray using formula 
        for (int i = 0; i < n; i++) 
            result += (arr[i] *  
                      (i + 1) * (n - i)); 
      
        // return all subarray sum 
        return result ; 
    } 
      
    // Driver code 
    static public void Main () 
    { 
        int []arr = {1, 2, 3} ; 
        int n = arr.Length; 
        Console.WriteLine("Sum of SubArray: " +  
                          SubArraySum(arr, n)); 
    } 
} 
  
// This code is contributed by akt_mit 

PHP

<?php 
// Efficient PHP program to compute  
// sum of subarray elements 
  
//function compute sum all sub-array 
function SubArraySum($arr , $n) 
{ 
    $result = 0; 
  
    // computing sum of subarray 
    // using formula 
    for ($i = 0; $i < $n; $i++) 
        $result += ($arr[$i] *  
                    ($i + 1) *  
                    ($n - $i)); 
  
    // return all subarray sum 
    return $result ; 
} 
  
// Driver Code 
$arr = array(1, 2, 3) ; 
$n = sizeof($arr); 
echo "Sum of SubArray : ", 
      SubArraySum($arr, $n) ,"\n"; 
  
  
#This code is contributed by aj_36 
?> 

Output :

Sum of SubArray : 20

Time complexity : O(n)

Reference :
https://www.quora.com/Can-we-find-the-sum-of-all-sub-arrays-of-an-integer-array-in-O-n-time

Sum of all subsequences of an array

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My Personal Notes

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

C++

Java

Python 3

C#

PHP

C++

Java

Python 3

C#

PHP

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