Sum of all substrings of a string representing a number

Given a integer represented as a string, we need to get the sum of all possible substrings of this string.

Examples:

Input  : num = “1234”
Output : 1670
Sum = 1 + 2 + 3 + 4 + 12 + 23 +
       34 + 123 + 234 + 1234 
    = 1670

Input  : num = “421”
Output : 491
Sum = 4 + 2 + 1 + 42 + 21 + 421 = 491

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

We can solve this problem using dynamic programming. We can write summation of all substrings on basis of digit at which they are ending in that case,
Sum of all substrings = sumofdigit[0] + sumofdigit[1] + sumofdigit[2] … + sumofdigit[n-1] where n is length of string.

Where sumofdigit[i] stores sum of all substring ending at ith index digit, in above example,

Example : num = "1234"
sumofdigit[0] = 1 = 1
sumofdigit[1] = 2 + 12  = 14
sumofdigit[2] = 3 + 23  + 123 = 149
sumofdigit[3] = 4 + 34  + 234 + 1234  = 1506
Result = 1670

Now we can get the relation between sumofdigit values and can solve the question iteratively. Each sumofdigit can be represented in terms of previous value as shown below,

For above example,
sumofdigit[3] = 4 + 34 + 234 + 1234
           = 4 + 30 + 4 + 230 + 4 + 1230 + 4
           = 4*4 + 10*(3 + 23 +123)
           = 4*4 + 10*(sumofdigit[2])
In general, 
sumofdigit[i]  =  (i+1)*num[i] + 10*sumofdigit[i-1]

Using above relation we can solve the problem in linear time. In below code a complete array is taken to store sumofdigit, as each sumofdigit value requires just previous value, we can solve this problem without allocating complete array also.

C++

//  C++ program to print sum of all substring of 
// a number represented as a string 
#include <bits/stdc++.h> 
using namespace std; 
  
// Utility method to covert character digit to 
// integer digit 
int toDigit(char ch) 
{ 
    return (ch - '0'); 
} 
  
// Returns sum of all substring of num 
int sumOfSubstrings(string num) 
{ 
    int n = num.length(); 
  
    //  allocate memory equal to length of string 
    int sumofdigit[n]; 
  
    //  initialize first value with first digit 
    sumofdigit[0] = toDigit(num[0]); 
    int res = sumofdigit[0]; 
  
    //  loop over all digits of string 
    for (int i=1; i<n; i++) 
    { 
        int numi = toDigit(num[i]); 
  
        // update each sumofdigit from previous value 
        sumofdigit[i] = (i+1) * numi + 
                        10 * sumofdigit[i-1]; 
  
        // add current value to the result 
        res += sumofdigit[i]; 
    } 
  
    return res; 
} 
  
//  Driver code to test above methods 
int main() 
{ 
    string num = "1234"; 
    cout << sumOfSubstrings(num) << endl; 
    return 0; 
} 

Java

// Java program to print sum of all substring of 
// a number represented as a string 
import java.util.Arrays; 
  
class GFG{ 
  
    // Returns sum of all substring of num 
    public static int sumOfSubstrings(String num) 
    { 
        int n = num.length(); 
          
        //  allocate memory equal to length of string 
        int sumofdigit[] = new int[n]; 
       
        //  initialize first value with first digit 
        sumofdigit[0] = num.charAt(0)-'0'; 
        int res = sumofdigit[0]; 
       
        //  loop over all digits of string 
        for (int i = 1; i < n; i++) 
        { 
            int numi = num.charAt(i)-'0'; 
       
            // update each sumofdigit from previous value 
            sumofdigit[i] = (i+1) * numi + 
                            10 * sumofdigit[i-1]; 
       
            // add current value to the result 
            res += sumofdigit[i]; 
        } 
       
        return res; 
    } 
       
    //  Driver code to test above methods 
     public static void main(String[] args)  
    { 
        String num = "1234"; 
          
        System.out.println(sumOfSubstrings(num)); 
          
    } 
} 
// This code is contributed by Arnav Kr. Mandal. 

Python3

# Python program to print  
# sum of all substring of 
# a number represented as  
# a string 
  
# Returns sum of all  
# substring of num 
def sumOfSubstrings(num): 
    n = len(num) 
  
    # allocate memory equal  
    # to length of string 
    sumofdigit = [] 
  
    # initialize first value 
    # with first digit 
    sumofdigit.append(int(num[0])) 
    res = sumofdigit[0] 
  
    # loop over all 
    # digits of string 
    for i in range(1, n): 
        numi = int(num[i]) 
  
        # update each sumofdigit 
        # from previous value 
        sumofdigit.append((i + 1) * 
                        numi + 10 * 
                        sumofdigit[i - 1]) 
  
        # add current value 
        # to the result 
        res += sumofdigit[i] 
  
    return res 
  
# Driver code  
num = "1234"
print(sumOfSubstrings(num)) 
  
# This code is contributed  
# by Sanjit_Prasad 

C#

// C# program to print sum of  
// all substring of a number 
// represented as a string 
using System; 
  
class GFG{ 
  
    // Returns sum of all  
    // substring of num 
    public static int sumOfSubstrings(String num) 
    { 
        int n = num.Length; 
          
        // allocate memory equal to  
        // length of string 
        int []sumofdigit = new int[n]; 
      
        // initialize first value  
        // with first digit 
        sumofdigit[0] = num[0] - '0'; 
        int res = sumofdigit[0]; 
      
        // loop over all digits 
        // of string 
        for (int i = 1; i < n; i++) 
        { 
            int numi = num[i] - '0'; 
      
            // update each sumofdigit 
            // from previous value 
            sumofdigit[i] = (i + 1) * numi + 10  
                             * sumofdigit[i - 1]; 
      
            // add current value 
            // to the result 
            res += sumofdigit[i]; 
        } 
      
        return res; 
    } 
      
    // Driver code  
    public static void Main()  
    { 
        String num = "1234"; 
          
        Console.Write(sumOfSubstrings(num)); 
          
    } 
} 
  
// This code is contributed by Nitin Mittal. 

PHP

<?php 
// PHP program to print sum of all 
// substring of a number represented  
// as a string 
  
// Method to covert character 
// digit to integer digit 
function toDigit($ch) 
{ 
    return ($ch - '0'); 
} 
  
// Returns sum of all 
// substring of num 
function sumOfSubstrings($num) 
{ 
    $n = strlen($num); 
  
    // allocate memory equal to 
    // length of string 
    $sumofdigit[$n] = 0; 
  
    // initialize first value 
    // with first digit 
    $sumofdigit[0] = toDigit($num[0]); 
    $res = $sumofdigit[0]; 
  
    // loop over all digits of string 
    for($i = 1; $i < $n; $i++) 
    { 
        $numi = toDigit($num[$i]); 
  
        // update each sumofdigit  
        // from previous value 
        $sumofdigit[$i] = ($i + 1) * $numi + 10 *  
                              $sumofdigit[$i - 1]; 
  
        // add current value to the result 
        $res += $sumofdigit[$i]; 
    } 
  
    return $res; 
} 
  
    // Driver Code 
    $num = "1234"; 
    echo sumOfSubstrings($num) ; 
      
// This code is contributed by nitin mittal. 
?> 

Output:

Time Complexity : O(n) where n is length of input string.
Auxiliary Space : O(n)

Sum of all substrings of a string representing a number | Set 2 (Constant Extra Space)

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