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

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Given a string representing a number, we need to get the sum of all possible sub strings of this string.

Examples : 

Input: s = "6759"
Output: 8421
Explanation: sum = 6 + 7 + 5 + 9 + 67 + 75 +
59 + 675 + 759 + 6759
= 8421

Input: s = "16"
Output: 23
Explanation: sum = 1 + 6 + 16 = 23

Approach

The main idea is to analyze how each digit contributes to the sum of all substrings by observing its position and place value.

Let number be s = "6759".

1 10 100 1000
6 1 1 1 1
7 2 2 2
5 3 3
9 4

The above table indicates that, when all the substrings are converted further to the ones, tens, hundreds etc.. form, each index of the string will have some fixed occurrence. The 0'th index will have 1 occurrence each of ones, tens etc. The 1st will have 2, 2nd will have 3 and so on. Each digit at index i appears in exactly (i + 1) substrings ending at that digit. Additionally, its contribution is scaled by powers of 10 based on its position from the right.

So the total sum of all substrings can be expressed by evaluating how each digit contributes based on its position. For the string s = "6759", the sum becomes:

sum = 6*(1*1 + 1*10 + 1*100 + 1*1000) + 7*(2*1 + 2*10 + 2*100) +
5*(3*1 + 3*10) + 9*(4*1)
= 6*1*(1111) + 7*2*(111) + 5*3*(11) + 9*4*(1)
= 6666 + 1554 + 165 + 36
= 8421

Now, to handle the multiplication we will be having a multiplying factor which starts from 1. It's clear from the example that the multiplying factor(in reverse) is 1, 11, 111, ... and so on. So the multiplication will be based on three factors. number, its index, and a multiplying factor. 

// C++ program to print sum of all substring of
// a number represented as a string
#include <bits/stdc++.h>
using namespace std;

// Returns sum of all substring of num
int sumOfSubstrings(string s){
    
    int sum = 0; 
    
    // Here traversing the array in reverse
    // order.Initializing loop from last element.
    // mf is multiplying factor.
    int mf = 1;
    for (int i=s.size()-1; i>=0; i--){
        
        // Each time sum is added to its previous
        // sum. Multiplying the three factors as explained above.
        // s[i]-'0' is done to convert char to int.
        sum += (s[i]-'0')*(i+1)*mf;

        // Making new multiplying factor as explained above.
        mf = mf*10 + 1;
    }

    return sum;
}

//  Driver code
int main(){
    
    string s = "6759";
    cout << sumOfSubstrings(s) << endl;
    return 0;
}

// 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
    static int sumOfSubstrings(String s){
        
        // Initialize result
        int sum = 0; 
     
        // Here traversing the array in reverse
        // order.Initializing loop from last
        // element.
        // mf is multiplying factor.
        int mf = 1;
        for (int i = s.length() - 1; i >= 0; i --){
            
            // Each time sum is added to its previous
            // sum. Multiplying the three factors as
            // explained above.
            // s[i]-'0' is done to convert char to int.
            sum += (s.charAt(i) - '0') * (i + 1) * mf;
     
            // Making new multiplying factor as
            // explained above.
            mf = mf * 10 + 1;
        }
     
        return sum;
    }
    
         
    //  Driver Code
    public static void main(String[] args){
        
        String s = "6759";
            
        System.out.println(sumOfSubstrings(s));
            
    }
}

// This code is contributed by Arnav Kr. Mandal.

# Python3 program to print sum of all substring of
# a number represented as a string

# Returns sum of all substring of num
def sumOfSubstrings(s):
    
    # Initialize result
    sum = 0 

    # Here traversing the array in reverse
    # order.Initializing loop from last
    # element.
    # mf is multiplying factor.
    mf = 1
    for i in range(len(s) - 1, -1, -1):

        # Each time sum is added to its previous
        # sum. Multiplying the three factors as
        # explained above.
        # int(s[i]) is done to convert char to int.
        sum = sum + (int(s[i])) * (i + 1) * mf

        # Making new multiplying factor as
        # explained above.
        mf = mf * 10 + 1

    return sum

# Driver Code
if __name__=='__main__':
    s = "6759"
    print(sumOfSubstrings(s))

// 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
    static int sumOfSubstrings(string s){
        
        int sum = 0; 
        
        // Here traversing the array in reverse
        // order.Initializing loop from last
        // element.
        // mf is multiplying factor.
        int mf = 1;
        
        for (int i = s.Length - 1; i >= 0; i --){
            
            // Each time sum is added to its previous
            // sum. Multiplying the three factors as
            // explained above.
            // s[i]-'0' is done to convert char to int.
            sum += (s[i] - '0') * (i + 1) * mf;
    
            // Making new multiplying factor as
            // explained above.
            mf = mf * 10 + 1;
        }
    
        return sum;
    }
    
        
    // Driver Code
    public static void Main() {
        
        string s = "6759";
        Console.WriteLine(sumOfSubstrings(s));
        
    }
}

// This code is contributed by Sam007.

// Function returns sum of all substring of num
function sumOfSubstrings(s){

    let sum = 0;

    // Here traversing the array in reverse
    // order.Initializing loop from last
    // element.
    // mf is multiplying factor.
    let mf = 1;

    for (let i = s.length - 1; i >= 0; i--) {

        // Each time sum is added to its previous
        // sum. Multiplying the three factors as
        // explained above.
        // s[i]-'0' is done to convert char to int.
        sum += (s[i].charCodeAt() - "0".charCodeAt())
               * (i + 1) * mf;

        // Making new multiplying factor as
        // explained above.
        mf = mf * 10 + 1;
    }

    return sum;
}

// Driver Code
let s = "6759";
console.log(sumOfSubstrings(s));

8421

Time Complexity: O(n), where n is size of the s.
Auxiliary Space: O(1)