Sum of Digits Problem using Dynamic Programming

This is a C++ Program that Solves Sum of Digits Problem using Dynamic Programming technique.

Problem Description

Given a number n, find the sum of digits of all numbers from 1 to n.

Problem Solution

To solve this problem in an efficient manner, we will calculate the sum in a pattern. If we are given a number 8459, we will calculate the result in this manner –
(8*999) + ((7*8)/2)*1000 + 8*459 + 459

Expected Input and Output

Case-1:

n=5
sum =15 (1+2+3+4+5)

Case-2:

n=9
sum=45

Case-3:

n=10
sum=46 (notice that the contribution of the number 10 in the sum is  just 1 not 10)

Program/Source Code

Here is source code of the C++ Program to Solve Sum of Digits Problem. The C++ program is successfully compiled and run on a Linux system. The program output is also shown below.

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#include<iostream>
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#include<cmath>
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using namespace std;
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//this function calculates the sum of digits of numbers from 1 to 10^digits -1

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//if digits=2, sum of 1 to 99
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int sumXminusOne(int digits)
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{
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    int sum=0;
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    for(int i=1;i<=digits;i++)
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    {
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        //45 is the sum of digits of numbers from 1...9

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        sum=sum*10 + 45*pow(10,i-1);
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    }
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    return sum;
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}
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```
 
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//function to return sum of digits of all numbers from 1 to n

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int sumDigits(int n)
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{
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    if(n<10)
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        return n*(n+1)/2;
```
```
 
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    //digits is equal to number of digits in n minus one

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    int digits;
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    digits=log10(n);
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    int X=pow(10,digits);
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    //leftmost digit of the number
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    int left=n/X;
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    //sum of digits of all numbers form 1 to n

    int sum=left*sumXminusOne(digits) +

```
            ((left*(left-1))/2)*X +
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            left*(n%X +1) +
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            sumDigits(n%X);
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    return sum;
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}
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int main()
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{
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    int n;
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    cout<<"Enter the number "<<endl;
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    cin>>n;
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    cout<<"The sum of digits of all numbers from 1 to "<<n<<" is "<<endl;

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    cout<<sumDigits(n);
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    cout<<endl;
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    return 0;
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}
```

Program Explanation

In the main function, we take input for the value n and pass this value to the function sumDigits. This function will return the sum of digits of all integers from 1 to n to be displayed on the standard output.

Runtime Test Cases

 
Case-1:
$ g++ sum_digits.cpp 
$ ./a.out
Enter the number 
45
The sum of digits of all numbers from 1 to 45 is 
279

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