# Difference of two large numbers

Given two numbers as strings. The numbers may be very large (may not fit in long long int), the task is to find difference of these two numbers.

Examples:

```Input : str1 = "11443333311111111100",
str2 =     "1144422222221111"
Output : 11442188888888889989

Input :str1 = "122387876566565674",
str2 =     "31435454654554"
Output : 122356441111911120```

This is simple based on school mathematics. We traverse both strings from end, one by one subtract digits.
1) Reverse both strings.
2) Keep subtracting digits one by one from 0’th index (in reversed strings) to end of smaller string, append the diff if it’s positive to end of result. If difference(diff) is negative then add 10 and keep track of carry as 1 if it’s positive then carry is 0.
3) Finally reverse the result.

`// C++ program to find difference of two large numbers.`
`#include<bits/stdc++.h>`
`using` `namespace` `std;`
`// Returns true if str1 is smaller than str2.`
`bool` `isSmaller(string str1, string str2)`
`{`
`    ``// Calculate lengths of both string`
`    ``int` `n1 = str1.length(), n2 = str2.length();`
`    ``if` `(n1 < n2)`
`       ``return` `true``;`
`    ``if` `(n2 < n1)`
`       ``return` `false``;`
`    ``for` `(``int` `i=0; i<n1; i++)`
`       ``if` `(str1[i] < str2[i])`
`          ``return` `true``;`
`       ``else` `if` `(str1[i] > str2[i])`
`          ``return` `false``;`
`    ``return` `false``;`
`}`
`// Function for find difference of larger numbers`
`string findDiff(string str1, string str2)`
`{`
`    ``// Before proceeding further, make sure str1`
`    ``// is not smaller`
`    ``if` `(isSmaller(str1, str2))`
`        ``swap(str1, str2);`
`    ``// Take an empty string for storing result`
`    ``string str = ``""``;`
`    ``// Calculate length of both string`
`    ``int` `n1 = str1.length(), n2 = str2.length();`
`    ``// Reverse both of strings`
`    ``reverse(str1.begin(), str1.end());`
`    ``reverse(str2.begin(), str2.end());`
`    ``int` `carry = 0;`
`    ``// Run loop till small string length`
`    ``// and subtract digit of str1 to str2`
`    ``for` `(``int` `i=0; i<n2; i++)`
`    ``{`
`        ``// Do school mathematics, compute difference of`
`        ``// current digits`
`        ``int` `sub = ((str1[i]-``'0'``)-(str2[i]-``'0'``)-carry);`
`        ``// If subtraction is less then zero`
`        ``// we add then we add 10 into sub and`
`        ``// take carry as 1 for calculating next step`
`        ``if` `(sub < 0)`
`        ``{`
`            ``sub = sub + 10;`
`            ``carry = 1;`
`        ``}`
`        ``else`
`            ``carry = 0;`
`        ``str.push_back(sub + ``'0'``);`
`    ``}`
`    ``// subtract remaining digits of larger number`
`    ``for` `(``int` `i=n2; i<n1; i++)`
`    ``{`
`        ``int` `sub = ((str1[i]-``'0'``) - carry);`
`        ``carry = 0;`
`        ``str.push_back(sub + ``'0'``);`
`    ``}`
`    ``// reverse resultant string`
`    ``reverse(str.begin(), str.end());`
`    ``return` `str;`
`}`
`// Driver code`
`int` `main()`
`{`
`    ``string str1 = ``"12"``;`
`    ``string str2 = ``"198111"``;`
`    ``cout << findDiff(str1, str2);`
`    ``return` `0;`
`}`

Output:

```198099
```

Optimized solution
We can avoid the first two string reverse operations by traversing them from end. Below is optimized solution.

`// C++ program to find difference of two large numbers.`
`#include<bits/stdc++.h>`
`using` `namespace` `std;`
`// Returns true if str1 is smaller than str2,`
`// else false.`
`bool` `isSmaller(string str1, string str2)`
`{`
`    ``// Calculate lengths of both string`
`    ``int` `n1 = str1.length(), n2 = str2.length();`
`    ``if` `(n1 < n2)`
`        ``return` `true``;`
`    ``if` `(n2 > n1)`
`        ``return` `false``;`
`    ``for` `(``int` `i=0; i<n1; i++)`
`    ``{`
`        ``if` `(str1[i] < str2[i])`
`            ``return` `true``;`
`        ``else` `if` `(str1[i] > str2[i])`
`            ``return` `false``;`
`    ``}`
`    ``return` `false``;`
`}`
`// Function for finding difference of larger numbers`
`string findDiff(string str1, string str2)`
`{`
`    ``// Before proceeding further, make sure str1`
`    ``// is not smaller`
`    ``if` `(isSmaller(str1, str2))`
`        ``swap(str1, str2);`
`    ``// Take an empty string for storing result`
`    ``string str = ``""``;`
`    ``// Calculate lengths of both string`
`    ``int` `n1 = str1.length(), n2 = str2.length();`
`    ``int` `diff = n1 - n2;`
`    ``// Initially take carry zero`
`    ``int` `carry = 0;`
`    ``// Traverse from end of both strings`
`    ``for` `(``int` `i=n2-1; i>=0; i--)`
`    ``{`
`        ``// Do school mathematics, compute difference of`
`        ``// current digits and carry`
`        ``int` `sub = ((str1[i+diff]-``'0'``) -`
`                   ``(str2[i]-``'0'``) -`
`                   ``carry);`
`        ``if` `(sub < 0)`
`        ``{`
`            ``sub = sub+10;`
`            ``carry = 1;`
`        ``}`
`        ``else`
`            ``carry = 0;`
`        ``str.push_back(sub + ``'0'``);`
`    ``}`
`    ``// subtract remaining digits of str1[]`
`    ``for` `(``int` `i=n1-n2-1; i>=0; i--)`
`    ``{`
`        ``if` `(str1[i]==``'0'` `&& carry)`
`        ``{`
`            ``str.push_back(``'9'``);`
`            ``continue``;`
`        ``}`
`        ``int` `sub = ((str1[i]-``'0'``) - carry);`
`        ``if` `(i>0 || sub>0) ``// remove preceding 0's`
`            ``str.push_back(sub+``'0'``);`
`        ``carry = 0;`
`    ``}`
`    ``// reverse resultant string`
`    ``reverse(str.begin(), str.end());`
`    ``return` `str;`
`}`
`// Driver code`
`int` `main()`
`{`
`    ``string str1 = ``"88"``;`
`    ``string str2 = ``"1079"``;`
`    ``cout << findDiff(str1, str2);`
`    ``return` `0;`
`}`

Output:

```991
```

Time complexity : O(n1 + n2)

Disclaimer: This does not belong to TechCodeBit, its an article taken from the below
source and credits.
source and credits:http://www.geeksforgeeks.org/difference-of-two-large-numbers/
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