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• Difficulty Level : Medium
• Last Updated : 07 Jul, 2021

Given two bit sequences as strings, write a function to return the addition of the two sequences. Bit strings can be of different lengths also. For example, if string 1 is “1100011” and second string 2 is “10”, then the function should return “1100101”.

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Since the sizes of two strings may be different, we first make the size of a smaller string equal to that of the bigger string by adding leading 0s. After making sizes the same, we one by one add bits from rightmost bit to leftmost bit. In every iteration, we need to sum 3 bits: 2 bits of 2 given strings and carry. The sum bit will be 1 if, either all of the 3 bits are set or one of them is set. So we can do XOR of all bits to find the sum bit. How to find carry – carry will be 1 if any of the two bits is set. So we can find carry by taking OR of all pairs. Following is step by step algorithm.

1. Make them equal sized by adding 0s at the beginning of smaller string.
…..Boolean expression for adding 3 bits a, b, c
…..Sum = a XOR b XOR c
…..Carry = (a AND b) OR ( b AND c ) OR ( c AND a )

Following is implementation of the above algorithm.

## C++

 `#include ``using` `namespace` `std;`` ` `//adds the two-bit strings and return the result``string addBitStrings( string first, string second );`` ` `// Helper method: given two unequal sized bit strings, converts them to``// same length by adding leading 0s in the smaller string. Returns the``// the new length``int` `makeEqualLength(string &str1, string &str2)``{``    ``int` `len1 = str1.size();``    ``int` `len2 = str2.size();``    ``if` `(len1 < len2)``    ``{``        ``for` `(``int` `i = 0 ; i < len2 - len1 ; i++)``            ``str1 = ``'0'` `+ str1;``        ``return` `len2;``    ``}``    ``else` `if` `(len1 > len2)``    ``{``        ``for` `(``int` `i = 0 ; i < len1 - len2 ; i++)``            ``str2 = ``'0'` `+ str2;``    ``}``    ``return` `len1; ``// If len1 >= len2``}`` ` `// The main function that adds two-bit sequences and returns the addition``string addBitStrings( string first, string second )``{``    ``string result;  ``// To store the sum bits`` ` `    ``// make the lengths same before adding``    ``int` `length = makeEqualLength(first, second);`` ` `    ``int` `carry = 0;  ``// Initialize carry`` ` `    ``// Add all bits one by one``    ``for` `(``int` `i = length-1 ; i >= 0 ; i--)``    ``{``        ``int` `firstBit = first.at(i) - ``'0'``;``        ``int` `secondBit = second.at(i) - ``'0'``;`` ` `        ``// boolean expression for sum of 3 bits``        ``int` `sum = (firstBit ^ secondBit ^ carry)+``'0'``;`` ` `        ``result = (``char``)sum + result;`` ` `        ``// boolean expression for 3-bit addition``        ``carry = (firstBit & secondBit) | (secondBit & carry) | (firstBit & carry);``    ``}`` ` `    ``// if overflow, then add a leading 1``    ``if` `(carry)``        ``result = ``'1'` `+ result;`` ` `    ``return` `result;``}`` ` `// Driver program to test above functions``int` `main()``{``    ``string str1 = ``"1100011"``;``    ``string str2 = ``"10"``;`` ` `    ``cout << ``"Sum is "` `<< addBitStrings(str1, str2);``    ``return` `0;``}`

## Java

 `// Java implementation of above algorithm``class` `GFG``{`` ` `    ``// Helper method: given two unequal sized bit strings, ``    ``// converts them to same length by adding leading 0s ``    ``// in the smaller string. Returns the the new length``    ``// Using StringBuilder as Java only uses call by value``    ``static` `int` `makeEqualLength(StringBuilder str1, ``                               ``StringBuilder str2) ``    ``{``        ``int` `len1 = str1.length();``        ``int` `len2 = str2.length();``        ``if` `(len1 < len2)``        ``{``            ``for` `(``int` `i = ``0``; i < len2 - len1; i++)``                ``str1.insert(``0``, ``'0'``);``            ``return` `len2;``        ``} ``        ``else` `if` `(len1 > len2)``        ``{``            ``for` `(``int` `i = ``0``; i < len1 - len2; i++)``                ``str2.insert(``0``, ``'0'``);``        ``}`` ` `        ``return` `len1; ``// If len1 >= len2``    ``}`` ` `    ``// The main function that adds two-bit sequences ``    ``// and returns the addition``    ``static` `String addBitStrings(StringBuilder str1,``                                ``StringBuilder str2) ``    ``{``        ``String result = ``""``; ``// To store the sum bits`` ` `        ``// make the lengths same before adding``        ``int` `length = makeEqualLength(str1, str2);`` ` `        ``// Convert StringBuilder to Strings``        ``String first = str1.toString();``        ``String second = str2.toString();`` ` `        ``int` `carry = ``0``; ``// Initialize carry`` ` `        ``// Add all bits one by one``        ``for` `(``int` `i = length - ``1``; i >= ``0``; i--)``        ``{``            ``int` `firstBit = first.charAt(i) - ``'0'``;``            ``int` `secondBit = second.charAt(i) - ``'0'``;`` ` `            ``// boolean expression for sum of 3 bits``            ``int` `sum = (firstBit ^ secondBit ^ carry) + ``'0'``;`` ` `            ``result = String.valueOf((``char``) sum) + result;`` ` `            ``// boolean expression for 3-bit addition``            ``carry = (firstBit & secondBit) | ``                    ``(secondBit & carry) | ``                    ``(firstBit & carry);``        ``}``         ` `        ``// if overflow, then add a leading 1``        ``if` `(carry == ``1``)``            ``result = ``"1"` `+ result;``        ``return` `result;``    ``}`` ` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args) ``    ``{``        ``String str1 = ``"1100011"``;``        ``String str2 = ``"10"``;``        ``System.out.println(``"Sum is "` `+ ``        ``addBitStrings(``new` `StringBuilder(str1), ``                      ``new` `StringBuilder(str2)));``    ``}``}`` ` `// This code is contributed by Vivek Kumar Singh`

## Python3

 `# Python3 program for above approach`` ` `# adds the two-bit strings and return the result`` ` `# Helper method: given two unequal sized bit strings, ``# converts them to same length by adding leading 0s ``# in the smaller string. Returns the the new length``def` `makeEqualLength(str1, str2):`` ` `    ``len1 ``=` `len``(str1)     ``# Length of string 1``    ``len2 ``=` `len``(str2)     ``# length of string 2``    ``if` `len1 < len2:``        ``str1 ``=` `(len2 ``-` `len1) ``*` `'0'` `+` `str1``        ``len1 ``=` `len2``    ``elif` `len2 < len1:``        ``str2 ``=` `(len1 ``-` `len2) ``*` `'0'` `+` `str2``        ``len2 ``=` `len1``    ``return` `len1, str1, str2`` ` `def` `addBitStrings( first, second ):``    ``result ``=` `'' ``# To store the sum bits`` ` `    ``# make the lengths same before adding``    ``length, first, second ``=` `makeEqualLength(first, second)`` ` `    ``carry ``=` `0` `# initialize carry as 0`` ` `    ``# Add all bits one by one``    ``for` `i ``in` `range``(length ``-` `1``, ``-``1``, ``-``1``):``        ``firstBit ``=` `int``(first[i])``        ``secondBit ``=` `int``(second[i])`` ` `        ``# boolean expression for sum of 3 bits``        ``sum` `=` `(firstBit ^ secondBit ^ carry) ``+` `48``        ``result ``=` `chr``(``sum``) ``+` `result`` ` `        ``# boolean expression for 3 bits addition``        ``carry ``=` `(firstBit & secondBit) | \``                ``(secondBit & carry) | \``                ``(firstBit & carry)`` ` `        ``# if overflow, then add a leading 1``    ``if` `carry ``=``=` `1``:``        ``result ``=` `'1'` `+` `result``    ``return` `result`` ` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``str1 ``=` `'1100011'``    ``str2 ``=` `'10'``    ``print``(``'Sum is'``, addBitStrings(str1, str2))``     ` `# This code is contributed by``# chaudhary_19 (Mayank Chaudhary)`

## C#

 `// C# implementation of above algorithm``using` `System;``using` `System.Text;`` ` `class` `GFG``{`` ` `    ``// Helper method: given two unequal sized ``    ``// bit strings, converts them to same length ``    ``// by adding leading 0s in the smaller string. ``    ``// Returns the the new length Using StringBuilder ``    ``// as Java only uses call by value ``    ``static` `int` `makeEqualLength(StringBuilder str1, ``                               ``StringBuilder str2)``    ``{``        ``int` `len1 = str1.Length;``        ``int` `len2 = str2.Length;``        ``if` `(len1 < len2)``        ``{``            ``for` `(``int` `i = 0; i < len2 - len1; i++)``            ``{``                ``str1.Insert(0, ``'0'``);``            ``}``            ``return` `len2;``        ``}``        ``else` `if` `(len1 > len2)``        ``{``            ``for` `(``int` `i = 0; i < len1 - len2; i++)``            ``{``                ``str2.Insert(0, ``'0'``);``            ``}``        ``}`` ` `        ``return` `len1; ``// If len1 >= len2``    ``}`` ` `    ``// The main function that adds two-bit sequences ``    ``// and returns the addition ``    ``static` `string` `addBitStrings(StringBuilder str1, ``                                ``StringBuilder str2)``    ``{``        ``string` `result = ``""``; ``// To store the sum bits`` ` `        ``// make the lengths same before adding ``        ``int` `length = makeEqualLength(str1, str2);`` ` `        ``// Convert StringBuilder to Strings ``        ``string` `first = str1.ToString();``        ``string` `second = str2.ToString();`` ` `        ``int` `carry = 0; ``// Initialize carry`` ` `        ``// Add all bits one by one ``        ``for` `(``int` `i = length - 1; i >= 0; i--)``        ``{``            ``int` `firstBit = first[i] - ``'0'``;``            ``int` `secondBit = second[i] - ``'0'``;`` ` `            ``// boolean expression for sum of 3 bits ``            ``int` `sum = (firstBit ^ secondBit ^ carry) + ``'0'``;`` ` `            ``result = ((``char``) sum).ToString() + result;`` ` `            ``// boolean expression for 3-bit addition ``            ``carry = (firstBit & secondBit) | ``                       ``(secondBit & carry) | ``                        ``(firstBit & carry);``        ``}`` ` `        ``// if overflow, then add a leading 1 ``        ``if` `(carry == 1)``        ``{``            ``result = ``"1"` `+ result;``        ``}``        ``return` `result;``    ``}`` ` `    ``// Driver Code ``    ``public` `static` `void` `Main(``string``[] args)``    ``{``        ``string` `str1 = ``"1100011"``;``        ``string` `str2 = ``"10"``;``        ``Console.WriteLine(``"Sum is "` `+ ``                ``addBitStrings(``new` `StringBuilder(str1), ``                              ``new` `StringBuilder(str2)));``    ``}``}`` ` `// This code is contributed by kumar65`

Output:

` Sum is 1100101`