## C++, divide_and_conquer, karatsuba_algorithm_for_fast_multiplication.cpp

``````/**
* @file
* @brief Implementation of the [Karatsuba algorithm for fast
* multiplication](https://en.wikipedia.org/wiki/Karatsuba_algorithm)
* @details
* Given two strings in binary notation we want to multiply them and return the
* value Simple approach is to multiply bits one by one which will give the time
* complexity of around O(n^2). To make it more efficient we will be using
* Karatsuba' algorithm to find the product which will solve the problem
* O(nlogn) of time.
* @author [Swastika Gupta](https://github.com/Swastyy)
*/

#include <cassert>   /// for assert
#include <cstring>   /// for string
#include <iostream>  /// for IO operations
#include <vector>    /// for std::vector

/**
* @namespace divide_and_conquer
* @brief Divide and Conquer algorithms
*/
namespace divide_and_conquer {
/**
* @namespace karatsuba_algorithm
* @brief Functions for the [Karatsuba algorithm for fast
* multiplication](https://en.wikipedia.org/wiki/Karatsuba_algorithm)
*/
namespace karatsuba_algorithm {
/**
* @brief Helper function for the main function, that implements Karatsuba's
* algorithm for fast multiplication
* @param first the input string 1
* @param second the input string 2
* @returns the concatenated string
*/
std::string addStrings(std::string first, std::string second) {
std::string result;  // To store the resulting sum bits

int64_t len1 = first.size();
int64_t len2 = second.size();
int64_t length = std::max(len1, len2);
std::string zero = "0";
if (len1 < len2)  // make the string lengths equal
{
for (int64_t i = 0; i < len2 - len1; i++) {
zero += first;
first = zero;
}
} else if (len1 > len2) {
zero = "0";
for (int64_t i = 0; i < len1 - len2; i++) {
zero += second;
second = zero;
}
}
int64_t carry = 0;
for (int64_t i = length - 1; i >= 0; i--) {
int64_t firstBit = first.at(i) - '0';
int64_t secondBit = second.at(i) - '0';

int64_t sum = (firstBit ^ secondBit ^ carry) + '0';  // sum of 3 bits
std::string temp;
temp = std::to_string(sum);
temp += result;
result = temp;

carry = (firstBit & secondBit) | (secondBit & carry) |
(firstBit & carry);  // sum of 3 bits
}

if (carry) {
result = '1' + result;  // adding 1 incase of overflow
}
return result;
}
/**
* @brief The main function implements Karatsuba's algorithm for fast
* multiplication
* @param str1 the input string 1
* @param str2 the input string 2
* @returns the multiplicative number value
*/
int64_t karatsuba_algorithm(std::string str1, std::string str2) {
int64_t len1 = str1.size();
int64_t len2 = str2.size();
int64_t n = std::max(len1, len2);
std::string zero = "0";
if (len1 < len2) {
for (int64_t i = 0; i < len2 - len1; i++) {
zero += str1;
str1 = zero;
}
} else if (len1 > len2) {
zero = "0";
for (int64_t i = 0; i < len1 - len2; i++) {
zero += str2;
str2 = zero;
}
}
if (n == 0) {
return 0;
}
if (n == 1) {
return (str1[0] - '0') * (str2[0] - '0');
}
int64_t fh = n / 2;     // first half of string
int64_t sh = (n - fh);  // second half of string

std::string Xl = str1.substr(0, fh);   // first half of first string
std::string Xr = str1.substr(fh, sh);  // second half of first string

std::string Yl = str2.substr(0, fh);   // first half of second string
std::string Yr = str2.substr(fh, sh);  // second half of second string

// Calculating the three products of inputs of size n/2 recursively
int64_t product1 = karatsuba_algorithm(Xl, Yl);
int64_t product2 = karatsuba_algorithm(Xr, Yr);
int64_t product3 = karatsuba_algorithm(

return product1 * (1 << (2 * sh)) +
(product3 - product1 - product2) * (1 << sh) +
product2;  // combining the three products to get the final result.
}
}  // namespace karatsuba_algorithm
}  // namespace divide_and_conquer

/**
* @brief Self-test implementations
* @returns void
*/
static void test() {
// 1st test
std::string s11 = "1";
std::string s12 = "1010";
std::cout << "1st test... ";
assert(divide_and_conquer::karatsuba_algorithm::karatsuba_algorithm(
s11, s12) == 10);  // here the multiplication is 10
std::cout << "passed" << std::endl;

// 2nd test
std::string s21 = "11";
std::string s22 = "1010";
std::cout << "2nd test... ";
assert(divide_and_conquer::karatsuba_algorithm::karatsuba_algorithm(
s21, s22) == 30);  // here the multiplication is 30
std::cout << "passed" << std::endl;

// 3rd test
std::string s31 = "110";
std::string s32 = "1010";
std::cout << "3rd test... ";
assert(divide_and_conquer::karatsuba_algorithm::karatsuba_algorithm(
s31, s32) == 60);  // here the multiplication is 60
std::cout << "passed" << std::endl;
}

/**
* @brief Main function
* @returns 0 on exit
*/
int main() {
test();  // run self-test implementations
return 0;
}
``````