feat: Karatsuba algorithm (#1550)

* karatsuba algorithm

* Update karatsuba_algorithm_for_fast_multiplication.cpp

* Update karatsuba_algorithm_for_fast_multiplication.cpp

* Update karatsuba_algorithm_for_fast_multiplication.cpp

* Update karatsuba_algorithm_for_fast_multiplication.cpp

* Update karatsuba_algorithm_for_fast_multiplication.cpp

* Update karatsuba_algorithm_for_fast_multiplication.cpp

* updating DIRECTORY.md

* clang-format and clang-tidy fixes for 180f356d

* Update divide_and_conquer/karatsuba_algorithm_for_fast_multiplication.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update divide_and_conquer/karatsuba_algorithm_for_fast_multiplication.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update divide_and_conquer/karatsuba_algorithm_for_fast_multiplication.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* clang-format and clang-tidy fixes for 2b8d67ad

* Apply suggestions from code review

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: David Leal <halfpacho@gmail.com>
Co-authored-by: Ayaan Khan <ayaankhan98@gmail.com>

DIRECTORY.md

@@ -66,6 +66,9 @@
   * [Trie Tree](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/data_structures/trie_tree.cpp)
   * [Trie Using Hashmap](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/data_structures/trie_using_hashmap.cpp)
 
+## Divide And Conquer
+  * [Karatsuba Algorithm For Fast Multiplication](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/divide_and_conquer/karatsuba_algorithm_for_fast_multiplication.cpp)
+
 ## Dynamic Programming
   * [0 1 Knapsack](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/dynamic_programming/0_1_knapsack.cpp)
   * [Abbreviation](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/dynamic_programming/abbreviation.cpp)

divide_and_conquer/karatsuba_algorithm_for_fast_multiplication.cpp

@@ -0,0 +1,167 @@
+/**
+ * @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(
+        divide_and_conquer::karatsuba_algorithm::addStrings(Xl, Xr),
+        divide_and_conquer::karatsuba_algorithm::addStrings(Yl, Yr));
+
+    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;
+}