karatsuba algorithm

divide_and_conquer/karatsuba_algorithm_for_fast_multiplication.cpp

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+/**
+ * @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 <iostream>  /// for IO operations
+#include <vector>    /// for std::vector
+#include <string>    /// for string
+#include <cstring>   /// for string
+
+/**
+ * @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);
+    if (len1 < len2) // make the string lengths equal
+    {
+        for (int64_t i = 0 ; i < len2 - len1 ; i++)
+            first = '0' + first;
+    }
+    else if (len1 > len2)
+    {
+        for (int64_t i = 0 ; i < len1 - len2 ; i++)
+            second = '0' + second;
+    }
+    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
+ 
+        result = (char)sum + result;
+ 
+        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);
+    if (len1 < len2)
+    {
+        for (int64_t i = 0 ; i < len2 - len1 ; i++)
+            str1 = '0' + str1;
+    }
+    else if (len1 > len2)
+    {
+        for (int64_t i = 0 ; i < len1 - len2 ; i++)
+            str2 = '0' + str2;
+    }
+    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 << "Test 1... ";
+    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 << "Test 2... ";
+    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 << "Test 3... ";
+    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;
+}