/* * The Karatsuba algorithm is a multiplication algorithm developed by Anatolii Alexeevitch Karatsuba in 1960. * It operates in O(n^log2(3)) time (~ O(n^1.585)), with n being the number of digits of the numbers we are multiplying together. * Standard grade-school multiplication operates in O(n^2) time. Karatsuba's method is asymptotically much faster. * Normally, you can choose any base you want, but we will be using base 10 in this algorithm with m varying depending on the length of the inputs. * Specific details are included with an example in the comments before the actual method. * * @author Ayamin * */ public class Karatsuba { // Takes two integers and returns the maximum of them public static int max(int x, int y) { return (x>y)? x:y; } // Takes a string and an index. // The index in this case is the "m". It will count backwards from the last (least significant) digit and split the string there. // It will return a 2-element array of the split string. // For example: Given 12345 as the string and 2 as the index, it will split the string into the string array ["123", "45"]. // This is so the 123 can be written as 123 * 10^m, with m = 2 the index. public static String[] strCopy(long index, String string) { String first = "", last = ""; long actualIndex = string.length() - index; for (int i = 0; i<actualIndex; i++) { first+=string.charAt(i); } for (int i = (int)actualIndex; i<string.length(); i++) { last+=string.charAt(i); } return new String[] {first, last}; } // An exponent function. Works the same way as Math.pow, but with 64bit integers instead of double precision floats. public static long power(long x, long y) { if (y == 0) return 1; else { long answer = 1; for (int i = 1; i<=y; i++) { answer *= x; } return answer; } } /* * Take two numbers, x and y. * Example: 12345 and 6789. * Find a base b and power m to separate it into. * We'll pick base = 10, and m to be half the length of the digits of the numbers in this implementation of the algorithm. * In this case, m will be 2, so 10^2 = 100. We will split the 2 numbers using this multiplier. * The form we want is: * x = x1*b^m + x0 * y = y1*b^m + y0 * ---------- * Using the above example, * x1 = 123 * x0 = 45 * ---------- * y1 = 67 * y2 = 89 * ---------- * b = 10 and m = 2 * ---------- * Thus: * 12345 = 123 * 10^2 + 45 * 6789 = 67 * 10^2 + 89 * * * The recursive algorithm is as follows: * * If x<10 or y<10, return x*y. Single digit multiplication is the base case. * Otherwise: * Let z2 = karatsuba(x1, y1). x1 and y1 are the most significant digits, and are the local variables "high". * Let z0 = karatsuba(x0, y0). x0 and y0 are the least significant digits, and are the local variables "low". * Let z1 = karatsuba(x1+y0, x0+y1) - z0 - z2. * And the result is the following sum: * z2 * b^2m + z1 * b^m + z0 * * @param x The multiplicand. * @param y The multiplier. * @return The product. */ public static long karatsuba(long x, long y) { // Base case: single digit multiplication if (x<10 || y<10) { return x * y; } // Recursive case: Decompose the problem by splitting the integers and applying the algorithm on the parts. else { // Convert the numbers to strings so we can compute the # of digits of each number. // Note: We could also use floor(log10(n) + 1) to compute the #digits, but remember that we need to split the numbers too. String xString = Integer.toString((int)x); String yString = Integer.toString((int)y); // Local variables long m = max(xString.length(), yString.length()), // the maximum # of digits m2 = m/2, // the middle; if the number is odd, it will floor the fraction high1 = Integer.parseInt(strCopy(m2, xString)[0]), // the most significant digits. this is the scalar multiplier for b^m2 low1 = Integer.parseInt(strCopy(m2, xString)[1]), // the least significant digits. this is what is added on to high1*b^m2 high2 = Integer.parseInt(strCopy(m2, yString)[0]), // same for y low2 = Integer.parseInt(strCopy(m2, yString)[1]), // same for y // Three recursive calls z0 = karatsuba(low1, low2), // z0 = x0y0 z2 = karatsuba(high1, high2), // z2 = x1y1 z1 = karatsuba((low1 + high1), (low2 + high2)) - z2 - z0; // z1 = (x0 + y1)*(x1 + y0) - z2 - z0, courtesy of Karatsuba return (z2 * power(10, 2*m2) + (z1 * power(10, m2)) + z0); } } } | 
