It is a replacement for the grade school algorithm for multiplying numbers. of bigger digits. The Karatsuba Algorithm for fast multiplication algorithm using Divide and Conquer developed by Anatolii Alexeevitch Karatsuba in 1960. The first thing that hits the head what is it and why it is designed. Though there are 3 ways to multiply numbers :
- Third-grade school algorithm Method (Standard-way)
- Recursive algorithm Method
- Karatsuba Multiplication Method
Why the Karatsuba algorithm?
The goal of the algorithm is designed because the design space is surprisingly rich. Its time complexity is:
O(n^log2(3)) time (~ O(n^1.585))
Where n is the number of digits of the numbers multiplying. It is discussed by multiplying two big integer numbers to show internal working step by step. The goal is to reduce the space complexity for which the integer numbers terms will be broken down in such a way to x and y are broken into a set of digits as the logic behind it is divide and conquer. If the numbers are smaller there is no need to multiply, standard mutilation of two integers is preferred.
The algorithm is standardized for 4 digits for sake of understanding. One can multiply as many digits taken into sets.
Algorithm Steps:
- Compute starting set (a*c)
- Compute set after starting set may it be ending set (b*d)
- Compute starting set with ending sets
- Subtract values of step 3 from step2 from step1
- Pad up (Add) 4 zeros to the number obtained from Step1, step2 value unchanged, and pad up two zeros to vale obtained from step4.
An example illustrating the Karatsuba Algorithm:
Input: x = 1234, y = 5678
Processing: As per above inputs
x = 1234
y = 5678
a = 12, b = 34
c = 56, d = 78
Step 1
a * c = 172
Step 2
b * d = 2652
Step 3
(a+b)(c+d) = 134 * 36 = 6164
Step 4:
6164 – 2652 – 172 = 2840
Step 5:
1720000 + 2652 + 284000 = 7006652
Output: 7006652
The value obtained from step 5 is in fact the product obtained if standard school multiplication is carried on between these two numbers ‘x’ and ‘y’.
1720000 + 2652 + 284000 = 7006652
Implementation: Note not to grasp the formulas laid down for this algorithm rather understanding it in this way makes its way to better.
Java
// Java program to illustrate // karatsuba multiplication // Importing generic java libraries import java.io.*; class GFG { // Karatsuba multiplication class public static long karatmultiply(long x, long y) { // Finding length of both integer // numbers x and y int noOneLength = numLength(x); int noTwoLength = numLength(y); // Find maximum length from both numbers // using math library max function int maxNumLength = Math.max(noOneLength, noTwoLength); // For very small values // multiply directly int smallValue = 1 << 4; if (maxNumLength < smallValue) return x * y; // Rounding up the // divided Max length maxNumLength = (maxNumLength / 2) + (maxNumLength % 2); // Multiplier long maxNumLengthTen = (long)Math.pow(10, maxNumLength); // Compute the expressions long b = x / maxNumLengthTen; long a = x - (b * maxNumLengthTen); long d = y / maxNumLengthTen; long c = y - (d * maxNumLength); // Compute all mutilpying variables // needed to get the multiplication long z0 = karatmultiply(a, c); long z1 = karatmultiply(a + b, c + d); long z2 = karatmultiply(b, d); return z0 + ((z1 - z0 - z2) * maxNumLengthTen) + (z2 * (long)(Math.pow(10, 2 * maxNumLength))); } // Fucntion to calculate // length of the number public static int numLength(long n) { int noLen = 0; while (n > 0) { noLen++; n /= 10; } // Returning length of number n return noLen; } // Main driver function public static void main(String[] args) { // Showcasing karatsuba multiplication // case 1: Big integer lengths long product1 = karatmultiply(1345, 63456); System.out.println("Product 1 : " + product1); // Case 2: quite smaller integer length long product2 = karatmultiply(102, 213); System.out.println("Product 2 : " + product2); } } |
Product 1 : 85348320 Product 2 : 21726
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