From 1c978c52f1252fdaea68e01cb056ffb396589190 Mon Sep 17 00:00:00 2001 From: Saahil Mahato <115351000+saahil-mahato@users.noreply.github.com> Date: Fri, 11 Oct 2024 19:58:51 +0545 Subject: [PATCH] feat: add karatsuba multiplication (#5719) * feat: add karatsuba multiplication * fix: fallback size * fix: big integer instances --------- Co-authored-by: Alex Klymenko --- .../maths/KaratsubaMultiplication.java | 93 +++++++++++++++++++ .../maths/KaratsubaMultiplicationTest.java | 58 ++++++++++++ 2 files changed, 151 insertions(+) create mode 100644 src/main/java/com/thealgorithms/maths/KaratsubaMultiplication.java create mode 100644 src/test/java/com/thealgorithms/maths/KaratsubaMultiplicationTest.java diff --git a/src/main/java/com/thealgorithms/maths/KaratsubaMultiplication.java b/src/main/java/com/thealgorithms/maths/KaratsubaMultiplication.java new file mode 100644 index 000000000..298fcb7e8 --- /dev/null +++ b/src/main/java/com/thealgorithms/maths/KaratsubaMultiplication.java @@ -0,0 +1,93 @@ +package com.thealgorithms.maths; + +import java.math.BigInteger; + +/** + * This class provides an implementation of the Karatsuba multiplication algorithm. + * + *

+ * Karatsuba multiplication is a divide-and-conquer algorithm for multiplying two large + * numbers. It is faster than the classical multiplication algorithm and reduces the + * time complexity to O(n^1.585) by breaking the multiplication of two n-digit numbers + * into three multiplications of n/2-digit numbers. + *

+ * + *

+ * The main idea of the Karatsuba algorithm is based on the following observation: + *

+ * + * 
+ * Let x and y be two numbers:
+ * x = a * 10^m + b
+ * y = c * 10^m + d
+ *
+ * Then, the product of x and y can be expressed as:
+ * x * y = (a * c) * 10^(2*m) + ((a * d) + (b * c)) * 10^m + (b * d)
+ *
+ * + * The Karatsuba algorithm calculates this more efficiently by reducing the number of + * multiplications from four to three by using the identity: + * + * 
+ * (a + b)(c + d) = ac + ad + bc + bd
+ *
+ * + *

+ * The recursion continues until the numbers are small enough to multiply directly using + * the traditional method. + *

+ */ +public final class KaratsubaMultiplication { + + /** + * Private constructor to hide the implicit public constructor + */ + private KaratsubaMultiplication() { + } + + /** + * Multiplies two large numbers using the Karatsuba algorithm. + * + *

+ * This method recursively splits the numbers into smaller parts until they are + * small enough to be multiplied directly using the traditional method. + *

+ * + * @param x The first large number to be multiplied (BigInteger). + * @param y The second large number to be multiplied (BigInteger). + * @return The product of the two numbers (BigInteger). + */ + public static BigInteger karatsuba(BigInteger x, BigInteger y) { + // Base case: when numbers are small enough, use direct multiplication + // If the number is 4 bits or smaller, switch to the classical method + if (x.bitLength() <= 4 || y.bitLength() <= 4) { + return x.multiply(y); + } + + // Find the maximum bit length of the two numbers + int n = Math.max(x.bitLength(), y.bitLength()); + + // Split the numbers in the middle + int m = n / 2; + + // High and low parts of the first number x (x = a * 10^m + b) + BigInteger high1 = x.shiftRight(m); // a = x / 2^m (higher part) + BigInteger low1 = x.subtract(high1.shiftLeft(m)); // b = x - a * 2^m (lower part) + + // High and low parts of the second number y (y = c * 10^m + d) + BigInteger high2 = y.shiftRight(m); // c = y / 2^m (higher part) + BigInteger low2 = y.subtract(high2.shiftLeft(m)); // d = y - c * 2^m (lower part) + + // Recursively calculate three products + BigInteger z0 = karatsuba(low1, low2); // z0 = b * d (low1 * low2) + BigInteger z1 = karatsuba(low1.add(high1), low2.add(high2)); // z1 = (a + b) * (c + d) + BigInteger z2 = karatsuba(high1, high2); // z2 = a * c (high1 * high2) + + // Combine the results using Karatsuba's formula + // z0 + ((z1 - z2 - z0) << m) + (z2 << 2m) + return z2 + .shiftLeft(2 * m) // z2 * 10^(2*m) + .add(z1.subtract(z2).subtract(z0).shiftLeft(m)) // (z1 - z2 - z0) * 10^m + .add(z0); // z0 + } +} diff --git a/src/test/java/com/thealgorithms/maths/KaratsubaMultiplicationTest.java b/src/test/java/com/thealgorithms/maths/KaratsubaMultiplicationTest.java new file mode 100644 index 000000000..e184d9987 --- /dev/null +++ b/src/test/java/com/thealgorithms/maths/KaratsubaMultiplicationTest.java @@ -0,0 +1,58 @@ +package com.thealgorithms.maths; + +import static org.junit.jupiter.api.Assertions.assertEquals; + +import java.math.BigInteger; +import java.util.stream.Stream; +import org.junit.jupiter.params.ParameterizedTest; +import org.junit.jupiter.params.provider.Arguments; +import org.junit.jupiter.params.provider.MethodSource; + +/** + * Unit test class for {@link KaratsubaMultiplication} class. + * + *

+ * This class tests various edge cases and normal cases for the + * Karatsuba multiplication algorithm implemented in the KaratsubaMultiplication class. + * It uses parameterized tests to handle multiple test cases. + *

+ */ +class KaratsubaMultiplicationTest { + + /** + * Provides test data for the parameterized test. + * Each entry in the stream contains three elements: x, y, and the expected result. + * + * @return a stream of arguments for the parameterized test + */ + static Stream provideTestCases() { + return Stream.of( + // Test case 1: Two small numbers + Arguments.of(new BigInteger("1234"), new BigInteger("5678"), new BigInteger("7006652")), + // Test case 2: Two large numbers + Arguments.of(new BigInteger("342364"), new BigInteger("393958"), new BigInteger("134877036712")), + // Test case 3: One number is zero + Arguments.of(BigInteger.ZERO, new BigInteger("5678"), BigInteger.ZERO), + // Test case 4: Both numbers are zero + Arguments.of(BigInteger.ZERO, BigInteger.ZERO, BigInteger.ZERO), + // Test case 5: Single-digit numbers + Arguments.of(new BigInteger("9"), new BigInteger("8"), new BigInteger("72"))); + } + + /** + * Parameterized test for Karatsuba multiplication. + * + *

+ * This method runs the Karatsuba multiplication algorithm for multiple test cases. + *

+ * + * @param x the first number to multiply + * @param y the second number to multiply + * @param expected the expected result of x * y + */ + @ParameterizedTest + @MethodSource("provideTestCases") + void testKaratsubaMultiplication(BigInteger x, BigInteger y, BigInteger expected) { + assertEquals(expected, KaratsubaMultiplication.karatsuba(x, y)); + } +}