kotlin coding test course, extended Euclidean algorithm

One of the frequently encountered topics in coding tests is mathematical algorithms. Among them, the “Extended Euclidean Algorithm” is a powerful technique that not only finds the greatest common divisor (GCD) of two numbers but also provides solutions related to Bézout’s identity. In this article, I will explain the Extended Euclidean Algorithm and detail the process of solving problems using it.

What is the Extended Euclidean Algorithm?

The Extended Euclidean Algorithm is a method for finding integers x and y in the process of calculating the greatest common divisor gcd(a, b) of two integers a and b:

gcd(a, b) = ax + by

The above equation is known as Bézout’s identity, where x and y are integers. The Extended Euclidean Algorithm can be implemented using either a recursive method or an iterative method.

Problem Definition

The problem we will address involves given numbers, using the Extended Euclidean Algorithm to output both the greatest common divisor and Bézout’s coefficients.

Problem

Problem Description:

Given two integers A and B, find the following:

  • The greatest common divisor GCD(A, B), and
  • Find integers X, Y such that GCD(A, B) = AX + BY.

Input: Two integers A, B (-109 ≤ A, B ≤ 109)

Output: Print GCD(A, B), X, Y separated by spaces.

Problem Solving Process

1. Understanding the Basic Algorithm

First, to implement the Extended Euclidean Algorithm, one must understand the basic Euclidean algorithm. The Euclidean Algorithm is a famous method for finding the greatest common divisor of two numbers, following these steps:

  • While both A and B are not zero, execute A = A % B.
  • Then swap the values of A and B. Make A equal to B and B equal to the previous B.
  • Repeat until one of the numbers becomes zero.

The remaining number at the end is the greatest common divisor. By tracking the changes of x and y at each step, we can construct Bézout’s identity.

2. Implementing the Extended Euclidean Algorithm

Here is the code implementing the Extended Euclidean Algorithm in Kotlin:

fun extendedGCD(a: Int, b: Int): Triple {
    if (b == 0) {
        return Triple(a, 1, 0) // GCD = a, X = 1, Y = 0
    } else {
        val (gcd, x1, y1) = extendedGCD(b, a % b) // Recursive call
        val x = y1
        val y = x1 - (a / b) * y1
        return Triple(gcd, x, y) // Return GCD, X, Y
    }
}

The above function takes two integers a and b as input and returns the greatest common divisor along with the Bézout coefficients x and y.

3. Writing the Main Function

Now, let’s write code in the main function to read two numbers from the user, call the Extended Euclidean Algorithm, and print the results:

fun main() {
    val reader = Scanner(System.`in`)
    println("Please enter two integers:")
    val A = reader.nextInt()
    val B = reader.nextInt()
    
    val (gcd, x, y) = extendedGCD(A, B)
    println("GCD: $gcd, X: $x, Y: $y")
}

4. Complexity Analysis

The time complexity of the Extended Euclidean Algorithm is the same as that of the basic Euclidean algorithm. The time complexity of the Euclidean algorithm is O(log(min(A, B))). This makes the Extended Euclidean Algorithm very efficient as well.

5. Testing and Examples

Now let’s perform some tests using the implemented code. For example, let’s check the results when A = 30 and B = 21:

Please enter two integers:
30
21
GCD: 3, X: -1, Y: 2

The result above shows that 3 = 30 * (-1) + 21 * 2, confirming that Bézout’s identity holds.

Conclusion

In this posting, we explored how to find the greatest common divisor and Bézout’s identity of two numbers using the Extended Euclidean Algorithm. This algorithm can be applied to various problem-solving scenarios and is particularly useful in problems related to mathematical algorithms. Its low complexity makes it a great tool for achieving high scores in actual coding tests. I hope we can continue to explore various algorithms and problem-solving methods together in the future.