Swift Coding Test Course, Extended Euclidean Algorithm

Hello! Today we will learn about one of the mathematical algorithms, the Extended Euclidean Algorithm. This course is designed to help you prepare for algorithm exams by explaining problem-solving methods in detail based on algorithmic thinking, and it can be implemented in Swift.

1. What is the Euclidean Algorithm?

The Euclidean algorithm is a classical algorithm for finding the greatest common divisor (GCD) of two numbers \(a\) and \(b\). The process is as follows:

GCD(a, b) = GCD(b, a % b)

This expression continues to repeat until \(b\) becomes 0, and \(GCD(a, 0) = a\) holds. In other words, the value remaining in the first argument is the GCD of the two numbers.

2. What is the Extended Euclidean Algorithm?

The Extended Euclidean Algorithm builds upon the Euclidean Algorithm to find the following expression for two integers \(a\) and \(b\):

ax + by = gcd(a, b)

Here, \(x\) and \(y\) are integers, commonly known as Bézout’s identity. Using this algorithm, you can find not only the GCD of the given two numbers but also the coefficients for the GCD.

3. Problem Description

Now, let’s look at a problem that can be solved using the Extended Euclidean Algorithm:

Problem: Given two integers a and b, find x and y that satisfy ax + by = gcd(a, b).

Input Format

  • Two integers \(a\) and \(b\) are given. (1 ≤ a, b ≤ 1,000,000)

Output Format

  • Output GCD(a, b), x, and y separated by spaces.

Example Input

30 12

Example Output

6 1 -2

4. Algorithm Solution Process

4.1. Problem Analysis

To solve this problem, we need to find the GCD of the given two numbers and the coefficients x and y to find that GCD. For this, we need to implement the Extended Euclidean Algorithm.

4.2. Algorithm Flowchart

  1. Receive two numbers \(a\) and \(b\) as input.
  2. Find the GCD using the Euclidean Algorithm.
  3. Use the Extended Euclidean Algorithm to find the coefficients x and y for the linear combination.
  4. Output the result.

4.3. Swift Implementation

Now let’s implement the above process in Swift code:


func extendedGCD(a: Int, b: Int) -> (gcd: Int, x: Int, y: Int) {
    if b == 0 {
        return (a, 1, 0)
    } else {
        let (gcd, x1, y1) = extendedGCD(a: b, b: a % b)
        let x = y1
        let y = x1 - (a / b) * y1
        return (gcd, x, y)
    }
}

func main() {
    let input = readLine()!.split(separator: " ").map { Int($0)! }
    let a = input[0]
    let b = input[1]
    let (gcd, x, y) = extendedGCD(a: a, b: b)
    print("\(gcd) \(x) \(y)")
}

// Program entry point
main()

5. Example Problem Solution

Let’s use the above Swift code to solve the given problem.

Input

30 12

Output

6 1 -2

For the above input, the GCD is 6, which can be confirmed with \(30x + 12y = 6\) where \(x = 1\) and \(y = -2\).

6. Conclusion

Today, we learned about the Extended Euclidean Algorithm and how to solve a given problem using it. This algorithm plays an important role in computer science and cryptography. Remember that it can be effectively used in various applications!

I hope you continue to enhance your coding skills through various algorithm-related courses. If you have any questions or want to learn more, please leave a comment!