Category: Swift Coding Test

Introduction

Currently, understanding algorithms and data structures is essential in the IT industry. Especially for developers preparing for employment, it is crucial to have problem-solving skills based on this knowledge. In this course, we will take a closer look at the ‘Union-Find’ algorithm and explain the problem-solving process using it step by step.

What is Union-Find?

Union-Find is a data structure primarily used to handle operations on sets, supporting two main operations:

• Union: The operation of merging two sets
• Find: The operation of finding which set a particular element belongs to

This data structure is useful for solving the Disjoint Set problem. Union-Find increases efficiency using optimization techniques called ‘Path Compression’ and ‘Rank’.

Problem Definition

To define the problem, let’s set up a hypothetical situation.

Problem: Given N elements, write a program to determine whether multiple pairs of elements belong to the same set.

The number of elements N and the number of pairs M are provided. For each pair, if they belong to the ‘same set’, print “YES”, and if they belong to ‘different sets’, print “NO”.

Example Problem

Input:

5 4
1 2
1 3
2 3
4 5
1 4
            

Output:

NO
YES
            

In the above example, the first pair (1, 4) belongs to different sets, so it outputs “NO”, and the second pair (1, 4) belongs to the same set, so it outputs “YES”.

Implementing the Union-Find Algorithm

Now, let’s implement the Union-Find algorithm in Swift. We will write the code reflecting the basic concepts.

class UnionFind {
    private var parent: [Int]
    private var rank: [Int]
    
    init(size: Int) {
        self.parent = Array(0.. Int {
        if p != parent[p] {
            parent[p] = find(parent[p]) // Path Compression
        }
        return parent[p]
    }
    
    func union(_ p: Int, _ q: Int) {
        let rootP = find(p)
        let rootQ = find(q)
        
        if rootP != rootQ {
            if rank[rootP] > rank[rootQ] {
                parent[rootQ] = rootP
            } else if rank[rootP] < rank[rootQ] {
                parent[rootP] = rootQ
            } else {
                parent[rootQ] = rootP
                rank[rootP] += 1
            }
        }
    }
}
            

Problem Solving Process

To solve the problem, we will follow these steps.

1. Parse the input to obtain the number of elements N and the number of pairs M.
2. Use the UnionFind class to manage the sets of elements.
3. Perform the union operation for each pair to merge the sets.
4. Perform the find operation for each pair and print the results.

The code is as follows.

import Foundation

let input = readLine()!.split(separator: " ").map { Int($0)! }
let n = input[0]
let m = input[1]

let unionFind = UnionFind(size: n + 1)

for _ in 0..

Conclusion

In this course, we have implemented the theoretical background of the Union-Find algorithm and the problem-solving process in Swift. Union-Find is one of the very useful data structures for graph-related problems. By utilizing this data structure, complex problems can be effectively solved.

I hope you continue to explore various algorithms to aid in your coding test preparation, and I look forward to introducing other algorithms in the next course.

Problem Description

You are given an integer array numbers. Write a function to check if a specific integer target exists in this array. If it exists, return the index of that integer; if not, return -1.

func findTarget(numbers: [Int], target: Int) -> Int

Input

• numbers: An array of integers (1 ≤ length of numbers ≤ 10⁶)
• target: The integer you want to find (-10⁹ ≤ target ≤ 10⁹)

Output

If target exists, return the corresponding index; otherwise, return -1.

Examples

Example 1

Input: numbers = [1, 2, 3, 4, 5], target = 3
Output: 2

Example 2

Input: numbers = [5, 6, 7, 8, 9], target = 1
Output: -1

Solution Approach

To solve this problem, one must think of how to find the index of target within the array. A basic method is linear search using a loop, and if the array is sorted, binary search can be used.

1. Linear Search

Linear search is a method where you check each element one by one from the beginning to the end of the array to find target. The time complexity is O(n) depending on the length of the array.

2. Binary Search

Binary search is much more efficient when the array is sorted. This method reduces the search range by half based on the middle index of the array to find target. The time complexity in this case is O(log n).

Implementation

First, we will implement linear search and then also implement binary search.

Linear Search Implementation

func findTarget(numbers: [Int], target: Int) -> Int {
    for (index, number) in numbers.enumerated() {
        if number == target {
            return index
        }
    }
    return -1
}

Binary Search Implementation

func binarySearch(numbers: [Int], target: Int) -> Int {
    var left = 0
    var right = numbers.count - 1

    while left <= right {
        let mid = (left + right) / 2
        if numbers[mid] == target {
            return mid
        } else if numbers[mid] < target {
            left = mid + 1
        } else {
            right = mid - 1
        }
    }
    return -1
}

Test Cases

We will run a few cases to test the function we have written.

let numbers = [1, 2, 3, 4, 5]
let target1 = 3
let target2 = 6

print(findTarget(numbers: numbers, target: target1)) // 2
print(findTarget(numbers: numbers, target: target2)) // -1

Performance Evaluation

Linear search is simple but, in the worst case, requires searching through all elements, taking an average of O(n) time. In contrast, binary search performs better on sorted arrays and guarantees O(log n). Therefore, binary search is recommended for large datasets.

Conclusion

Through this problem, we learned how to find integers in an array using Swift. We understood how to solve the problem using basic linear search and also learned about improving performance with more efficient binary search. We encourage you to develop your skills in solving algorithm problems through continuous practice.

Problem Definition

The Travelling Salesman Problem (TSP) involves finding the shortest route for a salesman to visit n cities exactly once and return to the starting city, given the distances between each pair of cities. The inputs consist of the number of cities n and the distance matrix between those cities.

Input Format

• First line: Number of cities n (1 ≤ n ≤ 10)
• Next n lines: Distance matrix, where the i-th line contains the distances from the i-th city to each city (A[i][j] is the distance from city i to city j)

Output Format

Output the length of the shortest route that visits all cities and returns to the starting city.

Problem-Solving Approach

This problem belongs to the NP-Hard category, so it can be solved using dynamic programming techniques combined with pruning. This method allows us to only choose the least costly paths while constructing routes, rather than exploring all possible paths.

Step 1: Data Structure Design

First, declare a 2D array to store distance information between cities. Additionally, declare an array to check which cities have been visited and a variable to store the total distance so far.

Step 2: Using Bitmask and Recursive Functions

Use bitmasking to represent whether each city has been visited. Every time we move to the next city via a recursive function, check off the visited cities and calculate the distance of the path taken once all cities have been visited and the route returns to the starting city. Since visited cities can be easily represented with a bitmask, they can be processed efficiently.

Step 3: Finding the Optimal Route

Instead of exploring all possible paths, proceed by calculating the minimum distance among the currently selected paths. If the given path has visited all cities, compare it with the previous distance to update the minimum value.

Swift Code Implementation

Below is the code for finding the travelling salesman route using Swift.

let INF = Int.max
var dist: [[Int]] = []
var n = 0
var dp: [[Int]] = []
var visited: Int = 0
var ans: Int = INF

func tsp(pos: Int, visited: Int) -> Int {
    if visited == (1 << n) - 1 {
        return dist[pos][0] != 0 ? dist[pos][0] : INF
    }
    
    if dp[pos][visited] != -1 {
        return dp[pos][visited]
    }
    
    var minimum = INF
    for city in 0.. Int {
    visited = 1 // start from city 0
    dp = Array(repeating: Array(repeating: -1, count: 1 << n), count: n)
    let minimumDistance = tsp(pos: 0, visited: visited)
    return minimumDistance
}

// Example distance matrix
dist = [
    [0, 10, 15, 20],
    [10, 0, 35, 25],
    [15, 35, 0, 30],
    [20, 25, 30, 0]
]
n = dist.count

let result = findMinimumRoute()
print("The length of the minimum route is \(result).")

Code Analysis and Functionality

This code uses bitmasking to check visit completion and recursively explores possible routes to calculate the minimum route. Every time a city is visited, the total distance is updated, and it finally returns the distance of the route that goes back to the starting city after visiting all cities.

1. Initializing the dp Array

The dp array stores the minimum route distances based on each city and visit status. Initially, all elements are set to -1 to indicate an unvisited state.

2. Recursive Function tsp()

The recursive function tsp() takes the current position (pos) and the visited city bitmask (visited) as parameters to compute the optimal route. If all cities have been visited, it compares the cost of returning to the starting city and returns the minimum value.

3. Route Calculation

Bitmasking checks for cities that have not yet been visited and calculates the move to the next city, exploring all possibilities. The distances between cities are accumulated by calling `tsp()` for the entire route distance.

Conclusion

In this tutorial, we explored the Travelling Salesman Problem in detail. We discussed how to efficiently solve it using bitmasking and DP while writing a Swift code. I hope this problem, which frequently appears in coding tests, enhances your algorithm problem-solving skills.

I wish you the best in your coding journey! Thank you.