Swift Coding Test Course, Creating a Traveling Salesman Route

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.