Swift Coding Test Course, Salesman's Concerns

Problem Definition

The ‘Salesman’s Dilemma’ problem seeks to find the minimum route for a salesman to visit all given cities
once and return to the starting city, based on the distance information between the cities. This is an
optimization problem known as the Traveling Salesman Problem (TSP), which is
a form of graph theory. This problem is known to be NP-complete, and there are various methods to solve it.

Problem Description

The following conditions are given:

There are n cities.
Each city is represented by an integer from 1 to n.
The distances between the cities are given in a directed graph form, and the distance information is represented in a two-dimensional array.
The salesman must visit all cities once and return to the starting city.

Input Format

The first line contains the number of cities, n. The following n lines contain the distance matrix between cities.
The distance matrix is constructed as follows:

In the example above, the first line indicates there are 4 cities, and it represents the distances between each city.
For example, the distance between city 1 and city 2 is 10, and the distance between city 1 and city 3 is 15.

Output Format

Print the total distance of the minimum route.

Problem Solving Approach

This problem can be approached in various ways, but here we will describe a method using backtracking and dynamic programming (DP).
While backtracking can be used to explore all routes and find the minimum path, the computational workload increases exponentially as the number of cities grows.
Therefore, combining dynamic programming with memoization can enhance efficiency.

Algorithm Implementation

Initially, we can implement the solution by generating all possible paths using backtracking with permutations, and then calculating the distance of each path to update the minimum value.
Below is an example code implemented in Swift.

        func tsp(graph: [[Int]], visited: inout [Bool], currentIndex: Int, count: Int, cost: Int, ans: inout Int) {
            // If all cities have been visited
            if count == graph.count && graph[currentIndex][0] > 0 {
                ans = min(ans, cost + graph[currentIndex][0])
                return
            }
            for i in 0.. 0 {
                    visited[i] = true
                    tsp(graph: graph, visited: &visited, currentIndex: i, count: count + 1, cost: cost + graph[currentIndex][i], ans: &ans)
                    visited[i] = false
                }
            }
        }

        func findMinCost(graph: [[Int]]) -> Int {
            var visited = [Bool](repeating: false, count: graph.count)
            var ans = Int.max
            visited[0] = true
            tsp(graph: graph, visited: &visited, currentIndex: 0, count: 1, cost: 0, ans: &ans)
            return ans
        }

        let graph = [
            [0, 10, 15, 20],
            [10, 0, 35, 25],
            [15, 35, 0, 30],
            [20, 25, 30, 0]
        ]

        let result = findMinCost(graph: graph)
        print("Distance of the minimum route: \(result)")

Code Explanation

The code above is structured as follows:

tsp Function: Recursively explores all paths visiting the cities,
traversing all unvisited cities from the current city. If the cost of visiting a city is minimal,
it updates the minimum cost of the current path.
findMinCost Function: Acts as an initializer and calls the tsp function
while visiting the first city (0) to explore the entire path.

Complexity Analysis

The time complexity of this algorithm is O(n!). This is because it has to traverse each city and explore all paths.
It can be used when n is small, but it becomes inefficient as n increases.
Therefore, for more efficient methods, dynamic programming can be applied using bit masking to reduce complexity.
This method has a time complexity of O(n^2 * 2^n) and is practical for n below 20.

Conclusion

The ‘Salesman’s Dilemma’ problem is closely related to various optimization problems in reality and provides important insights when solving problems through effective algorithm design.
By tackling problems like this while preparing for coding tests, one can enhance their understanding of graph algorithms and dynamic programming, as well as develop systematic problem-solving skills.

Swift Coding Test Course, Salesman’s Concerns