Author: [Your Name]
Date: [Date]
1. Problem Overview
One of the algorithm problems frequently addressed in the hiring process is ‘Planning a Trip’. This problem involves selecting travel destinations while meeting various given conditions.
Through this process, we develop our ability to efficiently solve problems using data structures and algorithms. In this course, we will detail the process of solving this algorithm problem using the C# language.
2. Problem Description
Problem:
You are planning a trip. You have information about several cities and the distances between those cities.
Write a program to find the travel route that visits all cities and returns to the starting city with the least distance.
The cities are given in the form of a graph, and the distances between each city are represented in an array.
Input:
– Number of cities N (1 ≤ N ≤ 10)
– An N x N array indicating the distances between cities
– Starting city (0 ≤ starting city < N)
Output:
– The distance of the route that visits all cities with the minimum distance and returns to the starting city
3. Problem Analysis
This problem is a typical Traveling Salesman Problem (TSP), which is NP-hard.
It concerns finding the shortest route that visits all cities and returns to the starting city, and all cases can be explored using a brute force algorithm.
For a small number of cities (N ≤ 10), it is feasible to check all combinations, so we can solve the problem using combinations.
For this, we will use dynamic programming with recursion and bit masking techniques.
4. Algorithm Design
Our goal is to find the shortest path based on the given city and distance information.
We can design the algorithm in the following steps:
1. Process the input data to define the distance array and the number of cities.
2. Use recursive language to explore all possible paths.
3. Calculate the distances of each path to update the shortest distance information.
4. Compare the distance of the path that returns to the starting city after visiting all cities with the minimum distance to derive the result.
C# Code to Implement:
5. C# Code Implementation
using System;
class TravelPlan
{
static int N;
static int[,] distance;
static int minDistance = int.MaxValue;
static void Main(string[] args)
{
N = Convert.ToInt32(Console.ReadLine());
distance = new int[N, N];
for (int i = 0; i < N; i++)
{
var inputs = Console.ReadLine().Split();
for (int j = 0; j < N; j++)
{
distance[i, j] = Convert.ToInt32(inputs[j]);
}
}
int startCity = 0;
bool[] visited = new bool[N];
visited[startCity] = true;
FindShortestPath(startCity, visited, 0, 0, 1);
Console.WriteLine(minDistance);
}
static void FindShortestPath(int currentCity, bool[] visited, int currentDistance, int count)
{
if (count == N)
{
currentDistance += distance[currentCity, 0];
minDistance = Math.Min(minDistance, currentDistance);
return;
}
for (int nextCity = 0; nextCity < N; nextCity++)
{
if (!visited[nextCity] && distance[currentCity, nextCity] > 0)
{
visited[nextCity] = true;
FindShortestPath(nextCity, visited, currentDistance + distance[currentCity, nextCity], count + 1);
visited[nextCity] = false;
}
}
}
}
The code above takes the number of cities and the distance information between each city as input and outputs the minimum distance after exploring all paths recursively starting from the starting city.
It records the visitation status of each city while checking if all cities have been visited to update the result.
6. Code Analysis
Looking at the code, the FindShortestPath method explores paths moving to all unvisited cities from the current city.
In this process, the visited array is used to track visited cities. When all cities have been visited,
it calculates the distance of the path returning to the starting city and updates the minimum distance information.
This algorithm examines all possible paths through recursive calls and backtracking.
7. Test Cases
We can write several test cases to verify this algorithm.
For example, we can test with input like the following:
Input:
4
0 10 15 20
10 0 35 25
15 35 0 30
20 25 30 0
Output:
Minimum Distance: 80
This input data represents the distances between each city, and the expected output is 80.
This corresponds to the shortest route: 0 → 1 → 3 → 2 → 0.
8. Optimization Suggestions
The algorithm above is a simple implementation; however, it becomes inefficient as N increases due to the substantial increase in time complexity.
Therefore, we can consider methods to improve performance, such as using memoization to store results of subproblems.
For example, using bit masking and dynamic programming to pre-compute each state and store results can significantly reduce execution time by minimizing consecutive recursive calls.
9. Conclusion
In this course, we dealt with solving the problem of planning a trip using C#.
Through this process, you can enhance your algorithm problem-solving abilities and also learn optimization techniques utilizing memoization.
We hope you can further develop your logical thinking and coding skills by solving various algorithm problems.