Problem Description
There is a salesman with N cities, and this salesman must visit all cities once and return to the starting city.
The distance information between each city is given, and the goal is to find the shortest path.
To solve this problem, algorithms for shortest path exploration can be used, such as backtracking, dynamic programming, or bitmasking.
Input Format
The first line contains the number of cities N. Following this, an N x N matrix is given,
where matrix[i][j] represents the distance from city i to city j.
Note that matrix[i][i] is always 0.
Output Format
Output the minimum cost for the salesman to visit all cities once and return to the starting point.
Example
Input:
4
0 10 15 20
10 0 35 25
15 35 0 30
20 25 30 0
Output:
80
Problem Solving Process
This problem is a classic Traveling Salesman Problem (TSP), which involves finding the shortest path for the salesman to visit all cities and return to the starting point.
This problem is NP-complete, meaning that as the number of cities increases, the computational requirements to find a solution grow exponentially.
Therefore, various approaches can be considered.
1. Backtracking
Backtracking is a method that explores all paths recursively, searching for the optimal path. However, as the number of cities increases, the computation time also increases.
2. Dynamic Programming
The dynamic programming approach utilizes memoization techniques to store already computed shortest distances for paths, preventing redundant calculations.
Following this methodology, the minimum cost for specific sets of cities is stored, and sub-problems are solved to construct a solution for the overall problem.
3. Bitmasking
Bitmasking is a technique that represents each city with a bit to manage all visit states efficiently.
It allows management of which cities the salesman has visited effectively.
This method is particularly effective when combined with dynamic programming.
Algorithm Implementation
1. Dynamic Programming Using Bitmasking
import java.util.Arrays;
public class TravelingSalesman {
static final int INF = Integer.MAX_VALUE;
static int[][] dist;
static int[][] dp;
static int n;
public static void main(String[] args) {
// Example Input
n = 4; // Number of cities
dist = new int[][] {
{0, 10, 15, 20},
{10, 0, 35, 25},
{15, 35, 0, 30},
{20, 25, 30, 0}
};
// Initialize DP array
dp = new int[n][1 << n];
for (int[] row : dp) {
Arrays.fill(row, -1);
}
// Calculate shortest path
int result = tsp(0, 1);
System.out.println("Minimum cost: " + result);
}
static int tsp(int pos, int mask) {
if (mask == (1 << n) - 1) { // All cities visited
return dist[pos][0]; // Return to starting city
}
if (dp[pos][mask] != -1) {
return dp[pos][mask]; // Return memoized value
}
int ans = INF;
// Loop through all cities
for (int city = 0; city < n; city++) {
if ((mask & (1 << city)) == 0) { // If the city has not been visited
ans = Math.min(ans, dist[pos][city] + tsp(city, mask | (1 << city)));
}
}
return dp[pos][mask] = ans; // Memoize current state
}
}
Conclusion
This problem represents a classic route optimization problem where the salesman must visit each city once and return to the starting city, allowing for various algorithmic approaches.
The implementation utilizing bitmasking and dynamic programming allows for quick calculations when the number of cities is small and can also accommodate other techniques for handling multiple cities.
Developing algorithmic thinking through such problems is essential, as it will be very useful for coding interviews and actual projects.
References
- Introduction to Algorithms, by Thomas H. Cormen et al.
- Online Resources on Dynamic Programming and Traveling Salesman Problem