711975. Minimum travel time
Minimum Travel Time
Summary
The problem "Minimum travel time" is a classic algorithmic challenge that requires us to find the minimum time it takes to visit a set of cities. The key concept involved in this problem is graph theory, specifically finding the shortest path between nodes.
Detailed Explanation
We are given a list of cities and their corresponding coordinates (x, y). We need to find the minimum travel time required to visit each city exactly once and return to the starting point. This can be modeled as a traveling salesman problem (TSP).
To solve this problem, we can use a greedy algorithm that involves selecting the closest unvisited city at each step. The algorithm starts by choosing an arbitrary city as the first stop. Then, it repeatedly selects the closest unvisited city until all cities have been visited.
Here is a step-by-step breakdown of the solution:
- Initialize the current city and set it to the starting point.
- Create a list of unvisited cities.
- While there are still unvisited cities:
- Calculate the distance from the current city to each unvisited city using the Euclidean distance formula (sqrt((x2-x1)^2 + (y2-y1)^2)).
- Choose the closest unvisited city and set it as the new current city.
- Once all cities have been visited, calculate the total travel time by summing up the distances between each pair of consecutive cities.
Time complexity: O(n^2), where n is the number of cities. Space complexity: O(1), since we only need a constant amount of space to store the current city and unvisited cities list.
Optimized Solutions
Java
public class MinimumTravelTime {
public static double calculate(int[][] coordinates) {
int n = coordinates.length;
double totalDistance = 0.0;
boolean[] visited = new boolean[n];
for (int i = 0; i < n; i++) {
visited[i] = false;
}
int currentCity = 0;
while (true) {
visited[currentCity] = true;
int minDistance = Integer.MAX_VALUE;
int nextCity = -1;
for (int j = 0; j < n; j++) {
if (!visited[j]) {
double distance = calculateDistance(coordinates, currentCity, j);
if (distance < minDistance) {
minDistance = distance;
nextCity = j;
}
}
}
totalDistance += minDistance;
currentCity = nextCity;
}
return totalDistance;
}
private static double calculateDistance(int[][] coordinates, int city1, int city2) {
return Math.sqrt(Math.pow(coordinates[city2][0] - coordinates[city1][0], 2) +
Math.pow(coordinates[city2][1] - coordinates[city1][1], 2));
}
}