2019 Dec Silver Problem 3 Milk Visits

Problem Statement

The farmer John has a farm with N cows (1 ≤ N ≤ 10^5). He wants to visit each cow exactly once and return to his house. He has a map of the farm, which contains N points: the house, the cows, and possibly some other points.

John can move between any two points on the map in a straight line. He can also move from one point to another and back again in a single move. The time it takes to move between two points is the Euclidean distance between them.

John wants to find the shortest route that visits each cow exactly once and returns to his house.

Solution

The solution to this problem can be found using the Travelling Salesman Problem (TSP) algorithm. This algorithm can be used to find the shortest route that visits each cow exactly once and returns to the house.

In C++, the algorithm can be implemented as follows:

#include <vector>
#include <algorithm>

using namespace std;

// Function to find the shortest route
// that visits each cow exactly once
// and returns to the house
int tsp(vector<vector<int>>& graph, vector<bool>& visited, int currPos, int n, int count, int cost)
{
    // If all cows are visited
    if (count == n)
    {
        // Return cost of the route
        return cost + graph[currPos][0];
    }

    // Initialize result
    int res = INT_MAX;

    // Try all unvisited cows
    for (int i = 0; i < n; i++)
    {
        if (!visited[i])
        {
            // Mark cow as visited
            visited[i] = true;

            // Calculate cost of the route
            int newCost = cost + graph[currPos][i];

            // Recur for remaining cows
            res = min(res, tsp(graph, visited, i, n, count + 1, newCost));

            // Mark cow as unvisited
            visited[i] = false;
        }
    }

    return res;
}

int main()
{
    // Number of cows
    int n = 5;

    // Adjacency matrix
    vector<vector<int>> graph = { { 0, 10, 15, 20 },
                                  { 10, 0, 35, 25 },
                                  { 15, 35, 0, 30 },
                                  { 20, 25, 30, 0 } };

    // Visited array to keep track
    // of visited cows
    vector<bool> visited(n);

    // Initialize result
    int res = INT_MAX;

    // Find the shortest route
    res = tsp(graph, visited, 0, n, 0, 0);

    // Print result
    cout << res << endl;

    return 0;
}