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:

<syntaxhighlight lang="cpp">

1. include <vector>
2. 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;

} </syntaxhighlight>