Problem Description
You are given two groups of points where the first group has size1 points, the second group has size2 points, and size1 >= size2. The cost of the connection between any two points are given in a size1 x size2 matrix where cost[i][j] is the cost of connecting point i of the first group and point j of the second group. The groups are connected if each point in both groups is connected to one or more points in the opposite group. Return the minimum cost it takes to connect the two groups.
Key Insights
- Each point in the first group must be connected to at least one point in the second group, and vice versa.
- The problem can be approached using dynamic programming with bit manipulation to represent the connections.
- A bitmask can be used to represent different subsets of points in the second group.
Space and Time Complexity
Time Complexity: O(2^(size2) * size1)
Space Complexity: O(2^(size2))
Solution
The solution uses dynamic programming along with bit manipulation to efficiently compute the minimum cost. The idea is to define a DP array dp[mask] where mask is a bitmask representing the set of points in the second group that have been connected. The initial state of the DP array is set to infinity, and the base case is dp[0] = 0 (no connections made). We then iterate through all possible masks and update the DP array based on the costs of connecting points from the first group to the newly added points in the second group. The final result is found by taking the minimum value from the DP array that represents all points in the second group connected.