Solution Explanation: Maximum Number of Accepted Invitations

This problem can be efficiently solved using the Hungarian Algorithm, a combinatorial optimization algorithm that finds a maximum matching in a bipartite graph. A bipartite graph is a graph whose vertices can be divided into two disjoint sets, such that every edge connects a vertex in one set to a vertex in the other set. In this problem:

• Boys represent one set of vertices.
• Girls represent the other set of vertices.
• An edge exists between a boy and a girl if grid[i][j] == 1 (the i-th boy can invite the j-th girl).

The goal is to find the maximum number of edges (invitations) such that no boy invites more than one girl and no girl receives more than one invitation. This is precisely what the Hungarian algorithm solves.

Algorithm (Hungarian Algorithm):

The Hungarian algorithm iteratively searches for augmenting paths. An augmenting path is a path in the graph that starts at an unmatched boy and ends at an unmatched girl, alternating between unmatched boys and girls. Finding such a path allows us to increase the number of matched pairs (invitations).

The algorithm follows these steps:

1. Initialization: Create a match array to store the girl each boy is matched with (-1 indicates unmatched).

2. Iteration: For each boy:

   • Create a vis (visited) set to track visited girls during the search for an augmenting path.
   • Call a recursive function find(i) to search for an augmenting path starting from boy i.

3. Recursive find(i) function:

   • For each girl j that boy i can invite (grid[i][j] == 1):
     • If the girl hasn't been visited (j not in vis):
       • Mark the girl as visited (vis.add(j)).
       • If the girl is unmatched (match[j] == -1) or if we can find an augmenting path from the boy currently matched with this girl (find(match[j])):
         • Match the boy i with the girl j (match[j] = i).
         • Return True (augmenting path found).
   • Return False (no augmenting path found from this boy).

4. Counting Matches: After iterating through all boys, the number of matched pairs (ans) represents the maximum number of accepted invitations.

Time Complexity Analysis

The Hungarian algorithm has a time complexity of approximately O(m*n) in its optimized implementations, where 'm' is the number of boys and 'n' is the number of girls. This is because in the worst case, the algorithm might explore all possible edges in the bipartite graph. In practice, it often performs much faster due to the efficiency of augmenting path searches.

Space Complexity Analysis

The space complexity is O(m + n), primarily due to the match array and the vis set (in some implementations). These data structures store information about the matching and visited nodes during the search for augmenting paths.

Code Examples (with explanations):

The code examples provided earlier demonstrate the Hungarian Algorithm in Python, Java, C++, and Go. The structure and logic are very similar across languages; the main differences lie in syntax and data structure implementations. The core idea of iteratively finding augmenting paths remains consistent.