Maximum Number of Achievable Transfer Requests

Problem Description

We have n buildings numbered from 0 to n - 1. Each building has a number of employees. It’s transfer season, and some employees want to change the building they reside in.

You are given an array requests where requests[i] = [from_i, to_i] represents an employee's request to transfer from building from_i to building to_i.

All buildings are full, so a list of requests is achievable only if for each building, the net change in employee transfers is zero. This means the number of employees leaving is equal to the number of employees moving in.

Return the maximum number of achievable requests.

Key Insights

• Each request affects two buildings: one that loses an employee and one that gains.
• The sum of employees leaving and entering each building must balance out to zero for the requests to be achievable.
• We can utilize backtracking or bit manipulation to explore subsets of requests to find the maximum number of achievable requests.
• Given the constraints (n <= 20 and requests.length <= 16), a brute-force approach is feasible.

Space and Time Complexity

Time Complexity: O(2^m * n), where m is the number of requests and n is the number of buildings, due to exploring all subsets of requests.

Space Complexity: O(n) for maintaining the state of employee transfers in each building during the exploration.

Solution

To solve the problem, we can use a backtracking approach. We will explore all possible combinations of the requests and check if they can be fulfilled by ensuring that the net change in each building is zero. The algorithm proceeds as follows:

1. Iterate through all possible subsets of requests.
2. For each subset, maintain an array to track the net changes in employees for each building.
3. For each request in the subset, update the net changes.
4. After processing each subset, check if all buildings have a net change of zero.
5. Keep track of the maximum size of the valid subsets found.

This approach effectively evaluates all combinations of requests while ensuring that the conditions for a valid transfer are met.

Code Solutions