Problem Description

We are given n buildings, and an array of employee transfer requests between these buildings. Each request is represented by a pair [from, to], meaning an employee wants to transfer from one building to another. A request array is deemed achievable if the transfers can happen without altering the total number of employees in each building; that is to say, the incoming employees must balance the outgoing ones for every building. The objective is to return the maximum number of such achievable transfer requests.

To put it simply, we have to find the largest subset of the given requests that can be satisfied simultaneously, ensuring that every building ends up with the same number of employees it started with.

Flowchart Walkthrough

First, let's pin down the algorithm using the Flowchart. Here's a step-by-step walkthrough:

1. Is it a graph?

   • No: The problem does not involve graph-based relationships like nodes directly connected by edges. It's more about requests and state changes.

2. Need to solve for kth smallest/largest?

   • No: The problem is about maximizing the number of achievable requests, not finding an order-based kth element.

3. Involves Linked Lists?

   • No: This problem does not involve linear data structures like linked lists.

4. Does the problem have small constraints?

   • Yes: The constraints on the number of requests and buildings are typically small enough to consider trying all combinations, which makes brute force or backtracking feasible.

5. Brute force / Backtracking?

   • Yes: Given the problem's nature of permutations and combinations of requests that can be granted under certain constraints, backtracking is ideal to explore all possible configurations to maximize the achievability.

Conclusion: The flowchart suggests using the backtracking approach to try different combinations of transfer requests and find the maximum number that can be achieved given the constraints.

Intuition

The solution approach is based on the idea of checking every possible combination of requests, from no requests being satisfied to all of them being considered. To do this efficiently, a bitmask representation is utilized.

A bitmask approach involves creating a mask for each possible subset of requests. Each bit in the mask corresponds to a decision whether to include or exclude a particular request. The total number of bitmasks to check will be 2^len(requests), because that's how many subsets are possible.

The intuition behind using a bitmask is that it allows us to efficiently iterate over all subsets of requests, including the empty set and the set containing all requests. By incrementally checking each bitmask, we determine whether that particular combination of requests leads to a balanced transfer where each building's employee count remains unchanged.

Checking a mask involves updating a list of net change in employees for each building (cnt). If a request is included in the mask (indicated by the corresponding bit being 1), we decrement the employees count from the from building and increment it in the to building. At the end of this process, we check if all buildings' counts are zero. If they are, it means that particular combination is achievable.

The final step keeps track of the maximum number of requests that form a valid set (ans). This is done by comparing the number of requests in the current mask (cnt) with the highest number found so far. If the current mask is both a larger set and a valid transfer set, it updates the maximum found.

This brute force method ensures that all possible request combinations are checked, and the largest valid subset is found.

Learn more about Backtracking patterns.

Solution Approach

The implementation of the solution employs a brute force approach with a bit manipulation technique to iterate through all possible subsets of the transfer requests.

Here’s the step-by-step breakdown of the code:

1. Define a helper function check(mask: int) -> bool:

   • This function takes a bitmask as an argument, representing a subset of requests.
   • It creates an array cnt of size n, initialized with zeros. The array represents the net change in the number of employees in each building.
   • The function iterates over all requests and for each request, checks if the corresponding bit in the mask is set to 1. If it is, it means the request is included in the current subset and the function updates the cnt array by decrementing the count for the from building and incrementing it for the to building.
   • After iterating through all requests, it checks if all buildings have a net change of zero, and returns True if they do, indicating that the subset is achievable.

2. Initialize a variable ans with 0 to keep track of the maximum number of achievable requests found.

3. Iterate through all possible masks/subsets using a for-loop:

   • The range of the loop is 1 << len(requests), which gives us all possible combinations of including or excluding each request.
   • The variable mask represents the current subset of requests being considered.

4. Inside the loop, calculate the number of requests included in the current subset using mask.bit_count(), which returns the number of set bits (or 1s) in the bitmask.

5. If the current mask has more requests than ans (the previous maximum), and the check(mask) function confirms that the current subset is achievable, update ans with the count of the current subset.

6. After considering all possible subsets, return ans as the final result. This value represents the maximum number of requests that can be accommodated without changing the number of employees in each building.

Overall, the key data structures used in this approach are:

• An integer mask to represent a subset of requests and facilitate bit manipulation.
• An array cnt for tracking the net change in employees for each building within a subset.

This algorithm makes use of combinatorial logic (to generate all possible subsets of requests) and bitwise operations (to manage and evaluate these subsets efficiently). The time complexity of this approach is O(2^m * n) where m is the number of requests and n is the number of buildings, since for each of the 2^m subsets, we perform n operations to validate it.

Example Walkthrough

Let's take a small example to illustrate the solution approach.

Suppose we have n = 3 buildings and a list of 4 employee transfer requests, as follows:

requests = [[0, 1], [1, 2], [0, 2], [2, 0]]

Here's a step-by-step walkthrough using a bitmask approach for the given requests:

1. Initialize Maximum Achievable Requests (ans):

   We start by setting ans to 0, as we have not processed any requests yet.

2. Iterate Through All Possible Subsets:

   With 4 requests, there are 2^4 = 16 possible combinations of these requests, from no requests (bitmask 0000) to all requests (bitmask 1111).

3. Evaluate Each Subset (Bitmask):

   For each bitmask, we perform the following steps:

   • Bitmask 0000: No requests are included, thus ans remains 0.
   • Bitmask 0001: The subset includes just the last request [2, 0]. This is achievable as we can transfer one employee from building 2 to 0.
   • Bitmask 0010: The subset includes the request [0, 2]. This is also achievable in isolation, but ans remains at 1 because we already found a subset of 1 request.
   • ...and so on for each combination...
   • Bitmask 1011: This subset includes requests [0, 1], [1, 2], and [0, 2]. Upon checking, the count for each building after these transfers would be 0, so the subset is achievable. Since this subset has 3 transfers, ans is updated to 3.

4. Check Each Subset For Balance (Using check Function):

   When we evaluate each set, the check function will update an array cnt that tracks the net change in number of employees at each building, considering which requests are included in the current subset. If all entries in cnt are 0, it means that the current subset of requests is balanced and thus achievable.

5. Find Maximum Number of Achievable Requests:

   As we evaluate all possible subsets, we use the ans variable to keep track of the greatest number of requests in a balanced subset encountered so far. In our example, upon checking all subsets, the maximum achievable number is 3, which would be the final answer.

For this example, the final ans represents that there is a subset of 3 transfer requests which can be satisfied simultaneously, maintaining the balance of employees in all buildings.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code is primarily determined by two nested operations:

1. The outer loop, which iterates over all possible subsets of requests. There are len(requests) requests, and for each request, there are two possibilities (either the request is fulfilled or it is not). Therefore, there are 2^(len(requests)) possible subsets, resulting in a time complexity of O(2^m) for this loop, where m is the length of requests.

2. The inner function check(mask), which is called for each subset to verify whether choosing a particular subset of requests satisfies the balance criterion (every building ends up with the same number of people). This function iterates through all requests and then all buildings to ensure balance, contributing a complexity of O(m + n), where n is the number of buildings.

The combined effect of these operations leads to a total time complexity of O(2^m * (m + n)).

Space Complexity

The space complexity of the provided code can be analyzed by looking at the additional memory used by the algorithm. The key factors influencing space complexity here are:

1. The counter array cnt for building balances, which uses O(n) space, where n is the number of buildings.

2. The bit mask, which does not add significant space complexity, as it is just an integer value storing the current subset being checked.

Thus, the overall space complexity of the algorithm is O(n), as it's primarily dependent on the number of buildings to store the balance counter for each building.

Learn more about how to find time and space complexity quickly using problem constraints.