Maximum Number of Achievable Transfer Requests

Hard

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Bit ManipulationRecursion

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. For example if n = 3 and two employees are leaving building 0, one is leaving building 1, and one is leaving building 2, there should be two employees moving to building 0, one employee moving to building 1, and one employee moving to building 2.

Return the maximum number of achievable requests.

Example 1:

Input: n = 5, requests = [[0,1],[1,0],[0,1],[1,2],[2,0],[3,4]]
Output: 5
Explantion: Let's see the requests:
From building 0 we have employees x and y and both want to move to building 1.
From building 1 we have employees a and b and they want to move to buildings 2 and 0 respectively.
From building 2 we have employee z and they want to move to building 0.
From building 3 we have employee c and they want to move to building 4.
From building 4 we don't have any requests.
We can achieve the requests of users x and b by swapping their places.
We can achieve the requests of users y, a and z by swapping the places in the 3 buildings.

Example 2:

Input: n = 3, requests = [[0,0],[1,2],[2,1]]
Output: 3
Explantion: Let's see the requests:
From building 0 we have employee x and they want to stay in the same building 0.
From building 1 we have employee y and they want to move to building 2.
From building 2 we have employee z and they want to move to building 1.
We can achieve all the requests.

Example 3:

Input: n = 4, requests = [[0,3],[3,1],[1,2],[2,0]]
Output: 4

Constraints:

1 <= n <= 20
1 <= requests.length <= 16
requests[i].length == 2
0 <= from_i, to_i < n

Try coding it in LeetCode

Maximum Number of Achievable Transfer Requests

Hard

Asked by:

11 views

Topics:

Bit ManipulationRecursion

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. For example if n = 3 and two employees are leaving building 0, one is leaving building 1, and one is leaving building 2, there should be two employees moving to building 0, one employee moving to building 1, and one employee moving to building 2.

Return the maximum number of achievable requests.

Example 1:

Input: n = 5, requests = [[0,1],[1,0],[0,1],[1,2],[2,0],[3,4]]
Output: 5
Explantion: Let's see the requests:
From building 0 we have employees x and y and both want to move to building 1.
From building 1 we have employees a and b and they want to move to buildings 2 and 0 respectively.
From building 2 we have employee z and they want to move to building 0.
From building 3 we have employee c and they want to move to building 4.
From building 4 we don't have any requests.
We can achieve the requests of users x and b by swapping their places.
We can achieve the requests of users y, a and z by swapping the places in the 3 buildings.

Example 2:

Input: n = 3, requests = [[0,0],[1,2],[2,1]]
Output: 3
Explantion: Let's see the requests:
From building 0 we have employee x and they want to stay in the same building 0.
From building 1 we have employee y and they want to move to building 2.
From building 2 we have employee z and they want to move to building 1.
We can achieve all the requests.

Example 3:

Input: n = 4, requests = [[0,3],[3,1],[1,2],[2,0]]
Output: 4

Constraints:

1 <= n <= 20
1 <= requests.length <= 16
requests[i].length == 2
0 <= from_i, to_i < n

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the constraints on the number of buildings and the number of transfer requests? Are they small enough that a brute-force approach is not viable?
Are the building numbers guaranteed to be within a specific range, or can they be arbitrarily large?
If no transfer requests can be fulfilled, should I return 0, or is there another specified return value?
Are duplicate transfer requests allowed? If so, does fulfilling one transfer request twice count as two achieved transfers?
Is it possible for a building to request a transfer to itself (i.e., `from` and `to` are the same for a request)? How should I handle such cases?

Brute Force Solution

Approach

The brute force approach involves checking every possible combination of transfer requests. We explore each subset of requests to find the largest set where all transfers can be fulfilled simultaneously.

Here's how the algorithm would work step-by-step:

Consider all possible groups of transfer requests, starting from a group containing no requests at all.
For each group of requests, check if it's possible to fulfill all the requests in that group without violating any building's capacity.
To check if a group is possible, calculate the net change in capacity for each building if those transfers happen.
If any building ends up with a negative capacity after the transfers, the group is not possible, and we discard it.
If all buildings have non-negative capacities after the transfers, the group is possible, and we record the number of requests in that group.
After checking all possible groups, find the group with the largest number of requests that was possible.
The number of requests in that largest possible group is the maximum number of achievable transfer requests.

Code Implementation

def maximum_achievable_transfer_requests(number_of_buildings, requests):
    maximum_achieved_requests = 0

    # Iterate through all possible combinations of requests
    for i in range(1 << len(requests)):
        current_requests = []
        for j in range(len(requests)):
            if (i >> j) & 1:
                current_requests.append(requests[j])

        building_changes = [0] * number_of_buildings

        # Calculate net change for each building.
        for request in current_requests:
            building_changes[request[0]] -= 1
            building_changes[request[1]] += 1

        possible = True

        # Verify buildings never have negative capacities.
        for change in building_changes:
            if change < 0:
                possible = False
                break

        #If the subset is possible, update maximum requests.
        if possible:
            maximum_achieved_requests = max(maximum_achieved_requests, len(current_requests))

    return maximum_achieved_requests

Big(O) Analysis

Time Complexity

O(2^n * m) – The brute force approach considers all possible subsets of transfer requests. With n requests, there are 2^n possible subsets to examine. For each subset, we iterate through all m buildings to calculate the net change in capacity caused by the transfers in that subset. Thus, we have a cost proportional to 2^n for the subsets and m for checking each subset's feasibility against the building capacities. This gives a total time complexity of O(2^n * m).

Space Complexity

O(N) – The primary space complexity stems from calculating the net change in capacity for each building for every subset of requests. This calculation involves creating a temporary array of size N, where N is the number of buildings, to store the changes in capacity. Although it isn't explicitly stated that the balances are stored in an array, logically, to efficiently keep track of building capacity changes during subset evaluations, we assume an array of size N is being utilized. Thus the auxiliary space used is directly proportional to N, giving a space complexity of O(N).

Optimal Solution

Approach

This problem asks us to find the largest set of transfer requests we can fulfill without creating an imbalance. The key idea is to focus on tracking the net change in the number of employees at each building and accepting only request cycles that don't change that balance.

Here's how the algorithm would work step-by-step:

First, calculate the net change in employees for each building. For each transfer request, count one employee leaving the 'from' building and one employee arriving at the 'to' building.
After processing all requests, check if the net change for every building is zero. If it is, all requests can be granted.
If not all requests can be granted, look for cycles of requests where people are essentially being swapped between buildings. Imagine a closed loop where employees are only moving around *within* the loop.
To find the maximum number of achievable requests, first assume we grant *none* of the requests. Then, systematically try granting requests (or combinations of requests) that don't affect the overall balance (net change).
Specifically, try to identify cycles in the requests. For example, if building A sends someone to building B, building B sends someone to building C, and building C sends someone to building A, that's a cycle and can be fulfilled without affecting the balance.
Keep track of the largest number of requests that form valid cycles that can be granted at the same time. This will be your final answer.

Code Implementation

def maximum_requests(number_of_buildings, requests): 
    number_of_requests = len(requests)
    max_achievable_requests = 0

    for i in range(1 << number_of_requests):
        # Determine which requests are included in the current subset.
        current_requests = []
        count = 0
        for j in range(number_of_requests):
            if (i >> j) & 1:
                current_requests.append(requests[j])
                count += 1

        # Calculate the net change in employees for each building.
        net_change = [0] * number_of_buildings
        for start_building, end_building in current_requests:
            net_change[start_building] -= 1
            net_change[end_building] += 1

        # Check if the net change for every building is zero.
        is_valid = all(change == 0 for change in net_change)
        
        # If all requests can be granted, update max_achievable_requests
        if is_valid:
            max_achievable_requests = max(max_achievable_requests, count)

    return max_achievable_requests

Big(O) Analysis

Time Complexity

O(2^n) – Calculating the net change for each building takes O(n) time, where n is the number of transfer requests. The core of the algorithm involves finding cycles among the requests. This is done by considering all possible subsets of the requests to see if they form a cycle, as the problem requires maximizing the number of transfer requests that can be achieved. Since there are 2^n possible subsets of n requests, the time complexity is dominated by the subset generation and checking. This checking each subset will take O(n) as well, but 2^n dominates this value. Therefore, the overall time complexity is O(2^n).

Space Complexity

O(N) – The primary space complexity stems from calculating the net change in employees for each building, which implies creating an auxiliary data structure, such as an array or a hash map, to store these changes. This data structure needs to hold the net change for each of the N buildings, where N represents the number of buildings. Additionally, identifying and storing cycles of requests might require space proportional to the number of buildings or requests in the worst case. Therefore, the auxiliary space used is directly proportional to N, resulting in a space complexity of O(N).

Edge Cases

Case	How to Handle
Null or empty buildings array	Return 0 if the buildings array is null or empty, as there are no requests to process.
Empty requests array	Return 0 if the requests array is empty, as no transfers are being made.
Single building and multiple requests to/from it	The flow tracking mechanism will account for the building's net balance and the maximum achievable is requests that balance each other.
Maximum number of buildings and requests (n=20, requests = n*(n-1))	The algorithm must be efficient enough to handle the maximum input size, considering time and space complexity (e.g., use efficient data structures to prevent time out).
All requests are to the same building	Check balance constraints for each node and if there's no building with a negative balance, increment the count otherwise do not.
All requests are from the same building	The flow tracking mechanism will account for the building's net balance and the maximum achievable is requests that balance each other.
Cyclic dependencies where transfers create cycles of imbalance	The flow tracking will correctly represent a cycle and only valid transfers will be counted.
Integer overflow in calculating building balances	Ensure the data type used for calculating balances (e.g., long) can accommodate the maximum possible net flow for a building to prevent integer overflow.

Null or empty buildings array

How to Handle:

Return 0 if the buildings array is null or empty, as there are no requests to process.

Empty requests array

How to Handle:

Return 0 if the requests array is empty, as no transfers are being made.

Single building and multiple requests to/from it

How to Handle:

The flow tracking mechanism will account for the building's net balance and the maximum achievable is requests that balance each other.

Maximum number of buildings and requests (n=20, requests = n*(n-1))

How to Handle:

The algorithm must be efficient enough to handle the maximum input size, considering time and space complexity (e.g., use efficient data structures to prevent time out).

All requests are to the same building

How to Handle:

Check balance constraints for each node and if there's no building with a negative balance, increment the count otherwise do not.

All requests are from the same building

How to Handle:

The flow tracking mechanism will account for the building's net balance and the maximum achievable is requests that balance each other.

Cyclic dependencies where transfers create cycles of imbalance

How to Handle:

The flow tracking will correctly represent a cycle and only valid transfers will be counted.

Integer overflow in calculating building balances

How to Handle:

Ensure the data type used for calculating balances (e.g., long) can accommodate the maximum possible net flow for a building to prevent integer overflow.