#1601

Hard

auto_awesomeBacktracking search with pruning

LeetCode Problem Workspace

Maximum Number of Achievable Transfer Requests

Find the maximum number of achievable transfer requests between buildings with constraints on net employee transfers.

array backtracking bit manipulation enumeration

Problem Statement

You are given a set of buildings and a list of transfer requests between them. Each request indicates a transfer from one building to another. The challenge is to find the maximum number of transfer requests that can be successfully executed while satisfying the condition that each building has a net zero transfer (i.e., the number of employees leaving equals the number of employees arriving).

This problem can be approached using backtracking and pruning techniques, where the search space is reduced by eliminating infeasible transfer combinations early in the process. With constraints on building capacity and the nature of the problem, this problem tests your ability to manage computational complexity while trying to maximize the number of feasible transfers.

Examples

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 details omitted.

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 details omitted.

Example 3

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

Output: 4

Example details omitted.

Constraints

1 <= n <= 20
1 <= requests.length <= 16
requests[i].length == 2
0 <= fromi, toi < n

Solution Approach

Backtracking with Pruning

To solve this problem, a backtracking approach can be used to explore all potential transfer combinations. For each request, we try to either include or exclude it from the solution space. Pruning helps cut off branches that lead to infeasible solutions, such as when the net transfer in any building becomes non-zero.

Bit Manipulation for Efficient State Tracking

Bit manipulation helps efficiently represent and track the current state of employee transfers between buildings. Each building's net transfer can be stored as a bitmask, which simplifies the checking of valid solutions and aids in quickly identifying infeasible combinations.

Optimization through Branch Cutting

As the problem is NP-hard, optimizing the search process is crucial. Branch cutting is done when a building's transfer net becomes unbalanced, reducing the number of iterations needed to explore all combinations.

Complexity Analysis

Metric	Value
Time	O(2^(M * (M + N))
Space	O(N)

The time complexity is O(2^(M * (M + N))) due to the combination of backtracking and pruning. The space complexity is O(N), where N represents the number of buildings. Efficient pruning and bitmasking contribute to managing the large number of possible combinations.

What Interviewers Usually Probe

Candidate demonstrates understanding of backtracking with pruning techniques.
Candidate effectively utilizes bit manipulation for optimization.
Candidate identifies early pruning conditions to limit unnecessary exploration.

Common Pitfalls or Variants

Common pitfalls

Failure to implement proper pruning leads to excessive exploration of infeasible branches.
Not optimizing state representation, which can lead to slow performance.
Overlooking the importance of checking net transfers early during the backtracking process.

Follow-up variants

Consider optimizing the pruning strategy to handle larger inputs.
Explore dynamic programming approaches to compare with backtracking.
Investigate greedy methods for approximate solutions in specific cases.

FAQ

What is the maximum number of achievable transfer requests in the "Maximum Number of Achievable Transfer Requests" problem?

The problem asks for the maximum number of transfer requests that can be fulfilled while ensuring that each building has a net zero transfer, meaning the number of employees leaving is equal to the number arriving.

How does backtracking help solve the problem?

Backtracking is used to explore all possible combinations of transfer requests. By pruning branches that lead to invalid solutions (like imbalanced transfers), we efficiently narrow down the search space.

What is pruning in the context of this problem?

Pruning involves eliminating infeasible solutions during the backtracking process, such as when the net transfer for any building becomes unbalanced. This saves computation time by avoiding unnecessary exploration of invalid paths.

How can bit manipulation be used in solving this problem?

Bit manipulation allows for efficient tracking of the state of employee transfers across buildings. Each building's net transfer can be represented as a bitmask, which helps quickly check if a solution is valid.

What is the time complexity of the "Maximum Number of Achievable Transfer Requests" problem?

The time complexity is O(2^(M * (M + N))), where M is the number of transfer requests and N is the number of buildings. This arises due to the backtracking approach with pruning, which explores all possible transfer combinations.

terminal

Solution

Solution 1: Binary Enumeration

We note that the length of the room change request list does not exceed $16$. Therefore, we can use the method of binary enumeration to enumerate all room change request lists. Specifically, we can use a binary number of length $16$ to represent a room change request list, where the $i$-th bit being $1$ means the $i$-th room change request is selected, and $0$ means the $i$-th room change request is not selected.

class Solution:
    def maximumRequests(self, n: int, requests: List[List[int]]) -> int:
        def check(mask: int) -> bool:
            cnt = [0] * n
            for i, (f, t) in enumerate(requests):
                if mask >> i & 1:
                    cnt[f] -= 1
                    cnt[t] += 1
            return all(v == 0 for v in cnt)

        ans = 0
        for mask in range(1 << len(requests)):
            cnt = mask.bit_count()
            if ans < cnt and check(mask):
                ans = cnt
        return ans

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