#1376

Medium

auto_awesomeBinary-tree traversal and state tracking

LeetCode Problem Workspace

Time Needed to Inform All Employees

Calculate the time needed for the head of a company to inform all employees using tree traversal techniques.

tree depth first search breadth first search

Problem Statement

You are given a company structure with n employees, where each employee has a unique ID from 0 to n-1. The head of the company is identified by the headID, and each employee has a direct manager as given by the manager array. The manager of each employee is denoted by manager[i], with manager[headID] = -1 indicating that the head has no manager. The company structure forms a tree where every non-head employee has exactly one manager.

The head of the company wants to inform all employees about urgent news. Starting from the head, they will inform their subordinates, and those subordinates will inform their own subordinates, continuing in this fashion. The time each employee takes to inform their direct subordinates is given by the informTime array. Your task is to determine how long it takes for the news to reach all employees.

Examples

Example 1

Input: n = 1, headID = 0, manager = [-1], informTime = [0]

Output: 0

The head of the company is the only employee in the company.

Example 2

Input: n = 6, headID = 2, manager = [2,2,-1,2,2,2], informTime = [0,0,1,0,0,0]

Output: 1

The head of the company with id = 2 is the direct manager of all the employees in the company and needs 1 minute to inform them all. The tree structure of the employees in the company is shown.

Constraints

1 <= n <= 105
0 <= headID < n
manager.length == n
0 <= manager[i] < n
manager[headID] == -1
informTime.length == n
0 <= informTime[i] <= 1000
informTime[i] == 0 if employee i has no subordinates.
It is guaranteed that all the employees can be informed.

Solution Approach

Tree Representation and Depth-First Search (DFS)

The company's structure is a tree, where the head is the root. Perform a Depth-First Search (DFS) starting from the headID, calculating the time taken for each employee to inform their subordinates by propagating the informTime values. The total time will be the longest path from the head to any leaf node in this tree structure.

Breadth-First Search (BFS) for Level-wise Propagation

Alternatively, use a Breadth-First Search (BFS) approach where you start from the head and propagate the information level by level. Track the maximum time at each level of propagation, and the overall time will be the longest time taken for any employee to be informed.

State Tracking and Optimization

By maintaining an array of the maximum inform time for each employee, and updating this as you traverse the tree, you can optimize the solution to handle the large constraints efficiently. Whether using DFS or BFS, careful state tracking ensures you minimize redundant calculations.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the traversal approach used. Both DFS and BFS will have a time complexity of O(n), as they need to visit each employee once. The space complexity will also be O(n) due to the need to store the employee structure and track the inform times.

What Interviewers Usually Probe

Look for the candidate's ability to represent hierarchical structures as trees.
Assess their understanding of tree traversal methods, such as DFS and BFS.
Evaluate their optimization skills, particularly in handling large input sizes.

Common Pitfalls or Variants

Common pitfalls

Confusing the order of traversal can lead to incorrect timing calculations.
Forgetting to account for employees with no subordinates, whose inform time is 0.
Incorrectly propagating the inform time across levels or subordinates.

Follow-up variants

Modify the problem to include multiple heads of different departments.
Consider cases where some employees may already know the information.
Extend the problem to handle employees who can inform more than one subordinate simultaneously.

FAQ

How do I calculate the time needed to inform all employees?

You calculate the time by simulating the process using either Depth-First Search (DFS) or Breadth-First Search (BFS), starting from the head and considering the time it takes each employee to inform their subordinates.

What is the tree traversal method for solving the Time Needed to Inform All Employees problem?

Both Depth-First Search (DFS) and Breadth-First Search (BFS) are valid approaches for solving the problem, as the company's structure is a tree.

How do I handle the case where employees have no subordinates?

Employees with no subordinates will have an inform time of 0. Ensure this is handled when calculating the maximum time required for information to spread.

Can I solve the problem using both DFS and BFS?

Yes, both DFS and BFS are valid approaches, though the implementation details may vary. The key is tracking the maximum time required to inform all employees.

What is the time complexity for solving the Time Needed to Inform All Employees problem?

The time complexity is O(n) for both DFS and BFS, where n is the number of employees, as both algorithms visit each employee exactly once.

terminal

Solution

Solution 1: DFS

We first build an adjacent list $g$ according to the $manager$ array, where $g[i]$ represents all direct subordinates of employee $i$.

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class Solution:
    def numOfMinutes(
        self, n: int, headID: int, manager: List[int], informTime: List[int]
    ) -> int:
        def dfs(i: int) -> int:
            ans = 0
            for j in g[i]:
                ans = max(ans, dfs(j) + informTime[i])
            return ans

        g = defaultdict(list)
        for i, x in enumerate(manager):
            g[x].append(i)
        return dfs(headID)

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