Problem Description

In this problem, we are working with a record of tasks completed by different employees, each of whom has a unique identifier ranging from 0 to n - 1. There is a given 2D integer array named logs, where each element is a pair consisting of an employee id and a leave time. This array essentially tracks which employee finished which task and the time they finished it. The leave times are unique for each task.

Each task begins immediately after the previous one ends, and the first task (0th task) starts at time 0. The goal is to determine the employee id that worked on the task that took the longest time to complete. In case there are tasks that took the same maximum amount of time and were worked on by different employees, the employee with the smaller id is the one that should be returned.

Intuition

To solve this problem, we need to determine the duration that each employee spent on their task. Since the tasks are completed in a serial manner, the duration an employee spent on a task can be calculated by subtracting the finish time of the previous task from the finish time of the current task. The 0th task starts at time 0, so for this first task, the duration is simply the finishing time of the task since it started at time 0.

Now, we keep track of the maximum duration observed and the associated employee id. As we iterate through the logs array, we compare the current task's duration with what we've recorded as the maximum duration so far. If the current duration is greater, we update our records to now consider this as the max duration, and we update the "hardest worker" to be the current employee's id. If the current duration matches the maximum duration seen so far (a tie), we check the employee ids involved and update our "hardest worker" to be the employee with the smaller id.

The managing of the "last" task finish time is critical because it marks the start of the next task (other than the first task, which starts at time 0). After determining the duration for the current task, we update the "last" finish time to match the current task's finish time, effectively setting it up for the next comparison.

The given solution employs a for-loop to scan the logs list once, maintaining the current maximum duration and the employee with the smallest id that worked for this duration. Once the loop is complete, it returns the identifier of the "hardest worker" as the result.

Solution Approach

The solution approach uses a simple linear scan algorithm to determine which employee worked the longest on a single task. It makes use of basic comparison and arithmetic operations to track and update the task with the longest duration and the corresponding employee id.

Here is the step-by-step explanation of the solution algorithm:

1. Initialize three variables: last, mx, and ans. Set both last and mx (maximum task duration so far) to 0, and ans (the id of the employee who worked the longest) to the first employee's id.

2. Iterate over each entry in the logs list. Each entry is a pair [uid, t] which stands for an employee id (uid) and the time the task was completed (t).

3. For each entry, update the duration of the current task by calculating the difference between the current task's completion time t and the last task's completion time (stored in last). The expression for this is:

   t -= last

4. Now check if the current duration t is greater than the maximum duration mx or if there is a tie and the current employee's id is smaller than the one previously stored. If either condition is true:

   • Update mx to match the current duration t.
   • Update ans to the current employee's id (uid).

5. Update the last task completion time to account for the subsequent task's duration calculation:

   last += t

6. After processing all entries, return ans as the solution. This is the id of the employee who worked the most extended task.

The solution uses no additional data structures; it relies only on a few variables to track the information needed and operates with an O(n) time complexity where n is the number of entries in the logs list. The algorithm is efficient as it scans the logs only once, and the space complexity is constant O(1), as no extra space is needed regardless of the size of the input list logs.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach:

Suppose we have these logs for employees with durations in which they completed the tasks:

logs = [[0,3],[2,5],[0,10],[1,15]]

In this logs array, the first element [0,3] indicates that employee 0 finished a task at time 3. The second element [2,5] indicates that employee 2 finished the following task at time 5; and so on.

Now, let's apply the solution approach:

1. We initialize last, mx, and ans to 0 since no task has been completed yet, and we need a starting point.

2. We begin iteration with the first entry [0,3]. This task took 3 time units because it started at time 0. We set mx to 3 and ans to 0, since this is the longest duration we have seen so far, and it belongs to employee 0.

last = 0
mx = 3
ans = 0

3. We continue to the next entry [2,5]. The duration for this task is 5 - last which is 2 (5 - 3). This duration is not longer than mx which is 3, so we don't change mx or ans. We then update last to 5.

last = 5
mx = 3
ans = 0

4. The next entry is [0,10]. The duration for this task is 10 - last which is 5 (10 - 5). This duration is greater than the current mx which is 3, so we update mx to 5 and ans to 0. Then we update the last to 10.

last = 10
mx = 5
ans = 0

5. The last entry is [1,15]. The duration for this task is 15 - last which is 5. This is equal to our current mx. Now, we compare the employee ids; since 1 is larger than the current ans which is 0, we keep ans as 0. We update last to 15.

last = 15
mx = 5
ans = 0

6. We have exhausted all entries, so we finish the iteration, and the ans is the final result. Employee 0 is the one who worked the longest on a single task.

In this example, the final output would be 0, which indicates that employee 0 worked the longest time on a task, which took 5 time units to complete.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code is O(n), where n is the number of entries in the logs list. The reason is that the code iterates through each log entry exactly once, performing a constant amount of work for each entry (updating variables and simple comparisons).

Space Complexity

The space complexity of the code is O(1) because the code uses a fixed amount of space that does not depend on the input size. Variables last, mx, ans, and any other temporary variables used during iteration do not scale with the size of the input, as they are simply storage for current and maximum time differences as well as the corresponding worker ID.

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