Problem Description

The problem is about finding a mutual meeting time slot between two people given their individual schedules and a required meeting duration. Each person's schedule is represented by a list of non-overlapping time slots where a time slot is an array [start, end] showing availability from start to end. The goal is to find the earliest starting time slot that is available in both schedules and lasts at least for the given duration. If there's no such common time slot, we return an empty array.

Intuition

To solve this problem, we can use the two-pointer technique. Since no individual time slots overlap within a single person's schedule, we can sort both schedules by the starting times of the slots. We then compare the slots from both schedules to find overlapping slots.

We use two pointers i and j to traverse slots1 and slots2 respectively. In each iteration, we find the latest start time by taking the maximum of slots1[i][0] and slots2[j][0] and the earliest end time by taking the minimum of slots1[i][1] and slots2[j][1]. If the overlapping time between the latest start and earliest end is greater than or equal to the required duration, we have found a suitable time slot and return the start and end of this meeting slot.

If the overlap is not sufficient, we move the pointer forward in the list which has the earlier ending time slot, hoping to find a longer slot that might overlap with the other person's next slot. This process continues until we either find a suitable slot or exhaust all available slots in either list.

Learn more about Two Pointers and Sorting patterns.

Solution Approach

The provided Python solution follows a straightforward two-pointer approach to solve the problem:

1. Sorting the time slots: Both slots1 and slots2 are sorted based on their starting times. This ensures that we are always considering the earliest available slots first and eliminates the need for checking past time slots. Sorting is crucial as it sets up the structure for the two-pointer technique to work effectively.

2. Two-pointer Technique: Two pointers, i and j, are used to iterate through slots1 and slots2 respectively. At each step, i refers to the current slot in the first person's schedule, and j refers to the current slot in the second person's schedule.

3. Finding Overlaps: For the current pair of time slots pointed by i and j, we calculate the overlap by determining the maximum of the two start times and the minimum of the two end times. The variables start and end are used to record these values:

   start = max(slots1[i][0], slots2[j][0])
   end = min(slots1[i][1], slots2[j][1])

4. Checking Duration: We then check if the overlapping duration is greater than or equal to the required duration by subtracting start from end:

   if end - start >= duration:

   If the condition is met, we have found a valid time slot and can return [start, start + duration] as the solution.

5. Advancing the Pointers: If the overlapping time slot is not long enough, we need to discard the time slot that ends earlier and move forward. This decision is made by comparing the end times of the current time slots pointed by i and j. The pointer corresponding to the slot with the earlier end time is incremented:

   if slots1[i][1] < slots2[j][1]:
       i += 1
   else:
       j += 1

   This step ensures that we're always trying to find overlap with the nearest possible future slot.

6. Continuation and Termination: The above steps are continued in a loop until one of the lists is exhausted (i.e., i reaches the end of slots1 or j reaches the end of slots2). If no common time slots with sufficient duration are found by the end of either list, we return an empty array [] as specified.

This algorithm makes efficient use of the sorted structure of the time slots and the two-pointer technique to minimize the number of comparisons and quickly find the first suitable time slot, achieving a time complexity that is linear in the size of the input time slots after the initial sort.

Example Walkthrough

Let's consider an example to illustrate the solution approach:

• Person A's schedule (slots1): [[10, 50], [60, 120], [140, 210]]
• Person B's schedule (slots2): [[0, 15], [25, 50], [60, 70], [80, 100]]
• Required duration for the meeting: 8 minutes

1. Sorting the time slots: Both schedules are already sorted based on their starting times, eliminating the need for a sort operation in this example.

2. Two-pointer Technique: Initialize two pointers, i for slots1 and j for slots2. Initially, i = 0 and j = 0.

3. Finding Overlaps:

   • First Comparison:

     • slots1[i] = [10, 50] and slots2[j] = [0, 15]
     • The overlap is from 10 (max of 10 and 0) to 15 (min of 50 and 15), which is 5 minutes long. This is not enough for the 8-minute duration.
     • Move the pointer j forward because slots2[j][1] is less than slots1[i][1].

   • Second Comparison:

     • slots1[i] = [10, 50] and slots2[j] = [25, 50]
     • The overlap is from 25 to 50, which is 25 minutes long and suffices for the 8-minute duration.
     • We have found a suitable slot, so we can return the result [25, 25 + 8], which is [25, 33].

This result indicates that Person A and Person B can successfully schedule a meeting starting at 25 minutes past the hour and lasting until 33 minutes past the hour.

The method described above is efficient because it continuously seeks the earliest possible meeting time by looking for overlaps in the schedules and moves forward based on the end times of the current slots. As soon as a fitting time slot is found, it returns the result without unnecessary comparisons of later time slots, saving time and computation.

Solution Implementation

Time and Space Complexity

Time Complexity

The given algorithm primarily consists of two parts: sorting the time slots and then iterating through the sorted lists to find a common available duration.

First, we sort both slots1 and slots2. Assuming that slots1 has m intervals and slots2 has n intervals, the time taken to sort these lists using a comparison-based sorting algorithm, like Timsort (Python's default sorting algorithm), is O(m log m) for slots1 and O(n log n) for slots2.

After sorting, we have a while loop that runs until we reach the end of one of the slot lists. In the worst case, we'll compare each pair of slots once, which leads to a complexity of O(m + n) since each list is traversed at most once.

The overall time complexity is the sum of the complexities of sorting and iterating through the slot lists, which is O(m log m + n log n + m + n). However, the log factor dominates the linear factor in computational complexity analysis for large values. Hence, the time complexity simplifies to O(m log m + n log n).

Space Complexity

The space complexity of the algorithm is determined by the space we use to sort slots1 and slots2. Since the Timsort algorithm can require a certain amount of space for its operation, the space complexity is O(m+n) due to the auxiliary space used in sorting. All other operations use a constant amount of space, and no additional space is used that depends on the input size, so the space complexity remains O(m+n).

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