Maximize Event Attendance

Imagine you have a busy schedule filled with various events, each spanning a specific period. You want to attend as many events as possible, but you have a strict rule: you can only attend one event per day. This challenge asks you to devise an optimal strategy to maximize the total number of distinct events you can participate in, a common problem in resource scheduling and time management.

Problem Description

You will be given a list of events. Each event is represented by a pair of integers [startDay, endDay], indicating that the event is available to be attended on any day from startDay to endDay, inclusive. Your goal is to determine the maximum number of distinct events you can attend, given the constraint that you can only attend one event per day.

Key requirements:

• The input is an array or list of events, where each event is an array or tuple [startDay, endDay].
• The output should be a single integer representing the maximum count of events attended.
• An event [s, e] can be attended on any single day d such that s <= d <= e.
• If you choose to attend an event on a particular day d, you cannot attend any other event on that same day d.
• Each [startDay, endDay] pair in the input represents a distinct event, even if multiple events have identical start and end days.

Important edge cases to consider:

• Events with very short durations (e.g., [5, 5]).
• Events that overlap significantly.
• Events that do not overlap at all.
• A scenario where no events can be attended due to scheduling conflicts.
• A large number of events spanning a wide range of days.

Examples

Example 1:

Input: [[1, 2], [2, 3], [3, 4]]
Output: 3
Explanation:
We can attend event [1,2] on Day 1.
Then, attend event [2,3] on Day 2.
Finally, attend event [3,4] on Day 3.
All three events can be attended, each on a distinct day.

Example 2:

Input: [[1, 2], [1, 2], [3, 4], [1, 4]]
Output: 4
Explanation:
Let's label the events: E1=[1,2], E2=[1,2], E3=[3,4], E4=[1,4].
We can attend them as follows:
Day 1: Attend E1 ([1,2]).
Day 2: Attend E2 ([1,2]).
Day 3: Attend E4 ([1,4]). (E3 is also available, but E4 is chosen here. The specific choice among events ending on the same earliest day doesn't change the maximum count).
Day 4: Attend E3 ([3,4]).
In this schedule, all 4 distinct events are attended.

Constraints

• N: The number of events, where 1 <= N <= 10^5.
• startDay_i, endDay_i: The start and end days for an event, where 1 <= startDay_i <= endDay_i <= 10^5.
• The solution should be efficient enough to handle N events and days up to 10^5 within typical time limits (e.g., a few seconds). Aim for a time complexity around O(N log N) or O(N + MaxDay log N), where MaxDay is the maximum possible endDay.
• The space complexity should also be reasonable, ideally O(N) or O(MaxDay).

Notes

• Consider that sorting the events initially might be beneficial for processing them in a structured manner.
• A greedy approach, focusing on attending events that free up future days quickly, is often effective for scheduling problems.
• A priority queue (min-heap) can be a very useful data structure for efficiently managing events that are currently available and deciding which one to attend. Think about what property of an event you would prioritize in such a queue.
• The iteration for days should span the range from the earliest possible startDay to the latest possible endDay across all events.
• Be mindful of how to efficiently add events that become available on a given day and remove events that have already passed their endDay from consideration.