The Bulb Switcher III problem is an interesting coding challenge that involves simulating a sequential process of turning on bulbs and determining how many moments exist where all the bulbs that have been turned on are also shining brightly.

This problem can be found on coding platforms like LeetCode and is an excellent exercise for enhancing your understanding of arrays, simulation, and logical conditions. Let’s walk through this problem step-by-step and solve it using an efficient approach.

Problem Statement

You are given an array light where light[i] = x means that the x-th bulb is turned on in the i-th moment. Each bulb is initially turned off and only shines if it and all the bulbs to its left are turned on.

You need to return the number of moments when every bulb that has been turned on is shining.

Example:

Input: light = [2, 1, 3, 5, 4]
Output: 3
Explanation: 
- Moment 1: Bulb 2 is turned on, but bulb 1 is not on, so no bulbs are shining.
- Moment 2: Bulb 1 is turned on, and bulbs 1 and 2 are shining.
- Moment 3: Bulb 3 is turned on, and bulbs 1, 2, 3 are shining.
- Moment 4: Bulb 5 is turned on, but bulb 4 is not on, so bulbs 1, 2, 3 are still shining.
- Moment 5: Bulb 4 is turned on, and bulbs 1 to 5 are shining.
There are 3 moments (moments 2, 3, and 5) when all turned-on bulbs are shining.

Key Insights

This problem is about understanding when all the bulbs that have been turned on are continuously lit without any interruptions. A bulb can only shine when all the bulbs to its left (and including itself) are turned on.

To determine these moments:

• Keep track of the maximum bulb number turned on so far.
• Compare this maximum number to the number of bulbs that have been turned on at each step (i.e., the current moment).
• If the maximum bulb number matches the number of bulbs turned on, then all bulbs that have been turned on so far are shining.

Approach

1. Tracking Maximum Bulb:

• As you turn on each bulb, keep track of the maximum number of the bulb that has been turned on.

1. Counting Moments:

• At each moment, check if the maximum bulb number matches the number of bulbs turned on so far (i.e., the moment index).
• If it does, it means all bulbs that have been turned on are shining, and you can count this as a valid moment.

Solution in Python

Let’s implement the solution based on the approach discussed:

def numTimesAllBlue(light):
    max_bulb = 0  # Tracks the maximum bulb number turned on
    moments = 0   # Count of valid moments where all bulbs are shining

    for i, bulb in enumerate(light):
        max_bulb = max(max_bulb, bulb)

        # If the max bulb turned on equals the number of bulbs turned on so far
        if max_bulb == i + 1:
            moments += 1

    return moments

Explanation:

• max_bulb: This variable stores the maximum bulb number turned on at any given moment.
• moments: This is the counter that tracks how many times all bulbs that have been turned on are shining.
• Loop through the array: For each bulb that is turned on at moment i, we check if the maximum bulb number turned on matches the current moment i + 1. If it does, it means that all bulbs up to that point are shining.

Time Complexity

• Time Complexity: O(n), where n is the number of bulbs. We loop through the array once, making this a linear-time solution.
• Space Complexity: O(1) because we are only using a constant amount of extra space for the max_bulb and moments variables.

Edge Cases

1. All Bulbs Turn On Sequentially:

• Input: [1, 2, 3, 4, 5]
• Output: 5
• In this case, all bulbs are turned on in order, so each moment is a valid moment where all bulbs are shining.

1. No Valid Moments:

• Input: [5, 4, 3, 2, 1]
• Output: 0
• Here, bulbs are turned on in reverse order, so there is no moment where all bulbs that have been turned on are shining.

Conclusion

The Bulb Switcher III problem is a great exercise in logic and array manipulation. By keeping track of the maximum bulb turned on and comparing it with the current number of bulbs turned on, we can efficiently solve this problem in linear time.

This problem emphasizes the importance of understanding the relationship between elements in an array and how to derive meaningful patterns from them. The solution is both time-efficient and space-efficient, making it suitable for large input sizes.

LeetCode Reference: Bulb Switcher III

Let me know your thoughts or questions, and happy coding!