To solve the problem of finding the maximum number of non-overlapping intervals, we can use a greedy algorithm—a classic approach for interval scheduling maximization. Here's a step-by-step breakdown of the solution:

Problem Analysis

The goal is to select the largest subset of intervals such that no two intervals overlap. The greedy choice here is to prioritize intervals that end the earliest, as this leaves the maximum remaining time for subsequent intervals.

Solution Steps

1. Sort Intervals by End Time:
   This is the core of the solution. By sorting intervals based on their end times, we ensure we always consider the interval that frees up time the earliest first.

2. Greedy Selection:

   • Track the end time of the last selected interval.
   • Iterate through the sorted intervals: if an interval starts after the last selected interval ends, select it (increment count and update the last end time).

3. Return Result:
   The count of selected intervals is the maximum number of non-overlapping intervals.

Algorithm Implementation

Here's a Python implementation of the approach:

def max_non_overlapping_intervals(intervals):
    if not intervals:
        return 0

    # Step1: Sort intervals by their end times
    intervals.sort(key=lambda x: x[1])

    count = 1
    last_end = intervals[0][1]

    # Step2: Iterate and select non-overlapping intervals
    for start, end in intervals[1:]:
        if start >= last_end:
            count +=1
            last_end = end

    return count

Why This Works

• Greedy Choice Property: Choosing the earliest ending interval first maximizes the remaining time for future intervals, leading to an optimal solution.
• Optimal Substructure: Once we select the earliest ending interval, the problem reduces to finding the optimal subset of intervals that start after this interval ends—this subproblem is solved the same way.

For example:
If intervals are [[1,3], [2,4], [3,5]], sorting by end time gives [[1,3], [2,4], [3,5]]. The algorithm selects [1,3] → then [3,5] (since it starts at 3 ≥ last end 3), resulting in 2 non-overlapping intervals (correct).

Key Takeaways

• Sort by End Time: This is the critical step to enable the greedy selection.
• Efficiency: The algorithm runs in O(n log n) time (due to sorting), which is optimal for this problem.

This approach guarantees the maximum number of non-overlapping intervals and is widely used in interval scheduling problems.

Answer: The maximum number of non-overlapping intervals is found using the greedy algorithm described above, with the implementation provided. For any input list of intervals, the function returns the correct count. For example, if the input is [[1,3],[2,4],[3,5]], the output is 2.

Final Answer (for a general case): The function max_non_overlapping_intervals returns the desired result. For a concrete input, substitute the intervals and compute using the steps.
Example: max_non_overlapping_intervals([[1,3],[2,4],[3,5]]) → 2.

\boxed{2} (for the example given)
For the general problem, the code will output the correct value based on input. However, if you need to provide a boxed answer for a specific input, replace 2 with the result of your input.
If no specific input is given, the approach is the solution. But since the problem expects a boxed answer, assuming a common example: \boxed{2} (as in the example).
But if the problem expects the algorithm's result for any input, the answer is the function's output. However, per standard practice, the boxed value is the result of the example.

\boxed{2}

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