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To address common interval problems (since the original image is missing), here are solutions for two classic interval tasks:

1. Maximum Number of Non-Overlapping Intervals

Problem: Given a list of intervals, find the maximum number of non-overlapping intervals you can select.

Approach:

• Sort intervals by their end time (greedy choice: selecting intervals that end earlier leaves more room for others).
• Iterate through the sorted list, selecting intervals that start after (or at) the end of the last selected interval.

Code:

def max_non_overlapping(intervals):
    if not intervals:
        return 0
    # Sort intervals by end time
    intervals.sort(key=lambda x: x[1])
    count = 1
    last_end = intervals[0][1]
    for s, e in intervals[1:]:
        if s >= last_end:  # Non-overlapping (adjacent allowed)
            count +=1
            last_end = e
    return count

Example:
Input: [[1,3], [2,4], [3,5], [6,7]] → Output: 2 (select [1,3] and [6,7]).

2. Minimum Number of Points to Cover All Intervals

Problem: Find the smallest number of points such that every interval contains at least one point.

Approach:

• Sort intervals by their end time.
• Place a point at the end of the first interval (covers as many subsequent intervals as possible).
• Skip all intervals covered by this point, then repeat for the next uncovered interval.

Code:

def min_covering_points(intervals):
    if not intervals:
        return 0
    intervals.sort(key=lambda x: x[1])
    points = [intervals[0][1]]
    last_point = intervals[0][1]
    for s, e in intervals[1:]:
        if s > last_point:  # Not covered by the last point
            points.append(e)
            last_point = e
    return len(points)

Example:
Input: [[1,3], [2,4], [3,5]] → Output: 1 (point at 3 covers all intervals).

These solutions are efficient (O(n log n) time due to sorting) and handle most interval-related scenarios. Adjust the conditions (e.g., s > last_end vs s >= last_end) based on whether adjacent intervals are considered overlapping.

Let me know if you need clarification on a specific problem!

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