Process time intervals and detect overlaps/gaps

You are given n time intervals representing debug log segments. Each interval has a start time, an end time, and an associated text message.

• Each interval is represented as (start, end, text) .
• start and end are integers representing timestamps in seconds.
• Assume all intervals are half-open: they cover [start, end) , i.e., they include start and exclude end .
• The input intervals are not guaranteed to be in order and may overlap or leave gaps.

Task 1: Output ordered, merged intervals

Write a function that:

1. Takes as input a list of intervals [(start_i, end_i, text_i)] .
2. Sorts them by time and merges overlapping intervals.
   • When two or more intervals overlap in time, merge them into a single interval [min_start, max_end) where:
     • min_start is the smallest start time among those intervals.
     • max_end is the largest end time among those intervals.
   • For the merged interval's text, concatenate the texts of the merged intervals in chronological order, separated by a single space.
3. Returns the merged, non-overlapping intervals in ascending order of start time as a list of (start, end, merged_text) .

Example (one possible style of input/output):

• Input:
  • [(0, 5, "A"), (3, 10, "B"), (12, 15, "C")]
• Expected output:
  • [(0, 10, "A B"), (12, 15, "C")]

(You can assume that the concatenation order for texts is determined by the start time of each interval; if start times are equal, you may break ties arbitrarily.)

Task 2 (Follow-up): Detect overlaps and gaps

Using the same input model (original, unmerged intervals):

1. Determine whether any pair of intervals overlaps in time.
   • Two intervals [s1, e1) and [s2, e2) overlap if their intersection is non-empty, i.e., max(s1, s2) < min(e1, e2) .
2. Determine whether there exists any gap between the earliest start time and the latest end time.
   • A gap exists if, after ordering the intervals by start time, there is at least one pair of consecutive intervals where next.start > current.end .

Define and implement appropriate functions/methods to:

• Produce the ordered, merged list of intervals with concatenated texts (Task 1).
• Return booleans or equivalent indicators for:
  • "there is at least one overlap" and
  • "there is at least one gap" (Task 2).

State the expected time and space complexity of your approach in terms of n (the number of intervals).