Compute party hours per town and city gaps

You are given (1) a region table describing a hierarchy of locations and (2) a list of party attendance time intervals (all timestamps occur within a single calendar day).

Input

1. Region data (each row describes one town and the city/state it belongs to):

• town_id (string/int)
• town_name (string)
• city_id (string/int)
• city_name (string)
• state_id (string/int)
• state_name (string)

Assume each town_id belongs to exactly one city_id.

2. Party intervals (each row indicates that a given town is “at a party” during a time range on that day):

• town_id
• start_time (time within the day, e.g., minutes since 00:00)
• end_time (time within the day, end_time > start_time )

Notes:

• A town can have multiple party intervals in the same day.
• Party intervals for the same town may be unsorted and may overlap or touch.

Define the day as the window [day_start, day_end) (e.g., [00:00, 24:00)).

Task A (per-town party hours)

For each town_id, compute the total number of hours spent at a party during the day, counting the union of that town’s intervals (i.e., overlapping intervals should not be double-counted).

Return results as:

• town_id , party_hours (can be fractional)

Task B (city-level idle/gap time segments)

Using the same party intervals, compute idle (not-at-party) time segments for each town (within [day_start, day_end)), then group the idle segments by city_id.

Output for each city a set/list of time segments representing the idle periods contributed by its towns. (You should specify clearly whether you are returning:

• all town idle segments annotated with town, or
• merged/unioned idle segments at the city level.

If not specified otherwise, assume you should return all idle segments per town, grouped by city:

• city_id , town_id , idle_start , idle_end

Constraints / Expectations

• Times can be represented as integer minutes.
• Aim for an efficient approach (e.g., sort + merge per town).
• Clarify edge cases: no party intervals for a town, intervals touching at endpoints, intervals outside the day window (if present, clamp to the day window).