Solution Explanation:

This problem requires finding the largest time gap between consecutive visits for each user in the UserVisits table. The solution leverages SQL's window functions for efficient processing.

Approach:

1. Order Visits: The visits for each user need to be sorted chronologically. This is done implicitly within the LEAD window function using the ORDER BY visit_date clause.

2. LEAD Window Function: The core of the solution is the LEAD window function. LEAD(visit_date, 1, '2021-1-1') OVER (PARTITION BY user_id ORDER BY visit_date) does the following:

   • PARTITION BY user_id: This divides the data into groups based on each unique user_id. The LEAD function operates independently within each user's group.
   • ORDER BY visit_date: Within each user's group, the visits are ordered by their date.
   • LEAD(visit_date, 1): This retrieves the visit_date from the next row in the ordered partition. The 1 indicates that we're looking one row ahead.
   • '2021-1-1': This is the default value used if there is no next row (i.e., for the last visit of a user). The gap is calculated to 2021-01-01.

3. DATEDIFF: DATEDIFF(next_visit_date, current_visit_date) calculates the difference in days between the current visit and the next visit (or '2021-01-01'). This difference represents the window size.

4. MAX Aggregation: After calculating the day differences for all visits of each user, we use MAX(diff) to find the largest window size for each user. The GROUP BY 1 (which is equivalent to GROUP BY user_id) groups the results by user.

5. Ordering: Finally, ORDER BY 1 (or ORDER BY user_id) sorts the results by user ID.

Time Complexity:

The time complexity of this SQL query is dominated by the window function operations. Window functions typically have a time complexity of O(N log N) or O(N) depending on the database implementation and the specific window function used (where N is the total number of rows in the UserVisits table). The subsequent aggregation and sorting steps also contribute to the overall time complexity, but these are typically sublinear in nature compared to the window function processing. Therefore, the overall time complexity can be approximated as O(N log N) in the worst case.

Space Complexity:

The space complexity depends on the size of intermediate results generated by the window function and aggregation. It's largely determined by the size of the input data and is proportional to N (the number of rows in the input table). Therefore, the space complexity is O(N).

MySQL Code:

WITH
    T AS (
        SELECT
            user_id,
            DATEDIFF(
                LEAD(visit_date, 1, '2021-01-01') OVER (PARTITION BY user_id ORDER BY visit_date),
                visit_date
            ) AS diff
        FROM UserVisits
    )
SELECT user_id, MAX(diff) AS biggest_window
FROM T
GROUP BY user_id
ORDER BY user_id;

Other Database Systems (Conceptual):

The core logic remains the same for other SQL databases like PostgreSQL, SQL Server, and Oracle. The specific syntax for window functions and date/time functions might vary slightly, but the overall approach using LEAD, DATEDIFF (or equivalent), MAX, GROUP BY, and ORDER BY remains consistent. For instance, in PostgreSQL, you would use date('2021-01-01') instead of '2021-01-01' for the default date. The DATEDIFF function may be replaced with - operator for date subtraction depending on the database system.