Solution Explanation:

This problem requires finding the number of users who logged in for the first time on each date within the last 90 days from June 30th, 2019. The solution uses a SQL query to achieve this. Let's break down the MySQL solution step-by-step.

MySQL Solution:

The query uses a Common Table Expression (CTE) called T to simplify the process.

1. CTE T:

WITH
    T AS (
        SELECT
            user_id,
            MIN(activity_date) OVER (PARTITION BY user_id) AS login_date
        FROM Traffic
        WHERE activity = 'login'
    )

• SELECT user_id, MIN(activity_date) OVER (PARTITION BY user_id) AS login_date: This selects the user_id and calculates the minimum activity_date for each user where the activity is 'login'. The MIN(activity_date) OVER (PARTITION BY user_id) is a window function. It finds the minimum activity_date for each user_id (partitioning by user_id), effectively determining the first login date for each user. This minimum date is aliased as login_date.

• FROM Traffic WHERE activity = 'login': This specifies that we are selecting from the Traffic table and only considering rows where the activity is 'login'.

2. Main Query:

SELECT login_date, COUNT(DISTINCT user_id) AS user_count
FROM T
WHERE DATEDIFF('2019-06-30', login_date) <= 90
GROUP BY 1;

• SELECT login_date, COUNT(DISTINCT user_id) AS user_count: This selects the login_date and counts the distinct number of users (user_id) for each login_date. The count is aliased as user_count.

• FROM T: This selects the data from the CTE T which contains the user_id and their first login date.

• WHERE DATEDIFF('2019-06-30', login_date) <= 90: This filters the results to include only those login_dates that are within 90 days of '2019-06-30'. DATEDIFF calculates the difference in days between two dates.

• GROUP BY 1: This groups the results by the first column (login_date), so we get a count of users for each unique first login date.

Time Complexity Analysis:

The time complexity of this SQL query depends on the size of the Traffic table. The MIN() window function with PARTITION BY has a complexity that's typically O(N log N) or even O(N) depending on the database system's optimization strategies (where N is the number of rows in the Traffic table after filtering for 'login' activity). The GROUP BY operation also has a time complexity related to sorting or hashing, which is generally O(N log N) in the worst case. Therefore, the overall time complexity is dominated by these operations, making it approximately O(N log N) in the worst case. In practice, database optimizations might bring this closer to linear time for many scenarios.

Space Complexity Analysis:

The space complexity depends on the size of the intermediate results. The CTE T might store a significant amount of data if the table is large. The space used by the final result set is proportional to the number of unique login dates within the 90-day window, which is relatively small compared to the input table. Therefore, the space complexity is O(N) in the worst case (where N is the number of rows in the Traffic table), primarily due to the intermediate CTE. Again, this could be much smaller in practice if there are many duplicate logins.