Solution Explanation

This problem requires identifying accounts that have logged in from two different IP addresses within overlapping time intervals. The most efficient approach is using a self-join within a SQL query.

Approach: Self-Join

The core idea is to join the LogInfo table with itself (a self-join). This allows us to compare each login record with every other login record for the same account.

We filter the joined results based on these conditions:

1. Same Account ID: a.account_id = b.account_id: This ensures we only compare logins from the same account.

2. Different IP Addresses: a.ip_address != b.ip_address: This identifies logins from different IP addresses for the same account.

3. Overlapping Time Intervals: a.login BETWEEN b.login AND b.logout: This crucial condition checks if the login time of one record falls within the login and logout times of another record from the same account but a different IP address. This signifies simultaneous logins from different IPs.

Finally, we use DISTINCT to eliminate duplicate account IDs from the results.

Time Complexity Analysis

The time complexity of the self-join operation is dominated by the nested loop implicit in the join. In the worst case, it could be O(N^2), where N is the number of rows in the LogInfo table. However, the actual performance would heavily depend on the database's query optimizer and indexing. If appropriate indexes (e.g., on account_id, ip_address, and login) are present, the database engine can significantly improve performance. In practice, it's likely to be closer to O(N log N) or even O(N) with good indexing.

Code (MySQL)

SELECT DISTINCT a.account_id
FROM LogInfo a
JOIN LogInfo b ON a.account_id = b.account_id AND a.ip_address != b.ip_address
WHERE a.login BETWEEN b.login AND b.logout;

This SQL query performs the self-join, applies the filtering conditions, and returns the distinct account IDs that satisfy the criteria. The DISTINCT keyword ensures that each banned account is listed only once in the result.

Other Database Systems

The fundamental approach remains the same for other SQL-based database systems (PostgreSQL, SQL Server, Oracle, etc.). The syntax might vary slightly, but the core self-join logic with the three conditions remains the same. For instance, in PostgreSQL, the query would be almost identical.

This solution efficiently identifies accounts that need to be banned based on the specified criteria, leveraging the power of SQL's self-join capabilities. Proper indexing is key to optimizing the query's performance in larger datasets.