Solution Explanation and Code

This problem requires retrieving the number of Grand Slam titles won by each player from two tables: Players and Championships. The solution involves combining data from these tables and aggregating the results. We'll analyze two approaches.

Approach 1: UNION ALL, JOIN, and GROUP BY

This approach uses a series of SQL operations to efficiently achieve the desired result.

1. UNION ALL: First, we create a common table expression (CTE) called T. This CTE uses UNION ALL to stack the Wimbledon, Fr_open, US_open, and Au_open columns from the Championships table vertically. This effectively creates a single column containing all player IDs who won any Grand Slam title.

2. JOIN: Next, we perform an INNER JOIN between the T CTE and the Players table using player_id as the join key. This links the player IDs from the T CTE with their corresponding names from the Players table. Only players who have a matching player_id in both tables (i.e., players who won at least one Grand Slam) are included.

3. GROUP BY and COUNT: Finally, we group the results by player_id (and implicitly player_name since it's functionally dependent on player_id) and use COUNT(1) to count the number of rows for each player. This count represents the total number of Grand Slam titles won by each player.

Time Complexity: O(N log N + M log M), where N is the number of rows in the Championships table, and M is the number of rows in the Players table. This is dominated by the sorting done implicitly or explicitly during the GROUP BY operation. The UNION ALL and JOIN operations typically have a time complexity closer to O(N + M) in well-optimized database systems.

Space Complexity: O(N + M) because of the intermediate CTE and the potentially large result set.

MySQL Code:

WITH
    T AS (
        SELECT Wimbledon AS player_id
        FROM Championships
        UNION ALL
        SELECT Fr_open AS player_id
        FROM Championships
        UNION ALL
        SELECT US_open AS player_id
        FROM Championships
        UNION ALL
        SELECT Au_open AS player_id
        FROM Championships
    )
SELECT player_id, player_name, COUNT(1) AS grand_slams_count
FROM
    T
    JOIN Players USING (player_id)
GROUP BY 1;

Approach 2: CROSS JOIN, CASE, SUM, and HAVING

This approach uses a different strategy, avoiding the CTE.

1. CROSS JOIN: It starts with a CROSS JOIN between Championships and Players. This generates all possible combinations of rows from both tables, creating a much larger intermediate table.

2. CASE and SUM: For each row in this expanded table, it uses four CASE statements to check if the player_id matches any of the Grand Slam winner IDs (Wimbledon, Fr_open, US_open, Au_open). If a match is found, 1 is added; otherwise, 0 is added. The SUM() function then totals these values for each player. This directly counts the number of Grand Slam wins.

3. GROUP BY: The results are grouped by player_id.

4. HAVING: Finally, a HAVING clause filters out players with a grand_slams_count of 0, ensuring that only players with at least one Grand Slam win are included in the output.

Time Complexity: O(NM), where N is the number of rows in Championships and M is the number of rows in Players. The cross join creates a table of size NM, making this approach less efficient than Approach 1, especially with large datasets.

Space Complexity: O(NM) due to the intermediate table created by the cross join.

MySQL Code:

SELECT
    player_id,
    player_name,
    SUM(
        (CASE WHEN Wimbledon = player_id THEN 1 ELSE 0 END) +
        (CASE WHEN Fr_open = player_id THEN 1 ELSE 0 END) +
        (CASE WHEN US_open = player_id THEN 1 ELSE 0 END) +
        (CASE WHEN Au_open = player_id THEN 1 ELSE 0 END)
    ) AS grand_slams_count
FROM
    Championships
    CROSS JOIN Players
GROUP BY player_id
HAVING grand_slams_count > 0;

Recommendation: Approach 1 (using UNION ALL, JOIN, and GROUP BY) is generally more efficient and scalable for larger datasets because it avoids the computationally expensive cross join. Approach 2 is simpler to understand but can become very slow as the table sizes increase.