Solution Explanation

This problem requires cleaning and aggregating data from a Sales table. The goal is to present a report showing the lowercase, trimmed product names, the sales month (in 'YYYY-MM' format), and the total number of sales for each product in each month.

Approach:

The solution uses a common table expression (CTE) for better readability and modularity. The CTE, named t, performs the following actions:

1. Lowercase Conversion: LOWER(TRIM(product_name)) converts the product_name column to lowercase and removes any leading or trailing whitespace using the TRIM() function. This addresses the case-insensitivity and whitespace issues.

2. Date Formatting: DATE_FORMAT(sale_date, '%Y-%m') formats the sale_date column into the required 'YYYY-MM' format.

The main query then uses the results from the CTE to:

1. Group Data: GROUP BY 1, 2 groups the results by product_name (column 1) and sale_date (column 2). This allows us to count sales for each product in each month.

2. Count Sales: COUNT(1) AS total counts the number of rows in each group, providing the total sales for each product-month combination.

3. Order Results: ORDER BY 1, 2 orders the results first by product_name (ascending) and then by sale_date (ascending) as required.

MySQL Code:

WITH
    t AS (
        SELECT
            LOWER(TRIM(product_name)) AS product_name,
            DATE_FORMAT(sale_date, '%Y-%m') AS sale_date
        FROM Sales
    )
SELECT product_name, sale_date, COUNT(1) AS total
FROM t
GROUP BY 1, 2
ORDER BY 1, 2;

Time Complexity Analysis:

The time complexity is dominated by the GROUP BY and ORDER BY operations. In general, GROUP BY and ORDER BY have a time complexity of O(N log N) where N is the number of rows in the Sales table, because they typically involve sorting the data. The other operations (lowercase conversion, date formatting, and counting) have linear time complexity O(N). Therefore, the overall time complexity is O(N log N). The actual performance will depend on the specific database implementation and optimization strategies.

Space Complexity Analysis:

The space complexity is determined by the size of the intermediate CTE t and the final result set. In the worst case, if all product names are unique and each product is sold in many different months, the space complexity would be proportional to the number of rows in the Sales table (O(N)), as the CTE would need to store a copy of the data. The final result set will typically be smaller than N because the data is aggregated. Therefore the overall space complexity can be considered O(N) in the worst case.