Solution Explanation:

This problem requires calculating the running total of scores for each gender on each day. The solution leverages the power of window functions in SQL.

Approach:

The core idea is to use the SUM() OVER() window function. This function allows us to calculate aggregates (in this case, the sum) within a specified window of rows.

1. PARTITION BY gender: This clause divides the data into separate partitions based on the gender column. This ensures that the running total is calculated independently for each gender.

2. ORDER BY gender, day: This clause specifies the order within each partition. The running total is computed cumulatively, first ordered by gender and then by day. This ensures that the running total is calculated correctly based on the chronological order of days.

3. SUM(score_points): This calculates the sum of score_points within the defined window (each partition ordered by day).

4. AS total: This assigns an alias "total" to the running sum column.

Time Complexity Analysis:

The time complexity of the SQL query is determined by the sorting and aggregation steps within the window function. In most database systems that efficiently implement window functions (like MySQL), the complexity is approximately O(N log N) in the worst-case scenario, where N is the number of rows in the Scores table. This is because sorting is usually involved. However, in practice, well-optimized database systems can often achieve better performance than a purely O(N log N) implementation, especially with indexes.

Space Complexity Analysis:

The space complexity is O(N) in the worst case, because it needs to store the intermediate results during the window function calculation. However, the actual space used might be less than N depending on the database implementation and the size of the input data.

Code (MySQL):

SELECT
    gender,
    day,
    SUM(score_points) OVER (
        PARTITION BY gender
        ORDER BY gender, day
    ) AS total
FROM Scores;

This single query efficiently calculates and returns the desired running total for each gender on each day, ordered as requested. No other loops or complex logic are needed. The window function handles the aggregation and ordering elegantly.