Understanding the Problem and Requirements
The problem presented is a classic example of a “subset sum” problem, where we are given a table of numbers with an incrementing id column and a random positive non-zero number in each row. The goal is to write a query that returns all rows which add up to less than or equal to a given number.
We need to consider several rules:
- Rows must be “consumed” in order, even if a later row makes it a perfect match.
- We can use a row partially if there is more available than is necessary to add up to whatever is leftover on the number E.g. rows 1, 2, and 3 would be returned if your max number is 50 because 12 + 5 + 33 equals 50 and 90 is a partial result.
- If there are not enough rows to satisfy the amount, then return ALL the rows.
Choosing an Approach
This problem can be approached using various techniques, including brute force, dynamic programming, or even using advanced SQL features like common table expressions (CTEs) and window functions.
In this article, we will explore a solution using CTEs and window functions in SQL Server. We will break down the steps required to solve this problem and provide explanations for each part.
Setting Up the Table
First, let’s create a sample table with an incrementing id column and a random positive non-zero number in each row.
CREATE TABLE #t (id int, rand int);
INSERT INTO #t (id,rand)
VALUES (1,12),(2,5),(3,99),(4,87);
Step 1: Creating the CTEs
We will use two CTEs to solve this problem. The first CTE, dat, will calculate the cumulative sum of the random numbers in each row, ordered by the id column.
WITH dat
AS
(
SELECT id, rand, SUM(rand) OVER (ORDER BY id) as runsum
FROM #t
)
The second CTE, dat2, will calculate the previous cumulative sum for each row. This is necessary to allow us to use rows partially in our final calculation.
AS
(
SELECT id, rand,
SUM(rand) OVER (ORDER BY id) as runsum,
COALESCE(LAG(runsum,1) OVER (ORDER BY id),0) as prev_runsum
from dat
)
Step 2: Calculating the Final Results
Now that we have our two CTEs, we can calculate the final results. We will select all rows from dat2 where the target number is greater than or equal to the current cumulative sum, or if it’s not, then between the previous cumulative sum and the current one.
SELECT id, rand
FROM dat2
WHERE @target >= runsum
OR @target BETWEEN prev_runsum AND runsum;
Step 3: Handling Edge Cases
To handle edge cases where there are not enough rows to satisfy the amount, we will simply return all rows.
Putting it All Together
Now that we have our solution, let’s put it all together in a single query.
DECLARE @target int = 104;
WITH dat
AS
(
SELECT id, rand, SUM(rand) OVER (ORDER BY id) as runsum
FROM #t
),
dat2
as
(
SELECT id, rand,
SUM(rand) OVER (ORDER BY id) as runsum,
COALESCE(LAG(runsum,1) OVER (ORDER BY id),0) as prev_runsum
from dat
)
SELECT id, rand
FROM dat2
WHERE @target >= runsum
OR @target BETWEEN prev_runsum AND runsum;
Conclusion
In this article, we explored a solution to the subset sum problem using SQL Server CTEs and window functions. We broke down the steps required to solve this problem and provided explanations for each part.
We also discussed how to handle edge cases where there are not enough rows to satisfy the amount. By simply returning all rows in such cases, we ensure that our solution is robust and covers all possible scenarios.
Additional Considerations
While this solution works well for SQL Server, it may not be optimal for other RDBMS or programming languages. Depending on the specific requirements of your problem, you may need to use a different approach or technique.
Additionally, while CTEs and window functions can be powerful tools in solving subset sum problems, they may also have performance implications if used excessively. It’s essential to profile and optimize your queries to ensure they run efficiently.
Future Work
In future articles, we will explore additional techniques for solving subset sum problems, including dynamic programming and brute force approaches. We will also discuss advanced SQL features like window functions and CTEs in more detail.
We will also explore how to extend this solution to cover more complex scenarios, such as handling multiple target numbers or optimizing performance for large datasets.
References
Note: The references provided are general resources and may not be specific to the solution presented in this article.
Last modified on 2024-01-28