Infinite Loops in Recursive CTEs with UNION ALL in SQLite

Issue Overview: Infinite Loops in Recursive CTEs with UNION ALL

The core issue revolves around the behavior of recursive Common Table Expressions (CTEs) in SQLite, specifically when using UNION ALL versus UNION. The user attempted to implement the Sieve of Eratosthenes algorithm using recursive CTEs to generate prime numbers up to a specified limit. While the query works as expected with UNION, it falls into an infinite loop when UNION ALL is used. The user is puzzled as to why the infinite loop occurs only for the second generate_series call and not the first.

The query involves three CTEs: setup, gen, and comp. The setup CTE generates a series of odd numbers starting from 3 up to a specified maximum prime (:MAX_PRIME). The gen CTE calculates parameters for generating composite numbers, and the comp CTE recursively generates these composite numbers. The comp CTE uses a recursive union to generate the composite numbers, and the user observed that changing the union type from UNION to UNION ALL causes an infinite loop in the second generate_series call.

The user’s confusion stems from the fact that the first generate_series call (for num1=3) works correctly, but the second call (for num1=5) results in an infinite loop. This behavior is unexpected because both calls are structurally similar, and the user expected UNION ALL to behave consistently across all recursive steps.

Possible Causes: Recursive CTE Mechanics and UNION ALL Behavior

The issue arises due to the fundamental differences between UNION and UNION ALL in recursive CTEs, combined with the mechanics of recursion in SQLite. To understand the problem, we need to delve into how recursive CTEs work and how UNION and UNION ALL affect their execution.

Recursive CTE Mechanics

A recursive CTE consists of two parts: the base case and the recursive case. The base case initializes the result set, and the recursive case iteratively processes the result set until a termination condition is met. In SQLite, recursive CTEs are evaluated row-by-row, and each iteration builds upon the results of the previous iteration.

In the user’s query, the comp CTE is defined as follows:

comp(num1, num2, num3) AS (
    SELECT 1, 0, 0
    UNION ALL
    SELECT comp.num1 + 2 AS gg, comp.num2 + 1, value
    FROM generate_series(
        (SELECT start FROM gen WHERE gg = cand),
        (SELECT stop FROM gen WHERE gg = cand),
        (SELECT step FROM gen WHERE gg = cand)
    ), comp
    WHERE 1 LIMIT :MAX_PRIME
)

The base case initializes the CTE with (1, 0, 0), and the recursive case generates new rows by joining the generate_series function with the comp CTE itself. The LIMIT clause is intended to prevent infinite recursion, but it does not work as expected when UNION ALL is used.

UNION vs. UNION ALL

The key difference between UNION and UNION ALL is that UNION eliminates duplicate rows from the result set, while UNION ALL retains all rows, including duplicates. In the context of recursive CTEs, this difference has significant implications for termination conditions and result set growth.

When UNION is used, the recursive CTE stops iterating once all possible unique rows have been generated. This is because UNION ensures that each row is unique, and the CTE cannot produce new rows that have not already been included in the result set. As a result, the recursion terminates naturally.

However, when UNION ALL is used, the recursive CTE continues to generate rows indefinitely if the recursive case produces rows that are already in the result set. This is because UNION ALL does not eliminate duplicates, and the CTE keeps processing the same rows over and over again. In the user’s query, this leads to an infinite loop for the second generate_series call.

Why the First generate_series Call Works

The first generate_series call (for num1=3) works correctly because the recursive CTE generates a finite set of rows that do not overlap with the base case. The LIMIT clause effectively terminates the recursion after a fixed number of iterations, preventing an infinite loop.

However, the second generate_series call (for num1=5) falls into an infinite loop because the recursive case produces rows that overlap with the existing result set. Since UNION ALL does not eliminate duplicates, the CTE keeps processing the same rows indefinitely, leading to an infinite loop.

Troubleshooting Steps, Solutions & Fixes: Preventing Infinite Loops in Recursive CTEs

To resolve the issue, we need to ensure that the recursive CTE terminates correctly when UNION ALL is used. This can be achieved by modifying the query to prevent overlapping rows in the recursive case or by using a different approach to generate the composite numbers.

Solution 1: Use UNION Instead of UNION ALL

The simplest solution is to use UNION instead of UNION ALL in the recursive CTE. This ensures that duplicate rows are eliminated, and the recursion terminates naturally once all unique rows have been generated. However, this approach may not be suitable if the query requires retaining duplicate rows for some reason.

Solution 2: Add a Termination Condition

Another approach is to add an explicit termination condition to the recursive CTE. This can be done by introducing a counter or a flag that stops the recursion once a certain condition is met. For example, we can modify the comp CTE to include a counter that tracks the number of iterations and stops the recursion after a fixed number of steps.

comp(num1, num2, num3, iter) AS (
    SELECT 1, 0, 0, 0
    UNION ALL
    SELECT comp.num1 + 2 AS gg, comp.num2 + 1, value, iter + 1
    FROM generate_series(
        (SELECT start FROM gen WHERE gg = cand),
        (SELECT stop FROM gen WHERE gg = cand),
        (SELECT step FROM gen WHERE gg = cand)
    ), comp
    WHERE iter < :MAX_ITER LIMIT :MAX_PRIME
)

In this modified query, the iter column tracks the number of recursive iterations, and the WHERE clause ensures that the recursion stops after a specified number of steps (:MAX_ITER). This prevents the infinite loop while allowing the use of UNION ALL.

Solution 3: Restructure the Query to Avoid Overlapping Rows

A more robust solution is to restructure the query to avoid generating overlapping rows in the recursive case. This can be achieved by ensuring that each recursive step produces a distinct set of rows that do not overlap with the existing result set. For example, we can modify the comp CTE to generate only new rows that have not been included in the result set.

comp(num1, num2, num3) AS (
    SELECT 1, 0, 0
    UNION ALL
    SELECT comp.num1 + 2 AS gg, comp.num2 + 1, value
    FROM generate_series(
        (SELECT start FROM gen WHERE gg = cand),
        (SELECT stop FROM gen WHERE gg = cand),
        (SELECT step FROM gen WHERE gg = cand)
    ), comp
    WHERE value NOT IN (SELECT num3 FROM comp WHERE num3 > 0)
    LIMIT :MAX_PRIME
)

In this modified query, the WHERE clause ensures that only new values are added to the result set, preventing the infinite loop. This approach retains the use of UNION ALL while ensuring that the recursion terminates correctly.

Solution 4: Use a Non-Recursive Approach

If the recursive CTE proves too complex or error-prone, an alternative approach is to use a non-recursive query to generate the composite numbers. For example, we can use a self-join or a subquery to generate the composite numbers without relying on recursion.

WITH setup(series) AS (
    SELECT value FROM generate_series(3, :MAX_PRIME, 2)
),
gen(cand, start, stop, step) AS (
    SELECT series, series * series, :MAX_PRIME, series
    FROM setup
    WHERE series * series <= :MAX_PRIME
),
comp(num1, num2, num3) AS (
    SELECT 1, 0, 0
    UNION ALL
    SELECT g.cand, ROW_NUMBER() OVER (ORDER BY g.cand), s.value
    FROM gen g
    JOIN generate_series(g.start, g.stop, g.step) s
    ON 1=1
)
SELECT * FROM comp;

In this non-recursive query, the comp CTE generates the composite numbers using a self-join and the generate_series function. This approach avoids the complexities of recursion and ensures that the query terminates correctly.

Conclusion

The issue of infinite loops in recursive CTEs with UNION ALL arises due to the fundamental differences between UNION and UNION ALL in SQLite. By understanding the mechanics of recursive CTEs and the behavior of UNION ALL, we can implement solutions that prevent infinite loops and ensure correct query execution. Whether by using UNION, adding a termination condition, restructuring the query, or adopting a non-recursive approach, there are multiple ways to address the issue and achieve the desired results.