Recursive CTE concept in SQL - Time & Space Complexity

Time Complexity: Recursive CTE concept

O(n)

Understanding Time Complexity

Recursive CTEs run queries repeatedly to build results step-by-step.

We want to know how the work grows as the recursion goes deeper.

Scenario Under Consideration

Analyze the time complexity of the following recursive CTE.

WITH RECURSIVE Numbers AS (
  SELECT 1 AS num
  UNION ALL
  SELECT num + 1 FROM Numbers WHERE num < 6
)
SELECT * FROM Numbers;

This query generates numbers from 1 to 6 by repeatedly adding 1.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive step that adds 1 to the number.
• How many times: Once per number from 1 up to 6 (6 times).

How Execution Grows With Input

Each recursion adds one more number until the limit is reached.

Input Size (n)	Approx. Operations
10	About 10 steps (including base)
100	About 100 steps
1000	About 1000 steps

Pattern observation: The number of steps grows roughly the same as the input size.

Final Time Complexity

Time Complexity: O(n)

This means the work grows linearly as the recursion depth increases.

Common Mistake

[X] Wrong: "Recursive CTEs run in constant time regardless of input size."

[OK] Correct: Each recursive step adds work, so more input means more steps and more time.

Interview Connect

Understanding recursive CTE time helps you explain how queries scale and shows you can reason about repeated work.

Self-Check

"What if the recursive step added two numbers each time instead of one? How would the time complexity change?"

Practice

1. What is the main purpose of a Recursive CTE in SQL?

easy

A. To create temporary tables for one-time use

B. To join two unrelated tables

C. To optimize query performance by indexing

D. To repeatedly execute a query to process hierarchical or linked data

Solution

1. Step 1: Understand Recursive CTE definition

   A Recursive CTE runs a query repeatedly, starting from an anchor and then recursively processing related rows.

2. Step 2: Identify its main use

   This is mainly used to explore hierarchical or linked data stepwise, like organizational charts or folder trees.

3. Final Answer:

   To repeatedly execute a query to process hierarchical or linked data -> Option D

4. Quick Check:

   Recursive CTE = repeated query for linked data [OK]

Hint: Recursive CTEs repeat queries to explore linked data [OK]

Common Mistakes:

• Confusing recursive CTE with simple temporary tables
• Thinking recursive CTEs optimize indexes
• Assuming recursive CTEs join unrelated tables

2. Which of the following is the correct basic syntax to start a recursive CTE in SQL?

easy

A. WITH cte_name AS (SELECT ... UNION ALL SELECT ... FROM cte_name ...)

B. CREATE RECURSIVE cte_name AS SELECT ...

C. SELECT * FROM cte_name RECURSIVE

D. WITH RECURSIVE cte_name SELECT ... UNION SELECT ...

Solution

1. Step 1: Recall recursive CTE syntax

   The standard syntax starts with WITH cte_name AS (anchor_query UNION ALL recursive_query).

2. Step 2: Check options for correct form

   WITH cte_name AS (SELECT ... UNION ALL SELECT ... FROM cte_name ...) correctly shows the anchor and recursive parts combined with UNION ALL and references the CTE name in the recursive part.

3. Final Answer:

   WITH cte_name AS (SELECT ... UNION ALL SELECT ... FROM cte_name ...) -> Option A

4. Quick Check:

   Recursive CTE syntax = WITH + UNION ALL + self-reference [OK]

Hint: Recursive CTE uses WITH + UNION ALL + self-reference [OK]

Common Mistakes:

• Using CREATE instead of WITH
• Omitting UNION ALL
• Not referencing the CTE name recursively

3. Given the table Employees(id, manager_id) and this recursive CTE:

WITH RECURSIVE EmpCTE AS (
  SELECT id, manager_id FROM Employees WHERE id = 1
  UNION ALL
  SELECT e.id, e.manager_id FROM Employees e JOIN EmpCTE c ON e.manager_id = c.id
)
SELECT id FROM EmpCTE ORDER BY id;

What does this query return?

medium

A. All employees who have no manager

B. All employees who report directly or indirectly to employee with id 1

C. Only the employee with id 1

D. All employees in the table

Solution

1. Step 1: Analyze the anchor query

   The anchor selects employee with id = 1.

2. Step 2: Analyze the recursive part

   The recursive part joins Employees where manager_id matches the id from previous results, thus finding direct reports, then their reports, and so on.

3. Final Answer:

   All employees who report directly or indirectly to employee with id 1 -> Option B

4. Quick Check:

   Recursive CTE finds all subordinates under id 1 [OK]

Hint: Recursive join on manager_id finds all reports [OK]

Common Mistakes:

• Thinking it returns only the anchor row
• Assuming it returns employees without managers
• Believing it returns all employees regardless of manager

4. Identify the error in this recursive CTE:

WITH RECURSIVE Numbers AS (
  SELECT 1 AS num
  UNION ALL
  SELECT num + 1 FROM Numbers WHERE num < 5
)
SELECT * FROM Numbers;

medium

A. Missing anchor query

B. No termination condition in recursive part

C. Recursive CTE syntax is correct and will run successfully

D. Using UNION instead of UNION ALL causes infinite loop

Solution

1. Step 1: Check anchor query

   The anchor selects 1 AS num, which is correct.

2. Step 2: Check recursive part and termination

   The recursive part adds 1 to num and stops when num reaches 5, so it has a termination condition.

3. Final Answer:

   Recursive CTE syntax is correct and will run successfully -> Option C

4. Quick Check:

   Anchor + recursive with termination = valid recursive CTE [OK]

Hint: Check for termination condition and correct UNION ALL usage [OK]

Common Mistakes:

• Confusing UNION and UNION ALL effects
• Missing termination condition causing infinite recursion
• Assuming anchor query is missing

5. You want to calculate the factorial of 5 using a recursive CTE. Which query correctly computes this?

hard

A. WITH RECURSIVE Fact(n, fact) AS (
SELECT 1, 1
UNION ALL
SELECT n + 1, fact * (n + 1) FROM Fact WHERE n < 5
)
SELECT fact FROM Fact WHERE n = 5;

B. WITH Fact AS (
SELECT 1 AS n, 1 AS fact
UNION
SELECT n + 1, fact * n FROM Fact WHERE n <= 5
)
SELECT fact FROM Fact WHERE n = 5;

C. WITH RECURSIVE Fact(n, fact) AS (
SELECT 1, 1
UNION ALL
SELECT n + 1, fact * n FROM Fact WHERE n < 5
)
SELECT fact FROM Fact WHERE n = 5;

D. WITH RECURSIVE Fact(n, fact) AS (
SELECT 0, 1
UNION ALL
SELECT n + 1, fact * n FROM Fact WHERE n < 5
)
SELECT fact FROM Fact WHERE n = 5;

Solution

1. Step 1: Understand factorial calculation

   Factorial of n is product of all integers from 1 to n. We start with n=1, fact=1.

2. Step 2: Check recursive step correctness

   WITH RECURSIVE Fact(n, fact) AS ( SELECT 1, 1 UNION ALL SELECT n + 1, fact * (n + 1) FROM Fact WHERE n < 5 ) SELECT fact FROM Fact WHERE n = 5; correctly increments n by 1 and multiplies fact by (n+1), matching factorial definition up to 5.

3. Step 3: Verify termination and final selection

   It stops when n reaches 5 and selects fact at n=5, giving 120.

4. Final Answer:

   Option A correctly computes factorial of 5 using recursive CTE -> Option A

5. Quick Check:

   Factorial recursion: multiply by next number, stop at 5 [OK]

Hint: Multiply by n+1 each step, stop at target number [OK]

Common Mistakes:

• Starting from 0 causing multiplication by zero
• Using UNION instead of UNION ALL
• Incorrect multiplication order in recursive step