Recursive CTEs in MySQL - Time & Space Complexity

Time Complexity: Recursive CTEs

O(n)

Understanding Time Complexity

Recursive CTEs run queries that call themselves repeatedly to build results step by step.

We want to know how the time needed grows as the data or recursion depth grows.

Scenario Under Consideration

Analyze the time complexity of the following recursive CTE in MySQL.

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

This query generates numbers from 1 to 5 by repeatedly adding 1 until it reaches 5.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive step that adds 1 to the previous number.
• How many times: The recursion runs once for each number from 1 up to 5.

How Execution Grows With Input

Each recursive call adds one more number until the limit is reached.

Input Size (n)	Approx. Operations
10	10 recursive steps
100	100 recursive steps
1000	1000 recursive steps

Pattern observation: The number of operations grows directly with the number of recursive steps.

Final Time Complexity

Time Complexity: O(n)

This means the time needed grows linearly as the recursion depth or input size increases.

Common Mistake

[X] Wrong: "Recursive CTEs run all steps at once, so time stays the same no matter the input size."

[OK] Correct: Each recursive step depends on the previous one, so the query runs more steps as the input grows, increasing time.

Interview Connect

Understanding recursive CTE time helps you explain how queries scale and shows you can reason about complex SQL operations.

Self-Check

"What if we changed the recursion to add 2 each time instead of 1? How would the time complexity change?"

Practice

1. What is the main purpose of a recursive CTE in MySQL?

easy

A. To create temporary tables for storing large datasets

B. To repeatedly execute a query to explore hierarchical or connected data

C. To speed up simple SELECT queries without joins

D. To permanently store data in the database

Solution

1. Step 1: Understand recursive CTE concept

   A recursive CTE runs a base query and then repeatedly runs a recursive part referring back to itself to explore connected data.

2. Step 2: Identify the purpose

   This process is mainly used to handle hierarchical or tree-like data, such as organizational charts or folder structures.

3. Final Answer:

   To repeatedly execute a query to explore hierarchical or connected data -> Option B

4. Quick Check:

   Recursive CTE = hierarchical data exploration [OK]

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

Common Mistakes:

• Confusing recursive CTEs with temporary tables
• Thinking recursive CTEs speed up all queries
• Assuming recursive CTEs permanently store data

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

easy

A. WITH cte_name RECURSIVE (anchor_query UNION recursive_query) SELECT * FROM cte_name;

B. CREATE RECURSIVE TABLE cte_name AS (anchor_query UNION recursive_query);

C. WITH RECURSIVE cte_name AS (anchor_query UNION ALL recursive_query) SELECT * FROM cte_name;

D. SELECT RECURSIVE * FROM cte_name WHERE condition;

Solution

1. Step 1: Recall MySQL recursive CTE syntax

   MySQL uses the keyword WITH RECURSIVE followed by the CTE name and a query combining an anchor and recursive part with UNION ALL.

2. Step 2: Match syntax options

   WITH RECURSIVE cte_name AS (anchor_query UNION ALL recursive_query) SELECT * FROM cte_name; correctly shows this syntax, including UNION ALL and the final SELECT from the CTE.

3. Final Answer:

   WITH RECURSIVE cte_name AS (anchor_query UNION ALL recursive_query) SELECT * FROM cte_name; -> Option C

4. Quick Check:

   WITH RECURSIVE + UNION ALL = correct syntax [OK]

Hint: Look for WITH RECURSIVE and UNION ALL keywords [OK]

Common Mistakes:

• Using UNION instead of UNION ALL in recursive part
• Trying to CREATE RECURSIVE TABLE (not valid)
• Misplacing RECURSIVE keyword after CTE name

3. Given the table employees(id, manager_id, name), what will this query return?

WITH RECURSIVE subordinates AS (
  SELECT id, name FROM employees WHERE id = 2
  UNION ALL
  SELECT e.id, e.name FROM employees e
  JOIN subordinates s ON e.manager_id = s.id
)
SELECT * FROM subordinates;

medium

A. All employees who report directly or indirectly to employee with id 2

B. Only the employee with id 2

C. All employees who manage employee with id 2

D. Syntax error due to missing anchor query

Solution

1. Step 1: Analyze anchor query

   The anchor selects the employee with id = 2 as the starting point.

2. Step 2: Analyze recursive part

   The recursive part joins employees whose manager_id matches the current subordinates' id, thus finding direct reports, then their reports, and so on.

3. Step 3: Understand final output

   The query returns employee 2 and all employees reporting to them directly or indirectly.

4. Final Answer:

   All employees who report directly or indirectly to employee with id 2 -> Option A

5. Quick Check:

   Recursive join on manager_id finds all subordinates [OK]

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

Common Mistakes:

• Confusing manager and subordinate roles
• Thinking it returns only direct reports
• Assuming syntax error due to anchor query presence

4. Identify the error in this recursive CTE:

WITH RECURSIVE cte AS (
  SELECT id FROM items WHERE parent_id IS NULL
  UNION
  SELECT i.id FROM items i JOIN cte ON i.parent_id = cte.id
)
SELECT * FROM cte;

medium

A. Anchor query should use WHERE parent_id = 0 instead of IS NULL

B. Recursive part should use LEFT JOIN instead of JOIN

C. CTE name cannot be 'cte', must be more descriptive

D. Missing UNION ALL causes infinite recursion or wrong results

Solution

1. Step 1: Check recursive CTE syntax

   Recursive CTEs require UNION ALL to combine anchor and recursive parts to avoid duplicate elimination.

2. Step 2: Understand effect of UNION vs UNION ALL

   Using UNION removes duplicates each iteration, which can cause infinite recursion or incorrect results in recursive queries.

3. Final Answer:

   Missing UNION ALL causes infinite recursion or wrong results -> Option D

4. Quick Check:

   Recursive CTEs must use UNION ALL [OK]

Hint: Always use UNION ALL in recursive CTEs to avoid infinite loops [OK]

Common Mistakes:

• Using UNION instead of UNION ALL in recursive CTE
• Assuming NULL vs 0 matters for recursion logic
• Thinking JOIN type affects recursion correctness

5. You want to list all ancestors of an employee in a company hierarchy stored in employees(id, manager_id, name). Which recursive CTE query correctly finds all managers up the chain for employee with id 5?

hard

A. WITH RECURSIVE ancestors AS (
SELECT id, manager_id, name FROM employees WHERE id = 5
UNION ALL
SELECT e.id, e.manager_id, e.name FROM employees e
JOIN ancestors a ON e.id = a.manager_id
)
SELECT * FROM ancestors WHERE id != 5;

B. WITH RECURSIVE ancestors AS (
SELECT id, manager_id, name FROM employees WHERE id = 5
UNION ALL
SELECT e.id, e.manager_id, e.name FROM employees e
JOIN ancestors a ON e.manager_id = a.id
)
SELECT * FROM ancestors WHERE id != 5;

C. WITH RECURSIVE ancestors AS (
SELECT id, manager_id, name FROM employees WHERE manager_id = 5
UNION ALL
SELECT e.id, e.manager_id, e.name FROM employees e
JOIN ancestors a ON e.manager_id = a.id
)
SELECT * FROM ancestors;

D. WITH RECURSIVE ancestors AS (
SELECT id, manager_id, name FROM employees WHERE id = 5
UNION
SELECT e.id, e.manager_id, e.name FROM employees e
JOIN ancestors a ON e.id = a.manager_id
)
SELECT * FROM ancestors WHERE id != 5;

Solution

1. Step 1: Understand the goal

   We want to find all managers (ancestors) of employee 5, moving up the chain via manager_id.

2. Step 2: Analyze join condition

   To move up, join employees where employee id matches the manager_id of current ancestor (e.id = a.manager_id).

3. Step 3: Check anchor and recursive parts

   Anchor selects employee 5; recursive part finds their manager, then that manager's manager, etc.

4. Final Answer:

   WITH RECURSIVE ancestors AS (SELECT id, manager_id, name FROM employees WHERE id = 5 UNION ALL SELECT e.id, e.manager_id, e.name FROM employees e JOIN ancestors a ON e.id = a.manager_id) SELECT * FROM ancestors WHERE id != 5; -> Option A

5. Quick Check:

   Join on e.id = a.manager_id moves up hierarchy [OK]

Hint: Join on e.id = a.manager_id to move up the hierarchy [OK]

Common Mistakes:

• Reversing join condition to move down instead of up
• Using UNION instead of UNION ALL causing errors
• Starting anchor with manager_id = 5 instead of id = 5