Recursive CTE for graph traversal in PostgreSQL - Time & Space Complexity

Time Complexity: Recursive CTE for graph traversal

O(n + m)

Understanding Time Complexity

When using recursive queries to explore graphs, it's important to know how the work grows as the graph gets bigger.

We want to understand how the number of steps changes when the graph has more nodes and edges.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

WITH RECURSIVE traverse AS (
  SELECT id, parent_id
  FROM nodes
  WHERE id = 1
  UNION ALL
  SELECT n.id, n.parent_id
  FROM nodes n
  JOIN traverse t ON n.parent_id = t.id
)
SELECT * FROM traverse;

This query starts from node 1 and recursively finds all connected child nodes in the graph.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive join to find child nodes for each node found.
• How many times: Once for each node reachable from the start node.

How Execution Grows With Input

Each new node found triggers a join to find its children, so the work grows with the number of reachable nodes and edges.

Input Size (n)	Approx. Operations
10	About 10 node checks and their edges
100	About 100 node checks and their edges
1000	About 1000 node checks and their edges

Pattern observation: The operations grow roughly linearly with the number of nodes and edges visited.

Final Time Complexity

Time Complexity: O(n + m)

This means the time grows in proportion to the number of nodes (n) plus the number of edges (m) explored in the graph.

Common Mistake

[X] Wrong: "The recursive query runs in constant time regardless of graph size."

[OK] Correct: Each node and edge must be checked once, so the work grows as the graph grows.

Interview Connect

Understanding recursive queries and their time complexity helps you explain how databases handle graph data efficiently, a useful skill in many real-world tasks.

Self-Check

"What if the graph contains cycles and the query does not prevent revisiting nodes? How would the time complexity change?"

Practice

1. What is the main purpose of using a WITH RECURSIVE clause in PostgreSQL when working with graph data?

easy

A. To optimize query performance by indexing graph nodes

B. To create temporary tables for storing graph data

C. To repeatedly execute a query that references itself to explore connected nodes

D. To join two unrelated tables without conditions

Solution

1. Step 1: Understand the purpose of recursive CTEs

   Recursive CTEs allow a query to call itself repeatedly, which is useful for exploring connected data like graph nodes.

2. Step 2: Identify the correct use case

   Using WITH RECURSIVE lets you traverse graph structures step-by-step by referencing the CTE inside itself.

3. Final Answer:

   To repeatedly execute a query that references itself to explore connected nodes -> Option C

4. Quick Check:

   Recursive CTE = self-referencing query [OK]

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

Common Mistakes:

• Confusing recursive CTE with temporary tables
• Thinking it automatically optimizes performance
• Using it for unrelated table joins

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

easy

A. WITH path RECURSIVE AS (SELECT ... UNION ALL SELECT ... FROM path ...)

B. CREATE RECURSIVE TABLE path AS SELECT ...

C. SELECT RECURSIVE * FROM path

D. WITH RECURSIVE path AS (SELECT ... UNION ALL SELECT ... FROM path ...)

Solution

1. Step 1: Recall the syntax for recursive CTEs

   The correct syntax starts with WITH RECURSIVE cte_name AS ( ... ), where the CTE references itself inside.

2. Step 2: Match the syntax options

   WITH RECURSIVE path AS (SELECT ... UNION ALL SELECT ... FROM path ...) correctly uses WITH RECURSIVE path AS (SELECT ... UNION ALL SELECT ... FROM path ...). Other options misuse keywords or order.

3. Final Answer:

   WITH RECURSIVE path AS (SELECT ... UNION ALL SELECT ... FROM path ...) -> Option D

4. Quick Check:

   WITH RECURSIVE + CTE name + self-reference [OK]

Hint: Start with WITH RECURSIVE cte_name AS (...) [OK]

Common Mistakes:

• Swapping RECURSIVE and CTE name order
• Using CREATE instead of WITH
• Omitting UNION ALL between parts

3. Given the table edges(source INT, target INT) with data:
(1, 2), (2, 3), (3, 4), (2, 5)
What will the following query return?

WITH RECURSIVE path AS (SELECT source, target FROM edges WHERE source = 1 UNION ALL SELECT e.source, e.target FROM edges e JOIN path p ON e.source = p.target) SELECT * FROM path ORDER BY source, target;

medium

A. Rows: (1,2), (2,3), (2,5), (3,4)

B. Rows: (1,2), (2,3), (3,4)

C. Rows: (1,2), (2,5)

D. Empty result

Solution

1. Step 1: Identify the anchor and recursive parts

   The anchor selects edges where source=1: (1,2). The recursive part joins edges where source matches previous target.

2. Step 2: Trace recursive expansion

   From (1,2), next edges with source=2 are (2,3) and (2,5). From (2,3), next edge with source=3 is (3,4). No further edges from 5 or 4.

3. Final Answer:

   Rows: (1,2), (2,3), (2,5), (3,4) -> Option A

4. Quick Check:

   Recursive join expands all reachable edges [OK]

Hint: Follow edges from start node stepwise [OK]

Common Mistakes:

• Ignoring multiple edges from same source
• Stopping recursion too early
• Misordering results

4. Consider this recursive CTE:

WITH RECURSIVE path(node) AS (SELECT 1 UNION ALL SELECT node + 1 FROM path WHERE node < 3) SELECT * FROM path;

What is wrong if this query runs forever without stopping?

medium

A. Not selecting from the CTE in the final SELECT

B. Missing a proper termination condition in the recursive part

C. Using UNION instead of UNION ALL

D. Incorrect anchor query syntax

Solution

1. Step 1: Analyze the recursive condition

   The recursive part adds 1 to node while node < 3, but due to UNION ALL accumulating duplicates of rows satisfying the condition, recursion continues indefinitely.

2. Step 2: Identify why it loops infinitely

   If the termination condition is incorrect or not reached, recursion never stops. Here, the condition is present but ineffective against duplicates, causing infinite loop.

3. Final Answer:

   Missing a proper termination condition in the recursive part -> Option B

4. Quick Check:

   Infinite recursion = no stop condition [OK]

Hint: Check recursive WHERE clause stops recursion [OK]

Common Mistakes:

• Confusing UNION and UNION ALL effects
• Ignoring anchor query correctness
• Forgetting to select from CTE

5. You have a table employees(id INT, manager_id INT) representing a company hierarchy. How would you write a recursive CTE to find all employees under manager with id = 10, including indirect reports?

hard

A. WITH RECURSIVE subordinates AS (
SELECT id FROM employees WHERE manager_id = 10
UNION ALL
SELECT e.id FROM employees e JOIN subordinates s ON e.manager_id = s.id
) SELECT * FROM subordinates;

B. WITH RECURSIVE subordinates AS (
SELECT id FROM employees WHERE id = 10
UNION ALL
SELECT e.id FROM employees e JOIN subordinates s ON e.id = s.manager_id
) SELECT * FROM subordinates;

C. WITH RECURSIVE subordinates AS (
SELECT id FROM employees WHERE manager_id = 10
UNION ALL
SELECT e.manager_id FROM employees e JOIN subordinates s ON e.id = s.id
) SELECT * FROM subordinates;

D. WITH RECURSIVE subordinates AS (
SELECT id FROM employees WHERE manager_id = 10
UNION ALL
SELECT e.id FROM employees e JOIN subordinates s ON e.id = s.manager_id
) SELECT * FROM subordinates;

Solution

1. Step 1: Identify the anchor query

   The anchor selects employees directly reporting to manager 10: WHERE manager_id = 10.

2. Step 2: Understand the recursive join

   The recursive part joins employees whose manager_id matches any previously found employee id, thus finding indirect reports.

3. Final Answer:

   WITH RECURSIVE subordinates AS ( SELECT id FROM employees WHERE manager_id = 10 UNION ALL SELECT e.id FROM employees e JOIN subordinates s ON e.manager_id = s.id ) SELECT * FROM subordinates; -> Option A

4. Quick Check:

   Join on e.manager_id = s.id to find subordinates [OK]

Hint: Join on manager_id = previous employee id to find reports [OK]

Common Mistakes:

• Swapping join conditions (id vs manager_id)
• Starting anchor with wrong WHERE clause
• Selecting wrong columns in recursive part