How an index works (B-tree mental model) in SQL - Performance & Efficiency

Time Complexity: How an index works (B-tree mental model)

O(log n)

Understanding Time Complexity

We want to understand how fast searching is when using an index in a database.

Specifically, how the work grows as the data gets bigger.

Scenario Under Consideration

Analyze the time complexity of this query using a B-tree index.

SELECT * FROM employees
WHERE employee_id = 12345;
-- Assume employee_id is indexed with a B-tree

This query looks up one employee by their ID using a B-tree index.

Identify Repeating Operations

Look at what repeats when searching the B-tree.

• Primary operation: Comparing the search key to node keys in the B-tree.
• How many times: Once per tree level, moving down from root to leaf.

How Execution Grows With Input

As the number of employees grows, the tree gets taller slowly.

Input Size (n)	Approx. Operations
10	About 2 comparisons
100	About 3 comparisons
1000	About 4 comparisons

Pattern observation: The number of steps grows very slowly, adding just one step for about every 10 times more data.

Final Time Complexity

Time Complexity: O(log n)

This means the search time grows slowly as data grows, making lookups efficient even for large data.

Common Mistake

[X] Wrong: "Searching an index is as slow as scanning all rows one by one."

[OK] Correct: The B-tree index lets the database skip most rows by jumping through tree levels, so it does not check every row.

Interview Connect

Knowing how indexes speed up searches helps you explain database efficiency clearly and confidently.

Self-Check

"What if the index was a simple list instead of a B-tree? How would the time complexity change?"

Practice

1. What is the main purpose of a B-tree index in a database?

easy

A. To speed up data retrieval by organizing data in a balanced tree structure

B. To store data in a random order for faster insertion

C. To compress data to save disk space

D. To backup the database automatically

Solution

1. Step 1: Understand what a B-tree index does

   A B-tree index organizes data in a balanced tree structure to allow quick searching without scanning the entire table.

2. Step 2: Compare options with B-tree purpose

   Only To speed up data retrieval by organizing data in a balanced tree structure describes speeding up data retrieval using a balanced tree, which matches the B-tree index function.

3. Final Answer:

   To speed up data retrieval by organizing data in a balanced tree structure -> Option A

4. Quick Check:

   B-tree index = fast data lookup [OK]

Hint: B-tree indexes speed up searches by organizing data hierarchically [OK]

Common Mistakes:

• Thinking indexes store data randomly
• Confusing index with data compression
• Believing indexes backup data

2. Which SQL statement correctly creates a B-tree index on the column last_name of the table employees?

easy

A. CREATE INDEX idx_last_name ON employees USING HASH (last_name);

B. CREATE INDEX idx_last_name ON employees (last_name) USING BITMAP;

C. CREATE INDEX idx_last_name ON employees (last_name);

D. CREATE INDEX idx_last_name ON employees USING FULLTEXT (last_name);

Solution

1. Step 1: Identify default index type in SQL

   By default, CREATE INDEX creates a B-tree index unless specified otherwise.

2. Step 2: Check syntax correctness

   CREATE INDEX idx_last_name ON employees (last_name); uses correct syntax without specifying a different index type, so it creates a B-tree index on last_name.

3. Final Answer:

   CREATE INDEX idx_last_name ON employees (last_name); -> Option C

4. Quick Check:

   Default index type = B-tree [OK]

Hint: Default index type is B-tree unless stated otherwise [OK]

Common Mistakes:

• Using HASH or FULLTEXT without need
• Adding unsupported index types like BITMAP in standard SQL
• Incorrect syntax for index creation

3. Given a table products with a B-tree index on product_id, what will the query optimizer most likely do when running:
SELECT * FROM products WHERE product_id = 123;?

medium

A. Use the B-tree index to quickly locate the row with product_id 123

B. Perform a full table scan ignoring the index

C. Use a hash index lookup instead of the B-tree

D. Scan only the first 10 rows of the table

Solution

1. Step 1: Understand query condition and index

   The query searches for a specific product_id value, which matches the indexed column.

2. Step 2: Predict optimizer behavior with B-tree index

   The optimizer uses the B-tree index to quickly find the matching row without scanning the whole table.

3. Final Answer:

   Use the B-tree index to quickly locate the row with product_id 123 -> Option A

4. Quick Check:

   Indexed search = fast lookup [OK]

Hint: Queries on indexed columns use B-tree for fast lookup [OK]

Common Mistakes:

• Assuming full table scan always happens
• Confusing B-tree with hash index usage
• Thinking partial scan happens without index

4. You created a B-tree index on the email column of the users table, but queries filtering by email are still slow. What is a likely cause?

medium

A. The index was created on the wrong column

B. The query uses a function on email preventing index use

C. B-tree indexes do not speed up queries on text columns

D. The table has too few rows for the index to help

Solution

1. Step 1: Analyze query and index interaction

   If the query applies a function (like LOWER(email)) on the indexed column, the B-tree index cannot be used directly.

2. Step 2: Evaluate other options

   The index was created on the wrong column is unlikely if index was created correctly; C is false because B-tree indexes work on text; D is possible but less likely than B.

3. Final Answer:

   The query uses a function on email preventing index use -> Option B

4. Quick Check:

   Functions on indexed columns block B-tree use [OK]

Hint: Functions on indexed columns disable B-tree index use [OK]

Common Mistakes:

• Assuming B-tree indexes don't work on text
• Ignoring query transformations on columns
• Blaming index creation without checking query

5. You have a large table orders with columns order_id, customer_id, and order_date. You often query orders by customer_id and sort by order_date. How should you create an index to optimize these queries?

hard

A. Create a single-column B-tree index on order_date

B. Create separate B-tree indexes on customer_id and order_date

C. Create a hash index on customer_id

D. Create a composite B-tree index on (customer_id, order_date)

Solution

1. Step 1: Understand query pattern

   Queries filter by customer_id and sort by order_date, so both columns matter together.

2. Step 2: Choose index type for filtering and sorting

   A composite B-tree index on (customer_id, order_date) supports filtering by customer and sorting by date efficiently.

3. Final Answer:

   Create a composite B-tree index on (customer_id, order_date) -> Option D

4. Quick Check:

   Composite index matches filter + sort columns [OK]

Hint: Composite indexes speed up queries filtering and sorting on multiple columns [OK]

Common Mistakes:

• Creating index only on sort column
• Using separate indexes instead of composite
• Choosing hash index which doesn't support sorting