Finding gaps in sequences in SQL - Time & Space Complexity

Time Complexity: Finding gaps in sequences

O(n)

Understanding Time Complexity

When we look for missing numbers in a sequence stored in a database, we want to know how the time to find these gaps changes as the sequence grows.

We ask: How does the work increase when the list of numbers gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

WITH Numbered AS (
  SELECT id, ROW_NUMBER() OVER (ORDER BY id) AS rn
  FROM sequence_table
)
SELECT id + 1 AS gap_start
FROM Numbered
WHERE id + 1 <> (SELECT id FROM Numbered WHERE rn = Numbered.rn + 1)
;

This code finds gaps in a sequence by comparing each number with the next expected number.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Scanning each row in the sequence to compare with the next row.
• How many times: Once for each row in the sequence.

How Execution Grows With Input

As the number of rows grows, the database checks each row once to find gaps.

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

Pattern observation: The work grows roughly in direct proportion to the number of rows.

Final Time Complexity

Time Complexity: O(n)

This means the time to find gaps grows linearly as the sequence gets longer.

Common Mistake

[X] Wrong: "Finding gaps requires checking every possible number between the smallest and largest, so it must be much slower than just scanning the sequence."

[OK] Correct: The query only looks at existing rows once, comparing neighbors, so it does not check every missing number individually.

Interview Connect

Understanding how to find gaps efficiently shows you can work with sequences and spot missing data quickly, a useful skill in many data tasks.

Self-Check

"What if we changed the query to find gaps in an unsorted sequence without using ROW_NUMBER()? How would the time complexity change?"

Practice

1. What is the main purpose of using the LEAD() function when finding gaps in a sequence of numbers in SQL?

easy

A. To find the maximum number in the sequence

B. To compare each number with the next number in the sequence

C. To sum all numbers in the sequence

D. To sort the numbers in descending order

Solution

1. Step 1: Understand the role of LEAD()

   The LEAD() function allows you to look at the next row's value in the current row's context.

2. Step 2: Identify how LEAD() helps find gaps

   By comparing the current number with the next number, you can detect if there is a gap (difference greater than 1).

3. Final Answer:

   To compare each number with the next number in the sequence -> Option B

4. Quick Check:

   LEAD() compares current and next values [OK]

Hint: Use LEAD() to peek at the next number in sequence [OK]

Common Mistakes:

• Thinking LEAD() sums values
• Confusing LEAD() with aggregate functions
• Using LEAD() to sort data

2. Which of the following SQL snippets correctly uses LEAD() to find gaps in a sequence column named num in table numbers?

easy

A. SELECT num, LEAD(num) AS next_num FROM numbers;

B. SELECT num, LEAD(num) FROM numbers ORDER BY num;

C. SELECT num, LEAD(num) OVER (ORDER BY num) AS next_num FROM numbers;

D. SELECT num, LEAD(num) OVER () AS next_num FROM numbers;

Solution

1. Step 1: Check LEAD() syntax

   The correct syntax requires LEAD(column) OVER (ORDER BY column) to define the order for the window function.

2. Step 2: Identify the correct option

   SELECT num, LEAD(num) OVER (ORDER BY num) AS next_num FROM numbers; uses LEAD(num) OVER (ORDER BY num), which is the correct form to get the next number in order.

3. Final Answer:

   SELECT num, LEAD(num) OVER (ORDER BY num) AS next_num FROM numbers; -> Option C

4. Quick Check:

   LEAD() needs OVER with ORDER BY [OK]

Hint: Always include OVER(ORDER BY column) with LEAD() [OK]

Common Mistakes:

• Omitting OVER clause
• Not specifying ORDER BY inside OVER
• Using LEAD() without alias

3. Given the table numbers with column num containing values (1, 2, 4, 5, 7), what will the following query return?

SELECT num, LEAD(num) OVER (ORDER BY num) AS next_num FROM numbers;

medium

A. [{num:1, next_num:2}, {num:2, next_num:4}, {num:4, next_num:5}, {num:5, next_num:7}, {num:7, next_num:null}]

B. [{num:1, next_num:1}, {num:2, next_num:2}, {num:4, next_num:4}, {num:5, next_num:5}, {num:7, next_num:7}]

C. [{num:1, next_num:3}, {num:2, next_num:4}, {num:4, next_num:6}, {num:5, next_num:7}, {num:7, next_num:null}]

D. [{num:1, next_num:null}, {num:2, next_num:null}, {num:4, next_num:null}, {num:5, next_num:null}, {num:7, next_num:null}]

Solution

1. Step 1: Understand LEAD() output

   LEAD(num) returns the next row's num value ordered by num. So each row shows its own num and the next number.

2. Step 2: Apply to given data

   For values 1,2,4,5,7:
   1's next is 2
   2's next is 4
   4's next is 5
   5's next is 7
   7 has no next, so null

3. Final Answer:

   [{num:1, next_num:2}, {num:2, next_num:4}, {num:4, next_num:5}, {num:5, next_num:7}, {num:7, next_num:null}] -> Option A

4. Quick Check:

   LEAD() shifts values down by one [OK]

Hint: LEAD() shows next row's value or null if none [OK]

Common Mistakes:

• Assuming LEAD() repeats current value
• Expecting gaps filled automatically
• Confusing LEAD() with LAG()

4. The following query is intended to find gaps in the num sequence but returns no rows:

SELECT num FROM numbers WHERE LEAD(num) OVER (ORDER BY num) - num > 1;

What is the error?

medium

A. You cannot use window functions directly in WHERE clause

B. The ORDER BY inside LEAD() is missing

C. The subtraction operator is invalid here

D. LEAD() requires a GROUP BY clause

Solution

1. Step 1: Understand window function restrictions

   Window functions like LEAD() cannot be used directly in the WHERE clause because WHERE filters rows before window functions are applied.

2. Step 2: Correct approach

   Use a subquery or CTE to compute LEAD() first, then filter in an outer query.

3. Final Answer:

   You cannot use window functions directly in WHERE clause -> Option A

4. Quick Check:

   Window functions not allowed in WHERE [OK]

Hint: Use subquery or CTE to filter on window function results [OK]

Common Mistakes:

• Trying to filter window function in WHERE
• Missing OVER clause
• Expecting LEAD() to work like aggregate

5. You have a table orders with an integer column order_id that should be sequential but may have missing numbers. Which query correctly finds all missing order_id values between the minimum and maximum present?

hard

A. SELECT order_id FROM orders WHERE LEAD(order_id) - order_id > 1;

B. SELECT DISTINCT order_id + 1 FROM orders WHERE order_id + 1 NOT IN (SELECT order_id FROM orders);

C. SELECT order_id FROM orders WHERE order_id NOT IN (SELECT order_id FROM orders);

D. WITH seq AS (SELECT generate_series(MIN(order_id), MAX(order_id)) AS id FROM orders) SELECT id FROM seq LEFT JOIN orders ON seq.id = orders.order_id WHERE orders.order_id IS NULL;

Solution

1. Step 1: Generate full sequence range

   Use generate_series from minimum to maximum order_id to create all possible IDs.

2. Step 2: Find missing IDs by left join

   Left join this sequence to existing orders and select IDs where no matching order exists (NULL in orders).

3. Final Answer:

   WITH seq AS (SELECT generate_series(MIN(order_id), MAX(order_id)) AS id FROM orders) SELECT id FROM seq LEFT JOIN orders ON seq.id = orders.order_id WHERE orders.order_id IS NULL; -> Option D

4. Quick Check:

   Generate full range, then find missing with LEFT JOIN [OK]

Hint: Generate full range then find missing with LEFT JOIN [OK]

Common Mistakes:

• Using LEAD() alone misses multiple gaps
• Incorrect NOT IN subquery logic
• Assuming order_id + 1 always missing