Regular expressions for token patterns in Compiler Design - Time & Space Complexity

Time Complexity: Regular expressions for token patterns

O(m * n)

Understanding Time Complexity

We want to understand how the time to match tokens using regular expressions changes as the input grows.

How does the matching process scale when scanning longer source code?

Scenario Under Consideration

Analyze the time complexity of the following token matching using regular expressions.

for each token_pattern in token_patterns:
    if regex_match(token_pattern, input_string):
        return matched_token
return no_match
    

This code tries each token pattern one by one to find a match at the start of the input string.

Identify Repeating Operations

Look for repeated checks and scans.

• Primary operation: Matching each regular expression against the input string.
• How many times: Once per token pattern, scanning input characters as needed.

How Execution Grows With Input

As the input string gets longer, each regex may need to scan more characters to confirm a match or failure.

Input Size (n)	Approx. Operations
10	Checks each pattern scanning up to 10 chars
100	Checks each pattern scanning up to 100 chars
1000	Checks each pattern scanning up to 1000 chars

Pattern observation: The work grows roughly in proportion to the input length times the number of token patterns.

Final Time Complexity

Time Complexity: O(m * n)

This means the time grows linearly with both the number of token patterns (m) and the input length (n).

Common Mistake

[X] Wrong: "Matching tokens with regex is always constant time regardless of input size."

[OK] Correct: Regex matching scans input characters, so longer inputs take more time to check.

Interview Connect

Understanding how regex matching scales helps you explain lexer performance and design efficient tokenizers.

Self-Check

What if we used a single combined regex for all tokens instead of checking each pattern separately? How would the time complexity change?

Practice

1. What is the main purpose of using regular expressions in compiler design?

easy

A. To define patterns for recognizing tokens in source code

B. To execute arithmetic operations

C. To optimize machine code

D. To manage memory allocation

Solution

1. Step 1: Understand the role of regular expressions

   Regular expressions are used to describe patterns that match sequences of characters.

2. Step 2: Connect to compiler design

   In compiler design, these patterns help identify tokens like keywords, identifiers, and symbols.

3. Final Answer:

   To define patterns for recognizing tokens in source code -> Option A

4. Quick Check:

   Regular expressions = token patterns [OK]

Hint: Regular expressions match text patterns for tokens [OK]

Common Mistakes:

• Confusing regular expressions with code execution
• Thinking they optimize code performance
• Mixing up with memory management

2. Which of the following is the correct regular expression to match an identifier starting with a letter followed by letters or digits?

easy

A. [a-zA-Z][a-zA-Z0-9]*

B. [0-9][a-zA-Z]*

C. [a-zA-Z0-9]+[a-zA-Z]*

D. [0-9]+[a-zA-Z0-9]*

Solution

1. Step 1: Identify the pattern for an identifier

   An identifier starts with a letter, then can have letters or digits.

2. Step 2: Match the regular expression

   [a-zA-Z][a-zA-Z0-9]* matches one letter [a-zA-Z] followed by zero or more letters or digits [a-zA-Z0-9]*.

3. Final Answer:

   [a-zA-Z][a-zA-Z0-9]* -> Option A

4. Quick Check:

   Letter + letters/digits = [a-zA-Z][a-zA-Z0-9]* [OK]

Hint: Start with letter, then letters or digits [OK]

Common Mistakes:

• Starting identifier with a digit
• Using + instead of * incorrectly
• Allowing digits as first character

3. Given the regular expression \d{3}-\d{2}-\d{4}, what kind of token does it match?

medium

A. A date in format YYYY-MM-DD

B. A phone number with area code

C. A US Social Security Number format

D. An IP address

Solution

1. Step 1: Analyze the regular expression pattern

   \d{3} means exactly 3 digits, then a dash, then 2 digits, dash, then 4 digits.

2. Step 2: Identify the token format

   This matches the US Social Security Number format: 3 digits - 2 digits - 4 digits.

3. Final Answer:

   A US Social Security Number format -> Option C

4. Quick Check:

   \d{3}-\d{2}-\d{4} = SSN format [OK]

Hint: Count digits and dashes to identify pattern [OK]

Common Mistakes:

• Confusing with date formats
• Assuming phone number format
• Ignoring exact digit counts

4. Identify the error in this regular expression intended to match floating-point numbers: \d+\.\d*

medium

A. It requires at least one digit after decimal point

B. The dot is not escaped properly

C. It matches only integers

D. It does not allow numbers without decimal part

Solution

1. Step 1: Understand the pattern meaning

   \d+ means one or more digits, \. means a literal dot, \d* means zero or more digits after dot.

2. Step 2: Check what numbers it matches

   This matches numbers with a decimal point and optional digits after it, but does not match integers without a dot.

3. Final Answer:

   It does not allow numbers without decimal part -> Option D

4. Quick Check:

   Pattern requires dot, so no integers [OK]

Hint: Check if integers without dot are matched [OK]

Common Mistakes:

• Thinking dot is unescaped
• Assuming digits after dot are mandatory
• Confusing * and + quantifiers

5. How would you write a regular expression to match either the keyword if or else but not identifiers containing these as substrings?

hard

A. (if|else)+

B. \b(if|else)\b

C. if|else

D. [ifelse]+

Solution

1. Step 1: Understand the requirement

   We want to match exactly the words 'if' or 'else', not parts of longer words.

2. Step 2: Use word boundaries

   \b marks word boundaries, so \b(if|else)\b matches only whole words 'if' or 'else'.

3. Step 3: Check other options

   (if|else)+ allows repeated keywords, if|else matches substrings anywhere, [ifelse]+ matches any combination of letters i,f,e,l,s.

4. Final Answer:

   \b(if|else)\b -> Option B

5. Quick Check:

   Word boundaries ensure exact keyword match [OK]

Hint: Use \b for exact word matching [OK]

Common Mistakes:

• Not using word boundaries
• Using character classes instead of groups
• Allowing partial matches inside identifiers