Rabin Karp String Matching in DSA Python - Time & Space Complexity

Time Complexity: Rabin Karp String Matching

O(n * m) average, O(n * m) worst

Understanding Time Complexity

We want to understand how the time needed to find a pattern in a text grows as the text and pattern get bigger.

How does the Rabin Karp method handle this search efficiently?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

def rabin_karp(text, pattern):
    n, m = len(text), len(pattern)
    p_hash = hash(pattern)
    for i in range(n - m + 1):
        t_hash = hash(text[i:i+m])
        if t_hash == p_hash and text[i:i+m] == pattern:
            return i
    return -1

This code tries to find the first place where the pattern appears in the text using hashing to speed up comparisons.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Looping through the text to check each substring of pattern length.
• How many times: Approximately n - m + 1 times, where n is text length and m is pattern length.
• Secondary operation: Computing hash for each substring and comparing hashes.

How Execution Grows With Input

As the text grows, the number of substring checks grows roughly in a straight line with the text size.

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

Pattern observation: The work grows linearly with the text size, assuming pattern size is much smaller.

Final Time Complexity

Time Complexity: O(n * m) on average, O(n * m) in worst case

This means on average, the search grows roughly with the product of text and pattern sizes, and in worst case it can be slower.

Common Mistake

[X] Wrong: "Rabin Karp always runs in linear time no matter what."

[OK] Correct: Sometimes many hash collisions happen, causing extra checks that slow it down to quadratic time.

Interview Connect

Understanding Rabin Karp helps you see how hashing can speed up searching and when it might slow down, a useful skill for many coding problems.

Self-Check

"What if we used a rolling hash to avoid recomputing the hash from scratch each time? How would the time complexity change?"

Practice

1. What is the main idea behind the Rabin Karp String Matching algorithm?

easy

A. It uses hashing to find pattern matches efficiently.

B. It compares each character one by one without hashing.

C. It sorts the text before searching for the pattern.

D. It uses recursion to find the pattern in the text.

Solution

1. Step 1: Understand Rabin Karp's approach

   The algorithm uses a hash function to convert the pattern and text substrings into numbers.

2. Step 2: Use hashing for efficient matching

   By comparing hash values instead of characters, it quickly finds potential matches.

3. Final Answer:

   It uses hashing to find pattern matches efficiently. -> Option A

4. Quick Check:

   Hashing = Efficient matching [OK]

Hint: Remember: Rabin Karp uses hashing for fast pattern search [OK]

Common Mistakes:

• Thinking it compares characters directly without hashing
• Confusing it with sorting-based algorithms
• Assuming it uses recursion

2. Which of the following is the correct way to compute the rolling hash in Rabin Karp for the next substring?

easy

A. new_hash = old_hash * old_char + new_char

B. new_hash = old_hash + old_char - new_char

C. new_hash = (old_hash - old_char * base^(m-1)) * base + new_char

D. new_hash = old_hash / base + new_char * base^(m-1)

Solution

1. Step 1: Recall rolling hash formula

   Rolling hash removes the old char's contribution and adds the new char's value.

2. Step 2: Apply formula correctly

   It subtracts old_char * base^(m-1), multiplies by base, then adds new_char.

3. Final Answer:

   new_hash = (old_hash - old_char * base^(m-1)) * base + new_char -> Option C

4. Quick Check:

   Rolling hash formula = new_hash = (old_hash - old_char * base^(m-1)) * base + new_char [OK]

Hint: Subtract old char, multiply by base, add new char [OK]

Common Mistakes:

• Adding old_char instead of subtracting
• Dividing instead of multiplying by base
• Ignoring power of base for old_char

3. What will be the output of the following code snippet using Rabin Karp to find pattern 'abc' in text 'ababc'?

def rabin_karp(text, pattern):
    base = 256
    prime = 101
    m = len(pattern)
    n = len(text)
    pattern_hash = 0
    text_hash = 0
    h = 1

    for i in range(m-1):
        h = (h * base) % prime

    for i in range(m):
        pattern_hash = (base * pattern_hash + ord(pattern[i])) % prime
        text_hash = (base * text_hash + ord(text[i])) % prime

    for i in range(n - m + 1):
        if pattern_hash == text_hash:
            if text[i:i+m] == pattern:
                return i
        if i < n - m:
            text_hash = (base * (text_hash - ord(text[i]) * h) + ord(text[i + m])) % prime
            if text_hash < 0:
                text_hash += prime
    return -1

print(rabin_karp('ababc', 'abc'))

medium

A. 0

B. 1

C. -1

D. 2

Solution

1. Step 1: Calculate pattern and initial text hash

   Pattern 'abc' length 3, initial text substring 'aba'. Hashes differ, so move forward.

2. Step 2: Slide over text and compare hashes

   At index 1, substring 'bab' hash differs. At index 2, substring 'abc' hash matches pattern hash and strings match.

3. Final Answer:

   2 -> Option D

4. Quick Check:

   Pattern found at index 2 [OK]

Hint: Check hash then substring; match at index 2 [OK]

Common Mistakes:

• Returning first hash match without substring check
• Confusing index positions
• Ignoring negative hash adjustment

4. Identify the bug in this Rabin Karp implementation snippet:

text_hash = (base * (text_hash - ord(text[i]) * h) + ord(text[i + m])) % prime
if text_hash < 0:
    text_hash = text_hash - prime

medium

A. The negative hash correction should add prime, not subtract.

B. The subtraction of ord(text[i]) * h is incorrect.

C. The modulo operation should be done after adding prime.

D. The base should be added, not multiplied.

Solution

1. Step 1: Understand negative hash correction

   If hash is negative after modulo, add prime to make it positive.

2. Step 2: Identify incorrect operation

   Code subtracts prime instead of adding it, causing wrong hash values.

3. Final Answer:

   The negative hash correction should add prime, not subtract. -> Option A

4. Quick Check:

   Negative hash fix = add prime [OK]

Hint: Add prime if hash is negative, never subtract [OK]

Common Mistakes:

• Subtracting prime instead of adding
• Ignoring negative hash values
• Misplacing modulo operation

5. Given a large text and multiple patterns, which modification to Rabin Karp helps efficiently find all patterns in the text?

hard

A. Run Rabin Karp separately for each pattern without optimization.

B. Use a combined hash set of all patterns and check text substrings against it.

C. Sort all patterns and use binary search on text substrings.

D. Use recursion to search each pattern in the text.

Solution

1. Step 1: Understand multiple pattern search challenge

   Running Rabin Karp separately for each pattern is inefficient for many patterns.

2. Step 2: Use combined hashing approach

   Hash all patterns and store in a set. Then check each text substring's hash against this set for quick matches.

3. Final Answer:

   Use a combined hash set of all patterns and check text substrings against it. -> Option B

4. Quick Check:

   Combined hash set = efficient multi-pattern search [OK]

Hint: Hash all patterns once, check text hashes against set [OK]

Common Mistakes:

• Running separate searches for each pattern
• Sorting patterns instead of hashing
• Using recursion unnecessarily