Rabin Karp String Matching in DSA C++ - Time & Space Complexity

Time Complexity: Rabin Karp String Matching

O(n + m)

Understanding Time Complexity

We want to understand how the Rabin Karp algorithm's running time changes as the text and pattern sizes grow.

Specifically, how many steps does it take to find a pattern inside a larger text?

Scenario Under Consideration

Analyze the time complexity of the following Rabin Karp code snippet.

int rabinKarp(string text, string pattern) {
    int n = text.size();
    int m = pattern.size();
    int pHash = 0, tHash = 0, base = 256, mod = 101;
    int h = 1;

    for (int i = 0; i < m - 1; i++)
        h = (h * base) % mod;

    for (int i = 0; i < m; i++) {
        pHash = (base * pHash + pattern[i]) % mod;
        tHash = (base * tHash + text[i]) % mod;
    }

    for (int i = 0; i <= n - m; i++) {
        if (pHash == tHash) {
            bool match = true;
            for (int j = 0; j < m; j++) {
                if (text[i + j] != pattern[j]) {
                    match = false;
                    break;
                }
            }
            if (match) return i;
        }
        if (i < n - m) {
            tHash = (base * (tHash - text[i] * h) + text[i + m]) % mod;
            if (tHash < 0) tHash += mod;
        }
    }
    return -1;
}
    

This code searches for a pattern inside a text using hashing to speed up comparisons.

Identify Repeating Operations

Look at the loops and repeated checks:

• Primary operation: Sliding the pattern over the text and comparing hashes.
• How many times: The main loop runs about (n - m + 1) times, where n is text length and m is pattern length.
• Inside that, if hashes match, a character-by-character check runs up to m times.

How Execution Grows With Input

As the text size grows, the number of hash comparisons grows roughly linearly.

Input Size (n)	Approx. Operations
10	About 10 hash checks, few character checks if collisions occur
100	About 100 hash checks, some character checks on hash matches
1000	About 1000 hash checks, occasional character checks

Pattern size m affects character checks only when hashes match, which is rare with a good hash.

Final Time Complexity

Time Complexity: O(n + m)

This means the algorithm usually checks each character in the text and pattern once, making it efficient for large inputs.

Common Mistake

[X] Wrong: "Rabin Karp always compares every character in the text to the pattern."

[OK] Correct: Actually, it uses hash values to skip many character comparisons, only checking characters when hashes match.

Interview Connect

Understanding Rabin Karp's time complexity shows you how hashing can speed up searching tasks, a useful skill in many coding problems.

Self-Check

"What if the hash function causes many collisions? How would the time complexity change?"

Practice

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

easy

A. Compare each character one by one without hashing

B. Use hashing to find pattern matches efficiently

C. Sort the text and pattern before matching

D. Use recursion to find all pattern occurrences

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 and then verifies them.

3. Final Answer:

   Use hashing to find pattern matches efficiently -> Option B

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 always
• Assuming it sorts strings before matching
• Confusing it with recursive search methods

2. Which of the following is the correct way to update the rolling hash in Rabin Karp when moving from substring s[i..i+m-1] to s[i+1..i+m]?

easy

A. Add leftmost char's hash, multiply by base, subtract new char's hash

B. Multiply entire hash by base and add new char's hash only

C. Remove leftmost char's hash, multiply by base, add new char's hash

D. Recalculate hash from scratch for new substring

Solution

1. Step 1: Understand rolling hash update

   Remove the leftmost character's contribution, multiply remaining hash by base, then add new character's value.

2. Step 2: Apply formula for next substring

   This keeps hash calculation efficient without recomputing from scratch.

3. Final Answer:

   Remove leftmost char's hash, multiply by base, add new char's hash -> Option C

4. Quick Check:

   Rolling hash update = Remove + multiply + add [OK]

Hint: Rolling hash = remove old char + shift + add new char [OK]

Common Mistakes:

• Recalculating hash fully each time
• Adding instead of removing old char's hash
• Multiplying after adding new char instead of before

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

std::string text = "ababc";
std::string pattern = "abc";
// Assume correct Rabin Karp implementation
std::vector<int> matches = rabinKarpSearch(text, pattern);
for (int pos : matches) std::cout << pos << " ";

medium

A. 1

B. 0 2

C. No output

D. 2

Solution

1. Step 1: Identify pattern occurrences in text

   Pattern "abc" appears starting at index 2 in "ababc" (0-based indexing).

2. Step 2: Confirm no other matches

   Substring at index 0 is "aba", not "abc"; index 1 is "bab", not "abc".

3. Final Answer:

   2 -> Option D

4. Quick Check:

   Pattern found at index 2 [OK]

Hint: Check substring positions carefully for exact matches [OK]

Common Mistakes:

• Assuming match at index 0 incorrectly
• Confusing overlapping substrings
• Using 1-based indexing instead of 0-based

4. Identify the bug in this Rabin Karp rolling hash update snippet:

hash = (hash - text[i] * pow(base, m-1)) * base + text[i+m];

What is the likely problem?

medium

A. Missing modulo operation causing overflow

B. Wrong power used in subtraction term

C. Adding instead of subtracting old char's hash

D. Incorrect base multiplier order

Solution

1. Step 1: Check arithmetic operations in rolling hash

   The formula subtracts old char's hash, multiplies by base, adds new char's value.

2. Step 2: Identify missing modulo usage

   Without modulo, hash values can overflow integer limits causing wrong results.

3. Final Answer:

   Missing modulo operation causing overflow -> Option A

4. Quick Check:

   Modulo needed to prevent overflow [OK]

Hint: Always apply modulo after each hash update [OK]

Common Mistakes:

• Ignoring modulo leading to wrong hash values
• Using wrong power exponent
• Mixing addition and subtraction signs

5. Given a large text and multiple patterns, how can Rabin Karp be adapted to efficiently find all pattern occurrences?

hard

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

B. Run Rabin Karp separately for each pattern on the entire text

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

D. Use recursion to match patterns one by one

Solution

1. Step 1: Understand multi-pattern matching 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 slide over text, hashing substrings and checking set membership.

3. Step 3: Efficiently find all matches

   This reduces repeated work and speeds up multi-pattern search.

4. Final Answer:

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

5. Quick Check:

   Combined hashing = efficient multi-pattern search [OK]

Hint: Hash all patterns once, then check text substrings [OK]

Common Mistakes:

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