KMP Pattern Matching Algorithm in DSA Python - Time & Space Complexity

Time Complexity: KMP Pattern Matching Algorithm

O(n + m)

Understanding Time Complexity

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

Specifically, how fast can it find a pattern inside a text?

Scenario Under Consideration

Analyze the time complexity of the following KMP pattern matching code.

def compute_lps(pattern):
    lps = [0] * len(pattern)
    length = 0
    i = 1
    while i < len(pattern):
        if pattern[i] == pattern[length]:
            length += 1
            lps[i] = length
            i += 1
        else:
            if length != 0:
                length = lps[length - 1]
            else:
                lps[i] = 0
                i += 1
    return lps
  

This code builds the longest prefix suffix (LPS) array for the pattern, which helps skip unnecessary comparisons.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop that processes each character of the pattern once.
• How many times: At most twice per character, but overall linear to pattern length.

How Execution Grows With Input

The number of operations grows roughly in a straight line with the pattern length.

Input Size (pattern length)	Approx. Operations
10	About 20
100	About 200
1000	About 2000

Pattern observation: Doubling the pattern size roughly doubles the work done.

Final Time Complexity

Time Complexity: O(n + m)

This means the algorithm checks each character of the text and pattern at most once, making it very efficient.

Common Mistake

[X] Wrong: "The KMP algorithm has a nested loop causing O(n*m) time like naive search."

[OK] Correct: KMP cleverly uses the LPS array to avoid rechecking characters, so it never goes back in the text, keeping time linear.

Interview Connect

Understanding KMP's time complexity shows you can analyze efficient algorithms that improve over simple methods, a key skill in interviews.

Self-Check

"What if the pattern is much longer than the text? How would that affect the time complexity?"

Practice

1. What is the main purpose of the KMP (Knuth-Morris-Pratt) pattern matching algorithm?

easy

A. To calculate the factorial of a number recursively

B. To sort an array of numbers in ascending order

C. To reverse a string in linear time

D. To find occurrences of a pattern string within a text efficiently

Solution

1. Step 1: Understand the goal of KMP algorithm

   KMP is designed to search for a pattern inside a larger text efficiently without rechecking characters unnecessarily.

2. Step 2: Compare with other options

   Sorting, reversing strings, and factorial calculations are unrelated to pattern searching.

3. Final Answer:

   To find occurrences of a pattern string within a text efficiently -> Option D

4. Quick Check:

   KMP = Pattern search [OK]

Hint: KMP is about fast pattern search in text [OK]

Common Mistakes:

• Confusing KMP with sorting algorithms
• Thinking KMP reverses strings
• Assuming KMP calculates mathematical functions

2. Which of the following is the correct way to initialize the prefix table (lps array) in KMP algorithm in Python?

easy

A. lps = [0] * len(pattern)

B. lps = []

C. lps = [1] * len(pattern)

D. lps = None

Solution

1. Step 1: Understand the prefix table initialization

   The prefix table (lps array) stores lengths of longest proper prefix which is also suffix for each prefix of the pattern. It must be initialized with zeros for all positions.

2. Step 2: Check each option

   lps = [0] * len(pattern) correctly initializes lps with zeros for the entire pattern length. lps = [] is empty list, which is incomplete. lps = [1] * len(pattern) initializes with ones, which is incorrect. lps = None sets None, which is invalid.

3. Final Answer:

   lps = [0] * len(pattern) -> Option A

4. Quick Check:

   Prefix table starts with zeros [OK]

Hint: Prefix table starts with zeros for all pattern chars [OK]

Common Mistakes:

• Initializing lps with ones instead of zeros
• Leaving lps empty
• Assigning None to lps

3. What will be the output of the following Python code snippet using KMP algorithm?

def compute_lps(pattern):
    lps = [0] * len(pattern)
    length = 0
    i = 1
    while i < len(pattern):
        if pattern[i] == pattern[length]:
            length += 1
            lps[i] = length
            i += 1
        else:
            if length != 0:
                length = lps[length - 1]
            else:
                lps[i] = 0
                i += 1
    return lps

pattern = "ABABC"
lps = compute_lps(pattern)
print(lps)

medium

A. [1, 0, 1, 0, 1]

B. [0, 0, 1, 2, 0]

C. [0, 1, 2, 3, 4]

D. [0, 0, 0, 0, 0]

Solution

1. Step 1: Compute lps array for pattern "ABABC"

   Index 0: lps[0] = 0 (always zero)
   Index 1: pattern[1] = 'B' != pattern[0] = 'A', lps[1] = 0
   Index 2: pattern[2] = 'A' == pattern[0] = 'A', lps[2] = 1
   Index 3: pattern[3] = 'B' == pattern[1] = 'B', lps[3] = 2
   Index 4: pattern[4] = 'C' != pattern[2] = 'A', lps[4] = 0

2. Step 2: Final lps array

   The computed lps array is [0, 0, 1, 2, 0].

3. Final Answer:

   [0, 0, 1, 2, 0] -> Option B

4. Quick Check:

   lps for "ABABC" = [0,0,1,2,0] [OK]

Hint: Match prefix and suffix lengths stepwise [OK]

Common Mistakes:

• Confusing lps values with pattern indices
• Assuming all zeros or all increasing values
• Mixing up characters when comparing

4. Identify the error in the following KMP prefix table computation code snippet:

def compute_lps(pattern):
    lps = [0] * len(pattern)
    length = 0
    i = 0
    while i < len(pattern):
        if pattern[i] == pattern[length]:
            length += 1
            lps[i] = length
            i += 1
        else:
            if length != 0:
                length = lps[length - 1]
            else:
                lps[i] = 0
                i += 1
    return lps

medium

A. The loop should start with i = 1, not i = 0

B. The lps array should be initialized with ones

C. The length variable should start at 1

D. The function should return pattern instead of lps

Solution

1. Step 1: Analyze the loop initialization

   In KMP prefix table computation, the first character's lps is always zero and we start comparing from the second character, so i should start at 1.

2. Step 2: Check the given code

   The code starts with i = 0, which causes comparing pattern[0] with pattern[0] immediately, leading to incorrect lps values and infinite loop risk.

3. Final Answer:

   The loop should start with i = 1, not i = 0 -> Option A

4. Quick Check:

   Start i at 1 for prefix table [OK]

Hint: Start prefix loop from second character (i=1) [OK]

Common Mistakes:

• Starting loop at zero causing wrong comparisons
• Initializing lps with ones instead of zeros
• Returning wrong variable

5. Given the text "ABABDABACDABABCABAB" and pattern "ABABCABAB", how many times does the pattern appear in the text using KMP algorithm?

hard

A. 2

B. 3

C. 1

D. 0

Solution

1. Step 1: Understand the pattern and text

   The pattern "ABABCABAB" is searched inside the text "ABABDABACDABABCABAB" using KMP which efficiently finds all occurrences.

2. Step 2: Identify pattern occurrences

   By scanning the text:
   - Occurrence starts at index 10 (substring "ABABCABAB")
   So pattern appears 1 time.

3. Final Answer:

   1 -> Option C

4. Quick Check:

   Pattern appears once in text [OK]

Hint: Count all pattern matches including overlaps [OK]

Common Mistakes:

• Counting only first occurrence
• Ignoring overlapping matches
• Misreading pattern or text