KMP Algorithm Explained: A Beginner’s Guide

The Knuth-Morris-Pratt (KMP) algorithm is a powerful string-searching algorithm that allows for efficient location of a substring (pattern) within a larger string (text). Unlike naive string searching, which might repeatedly backtrack through the text, KMP leverages pre-computed information about the pattern to avoid redundant comparisons. This results in a linear time complexity, making it significantly faster, especially for patterns with repeating substrings. This article will provide a comprehensive beginner’s guide to the KMP algorithm, covering its core concepts, implementation, and practical applications.

1. The Problem: String Searching

The fundamental problem that KMP solves is finding occurrences of a pattern within a text. Imagine you have a large text document and you want to locate all instances of a specific word or phrase. The naive approach involves comparing the pattern character by character with the text, starting from the beginning of the text. If a mismatch occurs, the pattern is shifted one position to the right, and the comparison process restarts from the beginning of the pattern. This can be highly inefficient, especially when the pattern has repeating substrings.

2. The KMP Solution: Leveraging Prefixes and Suffixes

The key insight behind KMP is the observation that when a mismatch occurs, we often don’t need to restart the comparison from the beginning of the pattern. Instead, we can utilize information about the pattern itself to shift the pattern by a specific amount, skipping redundant comparisons. This information is encoded in the “Longest Proper Prefix Suffix” (LPS) array.

3. Understanding the LPS Array

The LPS array is the heart of the KMP algorithm. For each index i in the pattern, lps[i] stores the length of the longest proper prefix of the pattern (excluding the entire pattern itself) that is also a suffix of the pattern up to index i.

Let’s break this down with an example:

Pattern: ABABCABAB

Index (i)	Pattern[0..i]	Longest Proper Prefix Suffix (LPS[i])
0	A	0
1	AB	0
2	ABA	1
3	ABAB	2
4	ABABC	0
5	ABABCA	1
6	ABABCAB	2
7	ABABCABA	3
8	ABABCABAB	4

Explanation:

• LPS[0] is always 0. A single character cannot have a proper prefix.
• LPS[1] (Pattern “AB”): No proper prefix of “AB” is also a suffix. Hence, LPS[1] = 0.
• LPS[2] (Pattern “ABA”): The longest proper prefix “A” is also a suffix. Hence, LPS[2] = 1.
• LPS[3] (Pattern “ABAB”): The longest proper prefix “AB” is also a suffix. Hence, LPS[3] = 2.
• LPS[4] (Pattern “ABABC”): No proper prefix is also a suffix. Hence, LPS[4] = 0.
• …and so on.

4. Constructing the LPS Array

The LPS array is constructed efficiently in O(m) time, where m is the length of the pattern. The process involves comparing the pattern with itself and keeping track of the length of the longest proper prefix suffix found so far.

Here’s a step-by-step algorithm for constructing the LPS array:

1. Initialize lps[0] = 0.
2. Initialize length = 0 (length of the current longest proper prefix suffix).
3. Iterate from i = 1 to m-1:
   a. If pattern[i] == pattern[length]:
   Increment length.
   Set lps[i] = length.
   Increment i.
   b. If pattern[i] != pattern[length]:
   If length > 0:
   Set length = lps[length - 1]. (This is the crucial step that avoids redundant comparisons)
   Else:
   Set lps[i] = 0.
   Increment i.

5. The KMP Search Algorithm

Once the LPS array is constructed, the KMP search algorithm can efficiently locate occurrences of the pattern within the text.

Here’s the algorithm:

1. Initialize i = 0 (index for text) and j = 0 (index for pattern).
2. Iterate through the text while i < n (where n is the length of the text):
   a. If text[i] == pattern[j]:
   Increment both i and j.
   b. If j == m (pattern found):
   Print the index i - j (starting index of the match).
   Set j = lps[j - 1] (prepare for the next potential match).
   c. If text[i] != pattern[j] (mismatch):
   If j > 0:
   Set j = lps[j - 1] (shift the pattern).
   Else:
   Increment i.

6. Time Complexity Analysis

The KMP algorithm has a time complexity of O(n + m), where n is the length of the text and m is the length of the pattern. This linear time complexity is a significant improvement over the naive approach, which can have a worst-case time complexity of O(n*m).

7. Implementation in Python

“`python
def compute_lps_array(pattern):
m = len(pattern)
lps = [0] * m
length = 0
i = 1
while i < m:
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

def kmp_search(text, pattern):
n = len(text)
m = len(pattern)
lps = compute_lps_array(pattern)
i = 0
j = 0
while i < n:
if text[i] == pattern[j]:
i += 1
j += 1
if j == m:
print(“Pattern found at index”, i – j)
j = lps[j – 1]
elif i < n and text[i] != pattern[j]:
if j != 0:
j = lps[j – 1]
else:
i += 1

Example usage:

text = “ABABDABACDABABCABAB”
pattern = “ABABCABAB”
kmp_search(text, pattern) # Output: Pattern found at index 10
“`

8. Applications of the KMP Algorithm

The KMP algorithm has various applications in computer science, including:

• Text editors: Finding and replacing text.
• Intrusion detection systems: Identifying malicious patterns in network traffic.
• Bioinformatics: Searching for specific DNA sequences within a genome.
• Data compression: Identifying repeating patterns for compression.

9. Conclusion

The KMP algorithm is a powerful and efficient string-searching algorithm that significantly improves upon the naive approach. By pre-computing the LPS array, KMP avoids redundant comparisons and achieves linear time complexity. This makes it a valuable tool for various applications requiring efficient string searching. Understanding the concepts behind the LPS array and the KMP search algorithm is crucial for any programmer dealing with string manipulation and pattern matching. This guide provides a solid foundation for beginners to grasp the intricacies of the KMP algorithm and its practical applications.