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

Time Complexity: KMP Pattern Matching Algorithm

O(n + m)

Understanding Time Complexity

We want to understand how the time taken by the KMP algorithm changes as the size of the text and pattern grow.

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

Scenario Under Consideration

Analyze the time complexity of the following KMP code snippet.

public void KMPSearch(String pat, String txt) {
    int M = pat.length();
    int N = txt.length();
    int[] lps = new int[M];
    computeLPSArray(pat, M, lps);
    int i = 0, j = 0;
    while (i < N) {
        if (pat.charAt(j) == txt.charAt(i)) {
            i++; j++;
        }
        if (j == M) {
            j = lps[j - 1];
        } else if (i < N && pat.charAt(j) != txt.charAt(i)) {
            if (j != 0) j = lps[j - 1];
            else i++;
        }
    }
}
    

This code searches for a pattern inside a text using the KMP algorithm, which uses a precomputed array to skip unnecessary checks.

Identify Repeating Operations

Look at the loops and repeated steps:

• Primary operation: The while loop that scans through the text once.
• How many times: The text is scanned once, and the pattern is scanned partially using the lps array to avoid repeats.
• The computeLPSArray function also loops through the pattern once to build the lps array.

How Execution Grows With Input

The algorithm processes each character in the text and pattern at most a few times.

Input Size (n)	Approx. Operations
Text length = 10, Pattern length = 3	About 13 operations
Text length = 100, Pattern length = 10	About 110 operations
Text length = 1000, Pattern length = 50	About 1050 operations

Pattern observation: The operations grow roughly in a straight line with the size of text plus pattern.

Final Time Complexity

Time Complexity: O(n + m)

This means the algorithm checks each character in the text and pattern only once, making it very efficient.

Common Mistake

[X] Wrong: "KMP is slower because it looks back and repeats work."

[OK] Correct: KMP uses the lps array to avoid repeating checks, so it never re-examines characters unnecessarily, keeping the time linear.

Interview Connect

Understanding KMP's time complexity shows you can analyze efficient algorithms that smartly avoid repeated work, a key skill in coding interviews.

Self-Check

"What if we removed the lps array and did a simple naive search? How would the time complexity change?"

Practice

1. What is the main purpose of the KMP (Knuth-Morris-Pratt) algorithm in string processing?

easy

A. To count the number of vowels in a string

B. To sort characters in a string alphabetically

C. To reverse a string in linear time

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

Solution

1. Step 1: Understand KMP's goal

   KMP algorithm is designed to search for a pattern inside a larger text efficiently.

2. Step 2: Compare with other options

   Sorting, reversing, or counting vowels are unrelated to pattern searching.

3. Final Answer:

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

4. Quick Check:

   KMP = Pattern Searching [OK]

Hint: KMP is about fast pattern search, not string modification [OK]

Common Mistakes:

• Confusing KMP with sorting algorithms
• Thinking KMP reverses strings
• Assuming KMP counts characters

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

easy

A. int[] lps = new int[pattern.length()];

B. int lps[] = new int[pattern.length() + 1];

C. int lps = new int[pattern.length()];

D. int[] lps = new int[pattern.length() - 1];

Solution

1. Step 1: Check array declaration syntax

   In Java, arrays are declared with type[], e.g., int[] lps.

2. Step 2: Check array size correctness

   Prefix array size equals pattern length, so new int[pattern.length()] is correct.

3. Final Answer:

   int[] lps = new int[pattern.length()]; -> Option A

4. Quick Check:

   Correct Java array init = int[] lps = new int[pattern.length()]; [OK]

Hint: Use int[] lps = new int[pattern.length()] for prefix array [OK]

Common Mistakes:

• Using incorrect array size (too big or too small)
• Wrong syntax like int lps instead of int[] lps
• Adding unnecessary +1 or -1 to length

3. What will be the output of the following Java code snippet using KMP to search pattern "ABABC" in text "ABABABCABABC"?

String text = "ABABABCABABC";
String pattern = "ABABC";
List<Integer> result = KMPSearch(pattern, text);
System.out.println(result);

Assume KMPSearch returns a list of starting indices where pattern is found.

medium

A. [1, 6]

B. [0, 5]

C. [2, 7]

D. [3, 8]

Solution

1. Step 1: Identify pattern occurrences in text

   Check text "ABABABCABABC" for "ABABC" starting indices: - At index 0: substring "ABABA" does not match fully - At index 2: substring "ABABC" matches fully - At index 5: substring "CABAB" does not match - At index 6: substring "ABABC" matches fully Actually, the pattern "ABABC" occurs at indices 2 and 7, but index 7 is out of range for the second match. Re-examining carefully: - At index 0: substring "ABABA" partial - At index 5: substring "CABAB" no - At index 6: substring "ABABC" matches - At index 7: substring "BABC" incomplete So matches at indices 2 and 6.

2. Step 2: Confirm no other matches

   Other indices do not match the full pattern exactly.

3. Final Answer:

   [2, 6] -> Option B

4. Quick Check:

   Pattern found at indices 2 and 6

Hint: Scan text for exact pattern start indices [OK]

Common Mistakes:

• Off-by-one errors in indexing
• Confusing partial matches as full matches
• Assuming matches start at 0 or 1 incorrectly

4. Consider this snippet of KMP prefix array (lps) computation in Java:

int len = 0;
int i = 1;
while (i < pattern.length()) {
  if (pattern.charAt(i) == pattern.charAt(len)) {
    len++;
    lps[i] = len;
    i++;
  } else {
    if (len != 0) {
      len = lps[len - 1];
    } else {
      lps[i] = 0;
      i++;
    }
  }
}

What is the bug in this code?

medium

A. Should increment i before assigning lps[i]

B. len should be reset to 0 instead of lps[len - 1]

C. No bug; code correctly computes lps array

D. Missing initialization of lps array before loop

Solution

1. Step 1: Analyze lps computation logic

   This code follows the standard KMP prefix array logic: - Increment len and assign lps[i] when chars match - Reset len using lps[len-1] when mismatch and len != 0 - Assign 0 and increment i when len == 0

2. Step 2: Verify no missing steps

   Initialization of lps array is assumed done before; loop logic is correct.

3. Final Answer:

   No bug; code correctly computes lps array -> Option C

4. Quick Check:

   Standard KMP lps logic = No bug; code correctly computes lps array [OK]

Hint: Standard KMP lps code is correct as given [OK]

Common Mistakes:

• Thinking i must increment before lps assignment
• Resetting len incorrectly to 0 instead of lps[len-1]
• Forgetting to initialize lps array before loop

5. Given the pattern "AABAACAABAA" and text "AABAACAABAABAABA", how many times will the KMP algorithm find the pattern in the text? Assume zero-based indexing and full matches only.

hard

A. 2

B. 1

C. 3

D. 4

Solution

1. Step 1: Search for pattern occurrences in text

   Check text "AABAACAABAABAABA" for pattern "AABAACAABAA": - At index 0: substring matches pattern fully - At index 7: substring "AABAACAABAA" matches fully - Other indices do not match fully

2. Step 2: Confirm matches

   Two full matches found at indices 0 and 7.

3. Final Answer:

   2 -> Option A

4. Quick Check:

   Pattern appears twice fully [OK]

Hint: Count only full pattern matches in text [OK]

Common Mistakes:

• Counting partial matches as full matches
• Missing the first match at index 0
• Assuming overlapping partial matches count