KMP Pattern Matching Algorithm in DSA Javascript - 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 this KMP pattern matching code.

function kmpSearch(text, pattern) {
  const lps = computeLPSArray(pattern);
  let i = 0, j = 0;
  while (i < text.length) {
    if (pattern[j] === text[i]) {
      i++; j++;
      if (j === pattern.length) return i - j;
    } else if (j !== 0) {
      j = lps[j - 1];
    } else {
      i++;
    }
  }
  return -1;
}

function computeLPSArray(pattern) {
  const lps = Array(pattern.length).fill(0);
  let length = 0, i = 1;
  while (i < pattern.length) {
    if (pattern[i] === pattern[length]) {
      length++; lps[i] = length; i++;
    } else if (length !== 0) {
      length = lps[length - 1];
    } else {
      lps[i] = 0; i++;
    }
  }
  return lps;
}
    

This code finds the first occurrence of a pattern inside a text using the KMP algorithm, which precomputes a helper array (LPS) to skip unnecessary checks.

Identify Repeating Operations

Look at the loops and repeated steps:

• Primary operation: Comparing characters of text and pattern.
• How many times: Each character in text and pattern is visited at most twice during the search and LPS computation.

How Execution Grows With Input

As the text length (n) and pattern length (m) grow, the number of operations grows roughly in a straight line.

Input Size (n, m)	Approx. Operations
n=10, m=3	About 20-30 comparisons
n=100, m=10	About 200-300 comparisons
n=1000, m=50	About 2000-3000 comparisons

Pattern observation: The operations grow roughly proportional to the sum of text and pattern lengths, not their product.

Final Time Complexity

Time Complexity: O(n + m)

This means the algorithm checks characters in the text and pattern only a few times, making it very efficient even for large inputs.

Common Mistake

[X] Wrong: "KMP takes O(n * m) time because it compares every pattern character with every text character."

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

Interview Connect

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

Self-Check

"What if the pattern was very long compared to 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 sort an array of numbers in ascending order

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

C. To reverse a string in linear time

D. To calculate the factorial of a number recursively

Solution

1. Step 1: Understand the problem KMP solves

   KMP is designed to find all occurrences of a pattern inside a text without rechecking characters unnecessarily.

2. Step 2: Compare with other options

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

3. Final Answer:

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

4. Quick Check:

   KMP = Pattern searching [OK]

Hint: KMP is about searching patterns fast in text [OK]

Common Mistakes:

• Confusing KMP with sorting algorithms
• Thinking KMP reverses strings
• Assuming KMP calculates factorials

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

easy

A. const lps = new Array(pattern.length).fill(null);

B. const lps = Array(pattern.length);

C. const lps = []; lps.length = pattern.length;

D. const lps = new Array(pattern.length).fill(0);

Solution

1. Step 1: Understand prefix array initialization

   The prefix array (lps) must be initialized with zeros to store lengths of longest prefix suffix.

2. Step 2: Check each option

   const lps = new Array(pattern.length).fill(0); correctly creates an array of the pattern's length filled with zeros. const lps = Array(pattern.length); creates an array but leaves elements undefined. const lps = []; lps.length = pattern.length; sets length but elements are undefined. const lps = new Array(pattern.length).fill(null); fills with null, which is incorrect.

3. Final Answer:

   const lps = new Array(pattern.length).fill(0); -> Option D

4. Quick Check:

   Prefix array = zeros [OK]

Hint: Prefix array must start with zeros for correct matching [OK]

Common Mistakes:

• Leaving prefix array elements undefined
• Filling prefix array with null instead of zero
• Not initializing prefix array length

3. What will be the output of the following JavaScript code snippet using KMP to find pattern 'ABABC' in text 'ABABABCABABCBC'?

function KMPSearch(pattern, text) {
  const lps = computeLPSArray(pattern);
  let i = 0, j = 0;
  const result = [];
  while (i < text.length) {
    if (pattern[j] === text[i]) {
      i++; j++;
      if (j === pattern.length) {
        result.push(i - j);
        j = lps[j - 1];
      }
    } else if (j !== 0) {
      j = lps[j - 1];
    } else {
      i++;
    }
  }
  return result;
}

function computeLPSArray(pattern) {
  const lps = new Array(pattern.length).fill(0);
  let length = 0, i = 1;
  while (i < pattern.length) {
    if (pattern[i] === pattern[length]) {
      length++;
      lps[i] = length;
      i++;
    } else if (length !== 0) {
      length = lps[length - 1];
    } else {
      lps[i] = 0;
      i++;
    }
  }
  return lps;
}

console.log(KMPSearch('ABABC', 'ABABABCABABCBC'));

medium

A. [1, 6]

B. [0, 5]

C. [2, 7]

D. [3, 8]

Solution

1. Step 1: Compute LPS array for pattern 'ABABC'

   LPS array is [0, 0, 1, 2, 0]. This helps skip characters on mismatch.

2. Step 2: Trace pattern matches in text 'ABABABCABABCBC'

   Pattern matches start at index 2 and again at index 7 in the text.

3. Final Answer:

   [2, 7] -> Option C

4. Quick Check:

   Match indices = [2, 7] [OK]

Hint: Use LPS to jump and find pattern start indices [OK]

Common Mistakes:

• Incorrect LPS array calculation
• Off-by-one errors in indices
• Confusing pattern length with text length

4. Identify the bug in this KMP prefix array computation code snippet:

function computeLPSArray(pattern) {
  const lps = new Array(pattern.length).fill(0);
  let length = 0;
  let i = 0;
  while (i < pattern.length) {
    if (pattern[i] === pattern[length]) {
      length++;
      lps[i] = length;
      i++;
    } else if (length !== 0) {
      length = lps[length - 1];
    } else {
      lps[i] = 0;
      i++;
    }
  }
  return lps;
}

medium

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

B. The length variable should be initialized to 1

C. The lps array should be filled with -1 instead of 0

D. The condition pattern[i] === pattern[length] should be reversed

Solution

1. Step 1: Check loop initialization

   The prefix array computation starts from index 1 because index 0 always has lps value 0.

2. Step 2: Identify impact of starting at i=0

   Starting at i=0 causes comparing pattern[0] with pattern[0] unnecessarily and overwrites lps[0], which should remain 0.

3. Final Answer:

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

4. Quick Check:

   Start loop at 1 for prefix array [OK]

Hint: Prefix array loop always starts at 1 [OK]

Common Mistakes:

• Starting loop at 0
• Initializing length incorrectly
• Filling lps with wrong values

5. Given a large text and a pattern, how can you modify the KMP algorithm to count overlapping occurrences of the pattern in the text? For example, pattern 'AAA' in text 'AAAAA' occurs 3 times (positions 0,1,2). Which change is correct?

hard

A. After a match, set j = lps[j - 1] instead of resetting j to 0

B. After a match, increment i by pattern length to skip matched part

C. After a match, reset both i and j to 0 to restart search

D. After a match, increment j by 1 to continue matching

Solution

1. Step 1: Understand overlapping matches

   Overlapping matches require continuing search from the longest proper prefix which is also suffix, not restarting.

2. Step 2: Apply correct jump after match

   Setting j = lps[j - 1] after a match allows the algorithm to find overlapping occurrences without missing any.

3. Final Answer:

   After a match, set j = lps[j - 1] instead of resetting j to 0 -> Option A

4. Quick Check:

   Use lps jump to count overlaps [OK]

Hint: Use lps jump after match to find overlaps [OK]

Common Mistakes:

• Resetting j to 0 after match
• Skipping characters by incrementing i too much
• Not handling overlaps correctly