KMP Pattern Matching Algorithm in DSA C# - 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.

// Compute longest prefix suffix (LPS) array
void ComputeLPSArray(string pattern, int[] lps) {
    int length = 0; // length of previous longest prefix suffix
    int 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++;
            }
        }
    }
}

// KMP search
void KMPSearch(string pattern, string text) {
    int[] lps = new int[pattern.Length];
    ComputeLPSArray(pattern, lps);
    int i = 0, j = 0;
    while (i < text.Length) {
        if (pattern[j] == text[i]) {
            i++; j++;
        }
        if (j == pattern.Length) {
            // pattern found at i-j
            j = lps[j - 1];
        } else if (i < text.Length && pattern[j] != text[i]) {
            if (j != 0) j = lps[j - 1];
            else i++;
        }
    }
}
    

This code finds all occurrences of a pattern inside a text efficiently using the KMP algorithm.

Identify Repeating Operations

Look at the loops and repeated steps:

• Primary operation: Comparing characters of pattern and text.
• 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 operations grow roughly in a straight line.

Input Size (n)	Approx. Operations
10 (text), 3 (pattern)	About 20 operations
100 (text), 10 (pattern)	About 220 operations
1000 (text), 50 (pattern)	About 1050 operations

Pattern observation: The operations increase roughly proportional to the sum of text and pattern lengths.

Final Time Complexity

Time Complexity: O(n + m)

This means the algorithm checks characters in the text and pattern only once or twice, making it very efficient.

Common Mistake

[X] Wrong: "KMP is slower than simple search because it does extra work building the LPS array."

[OK] Correct: The LPS array is built in linear time and helps avoid re-checking characters, so overall KMP is faster on large inputs.

Interview Connect

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

Self-Check

"What if we changed the pattern to be very long compared to the text? How would the time complexity change?"

Practice

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

easy

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

B. To sort an array of numbers in ascending order

C. To reverse a string in linear time

D. To calculate the factorial of a number recursively

Solution

1. Step 1: Understand the goal of KMP

   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 calculating factorials are unrelated to pattern searching.

3. Final Answer:

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

4. Quick Check:

   KMP = Pattern searching [OK]

Hint: KMP is all about fast pattern search 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 declare the prefix table (lps array) in C# for KMP algorithm?

easy

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

B. int lps = new int[pattern.Length];

C. int lps[] = new int[pattern.Length];

D. int[] lps = new int[pattern.Length];

Solution

1. Step 1: Understand array declaration in C#

   In C#, arrays are declared with type[], then new type[size]. So int[] lps = new int[pattern.Length]; is correct.

2. Step 2: Check other options

   int lps = new int[pattern.Length]; misses [] for array type, C uses invalid syntax, D uses C/C++ style which is invalid in C#.

3. Final Answer:

   int[] lps = new int[pattern.Length]; -> Option D

4. Quick Check:

   C# array syntax = int[] name = new int[size]; [OK]

Hint: Use int[] for arrays, not int alone [OK]

Common Mistakes:

• Missing [] in array type declaration
• Using C/C++ style array declaration in C#
• Trying to instantiate array without size

3. What will be the output of the following C# code snippet using KMP algorithm to find pattern "ABABC" in text "ABABABC"?

string text = "ABABABC";
string pattern = "ABABC";
List<int> matches = KMPSearch(pattern, text);
Console.WriteLine(string.Join(",", matches));

// Assume KMPSearch returns list of start indices where pattern matches text

medium

A. 0,2

B. 1

C. 2

D. No matches

Solution

1. Step 1: Analyze the text and pattern

   Text: A B A B A B C Pattern: A B A B C Check where pattern fully matches text starting index.

2. Step 2: Find matching index

   At index 0: text substring "ABABA" does not match "ABABC" (last char mismatch) At index 1: substring "BABAB" no match At index 2: substring "ABABC" matches pattern exactly.

3. Final Answer:

   2 -> Option C

4. Quick Check:

   Pattern found at index 2 only [OK]

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

Common Mistakes:

• Assuming multiple matches without checking full pattern
• Off-by-one errors in indexing
• Confusing partial matches as full matches

4. Identify the bug in this KMP prefix table (lps) computation code snippet in C#:

int[] ComputeLPS(string pattern) {
    int[] lps = new int[pattern.Length];
    int length = 0;
    int i = 1;
    while (i < pattern.Length) {
        if (pattern[i] == pattern[length]) {
            length++;
            lps[i] = length;
            i++;
        } else {
            if (length != 0) {
                length = lps[length];
            } else {
                lps[i] = 0;
                i++;
            }
        }
    }
    return lps;
}

medium

A. Incorrect update of length in else block: should be length = lps[length - 1];

B. Wrong initialization of lps array size

C. Using while loop instead of for loop causes infinite loop

D. Missing return statement at the end

Solution

1. Step 1: Understand lps update logic

   When mismatch occurs and length != 0, length should be updated to lps[length - 1], not lps[length].

2. Step 2: Check other parts

   Array size is correct, while loop is valid, return statement exists. So only length update is wrong.

3. Final Answer:

   Incorrect update of length in else block: should be length = lps[length - 1]; -> Option A

4. Quick Check:

   lps update uses length-1 index [OK]

Hint: Use length-1 index when updating length on mismatch [OK]

Common Mistakes:

• Using lps[length] instead of lps[length - 1]
• Confusing loop types causing infinite loops
• Forgetting to return lps array

5. Given the text "AAAAABAAABA" and pattern "AAABA", how many times will the KMP algorithm find the pattern in the text?

hard

A. 3

B. 2

C. 4

D. 1

Solution

1. Step 1: Identify all occurrences of pattern in text

   Text: A A A A A B A A A B A Pattern: A A A B A Check each possible start index:

2. Step 2: Check matches at each index

   Index 0: substring "AAAAA" no (pattern ends with B) Index 1: substring "AAAAB" no (pattern ends with A) Index 2: substring "AAABA" yes (match 1) Index 3: substring "AABAA" no Index 4: substring "ABAAA" no Index 5: substring "BAAAB" no Index 6: substring "AAABA" yes (match 2) Index 7: substring "AABA" incomplete Index 8: substring "ABA" incomplete Check carefully for overlapping matches.

3. Step 3: Re-check for missed matches

   Check index 1 carefully: substring "AAAAB" pattern is "AAABA" so no match. Check index 0 carefully: substring "AAAAA" pattern ends with B, no match. Check index 6: "AAABA" matches pattern exactly (match 2). Check index 2: "AAABA" matches pattern exactly (match 1). No other matches found.

4. Step 4: Check index 3 for overlapping match

   At index 3: substring "AABAA" does not match pattern "AAABA". At index 4: substring "ABAAA" no match. At index 5: substring "BAAAB" no match. At index 7: substring "AABA" incomplete. So total matches found: 2.

5. Final Answer:

   2 -> Option B

6. Quick Check:

   Pattern found twice in text [OK]

Hint: Check each substring carefully for full pattern match [OK]

Common Mistakes:

• Counting partial matches as full matches
• Missing overlapping matches
• Off-by-one errors in substring indexing