Question

Compute a table representing the KMP failure function for the pattern string "cgtacgttcgtac".

Short answer

Failure table: [0, 0, 0, 0, 1, 2, 3, 0, 0, 1, 2, 3, 4]

Step-by-step solution

- Initialize the Failure Table

Start by creating a table with indices representing each character in the pattern string 'cgtacgttcgtac'. Initialize the failure function array with 0s. The first character of the pattern always has a failure function value of 0.

- Set Up Initial Variables

Set the variables: let 'i' be the current position in the pattern starting from 1 (since the first position is 0), and 'j' be the length of the previous longest prefix suffix, starting from 0.

- Iterate Through the Pattern

For each character in the pattern string starting from index 1 to the end, do the following steps.

- Match Current Character

Check if the pattern character at position 'i' matches the pattern character at position 'j'. If they match, increment both 'i' and 'j' and set the failure function value at 'i' to 'j'.

- Handle Mismatch

If the characters do not match and 'j' is not 0, set 'j' to the failure function value at position 'j-1'. If 'j' is 0, set the failure function value at 'i' to 0, and increment 'i'.

- Continue Until End

Continue repeating Steps 4 and 5 until 'i' reaches the end of the pattern string.

- Final Table

The final failure function table for the pattern 'cgtacgttcgtac' will be computed.

Key concepts

Knuth-Morris-Pratt Algorithm

The Knuth-Morris-Pratt Algorithm (KMP) is a popular pattern matching algorithm used in computer science. It searches for occurrences of a 'pattern' within a 'text' efficiently by using a preprocessing step.

KMP stands out because it uses information from the pattern itself to prevent excessive backtracking in the text. This makes the search process faster.

The algorithm mainly involves two parts:

• Preprocessing the pattern to create a failure function table.
• Using this table to perform the actual pattern search within the text.

Creating the failure function table is crucial. It helps the algorithm decide where to continue looking in the text without re-examining previously matched characters in the pattern. This preprocessing ensures the time complexity of the search is linear, i.e., O(n + m), where 'n' is the length of the text and 'm' is the length of the pattern.

Failure Function Table

The failure function table is essential for the KMP algorithm. It helps manage the search by identifying the length of the longest proper prefix which is also a suffix for any prefix of the pattern.

Let's break down creating the failure function table step-by-step using the pattern 'cgtacgttcgtac':

• Initialize variables 'i' and 'j'. Start with 'i' at position 1 and 'j' at 0.
• Compare characters at positions 'i' and 'j' in the pattern. If they match, increment both 'i' and 'j', and set the failure function value at 'i' to 'j'.
• If there's a mismatch and 'j' isn't 0, update 'j' to the failure function value at 'j-1'. If 'j' is 0, assign the failure function value at 'i' to 0 and increment 'i'.
• Continue this process until 'i' goes through all positions of the pattern.

With these steps, you can build the failure function table for any pattern. For 'cgtacgttcgtac', the table would be [0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 2, 3]. This table helps the algorithm skip unnecessary comparisons.

Pattern Matching

Pattern matching is a fundamental task in computer science. It involves finding a specific sequence of characters, or 'pattern', within a larger sequence of characters, or 'text'.

Here's how KMP uses the failure function table for pattern matching:

• Start with 'i' indexing the text and 'j' indexing the pattern, both at 0.
• Compare characters at 'i' (in the text) and 'j' (in the pattern). If they match, increment both 'i' and 'j'.
• If 'j' equals the length of the pattern, it means the pattern is found at position 'i - j' in the text. Reset 'j' according to the failure table.
• If there's a mismatch and 'j' is not 0, set 'j' to the value from the failure function table. If 'j' is 0, increment 'i'.

The failure function table allows the algorithm to skip some characters in the text, making pattern matching more efficient. Instead of re-checking every character from the start, it continues from the known partial match length, thus saving time.