Question

Assuming that the characters in alphabet \(\Sigma\) can be enumerated and can be used to index arrays, give an \(O(m+|\Sigma|)\) -time method for constructing the last function from an \(m\) -length pattern string \(P\)

Short answer

Initialize an array, loop over pattern, update array with last occurrence indices, return array.

Step-by-step solution

- Initialize the Array

Create an array `last` where each index corresponds to a character in the alphabet \(\Sigma\). Initialize each entry in the array to -1. This array will store the last occurrence of each character in the pattern string `P`.

- Iterate Over the Pattern

Loop through each character in the pattern string `P` from the beginning to the end. The loop will run from index 0 to index `m-1`, where `m` is the length of the pattern string `P`.

- Update the Last Occurrence

For each character in the pattern string `P`, update its corresponding index in the `last` array with the current position (index) in the pattern string. This will record the last occurrence of each character.

- Return the Last Function

After processing all characters in the pattern string `P`, the `last` array will contain the last occurrence of each character in alphabet \(\Sigma\). Return this `last` function array.

Key concepts

Last Occurrence Function

The last occurrence function is used in various pattern-matching algorithms like the Boyer-Moore algorithm. It helps in determining the last position of each character in the pattern string. This is crucial for efficient string matching.
To create this function, follow these steps:
First, initialize an array named `last`, with each index corresponding to a character in the alphabet \(\backslash Sigma\). Set each element in this array to -1, indicating that no character has been encountered yet.
Then, iterate over each character in the pattern string `P`. For every character, update the corresponding index in the `last` array with its current position in the string `P`. By the end of this iteration, you'll have a fully constructed `last` array.
This function makes it easier to skip sections of the text while matching patterns, significantly speeding up the process.

Alphabet Enumeration

Alphabet enumeration involves mapping characters in your alphabet \(\backslash Sigma\) to indices in an array. This means that each character in the alphabet is assigned a unique index, which helps in index-based operations like array manipulations.
For instance, if your alphabet consists of characters [a, b, c, d], then you can enumerate these as [0, 1, 2, 3]. This indexed representation allows for quick lookups and updates, which is essential in pattern-matching algorithms.
To implement alphabet enumeration, you can use either a direct mapping for small alphabets or a hash table for larger and more complex alphabets. This will allow you to efficiently index characters and perform operations in constant time.

Time Complexity O(m+|Σ|)

The time complexity of constructing the last occurrence function is O(m+|Σ|). This means the time taken depends on both the length of the pattern string `m` and the size of the alphabet \(\backslash Sigma\).
To break this down:

• Initializing the `last` array takes O(|\( \backslash Sigma \)|) time because you need to set a value for each character in the alphabet.
• Iterating through the pattern string of length `m` takes O(m) time.

Therefore, the overall complexity is the sum of these two operations, resulting in O(m+|\( \backslash Sigma \)|).
This balanced time complexity ensures that the algorithm is efficient even for larger pattern strings and alphabets, making it a robust solution for real-world applications.