Question

Compute a table representing the last function used in the BM pattern matching algorithm for the pattern string "the quick brown fox jumped over a lazy cat" assuming the following alphabet (which starts with the space character): \\[ \Sigma=\left\\{_{\boldsymbol{u}}, \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{i}, \mathrm{j}, \mathrm{k}, \mathrm{l}, \mathrm{m}, \mathrm{n}, \mathrm{o}, \mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}, \mathrm{t}, \mathrm{u}, \mathrm{v}, \mathrm{w}, \mathrm{x}, \mathrm{y}, \mathrm{z}\right\\} \\]

Short answer

Use the last occurrence of each character in the pattern string to fill the table, updating the value as each character is encountered.

Step-by-step solution

Explain the BM Algorithm

The BM (Boyer-Moore) pattern matching algorithm uses a table to store the rightmost occurrence of each character in the pattern. This step will help minimize unnecessary comparisons by skipping sections of the text.

Identify the Alphabet

List out all the characters from the given alphabet starting with the space character. The alphabet is: \[\Sigma=\left\{\mathrm{[space]}, \,\mathrm{a}, \,\mathrm{b}, \,\mathrm{c}, \,\mathrm{d}, \,\mathrm{e}, \,\mathrm{f}, \,\mathrm{g}, \,\mathrm{h}, \,\mathrm{i}, \,\mathrm{j}, \,\mathrm{k}, \,\mathrm{l}, \,\mathrm{m}, \,\mathrm{n}, \,\mathrm{o}, \,\mathrm{p}, \,\mathrm{q}, \,\mathrm{r}, \,\mathrm{s}, \,\mathrm{t}, \,\mathrm{u}, \,\mathrm{v}, \,\mathrm{w}, \,\mathrm{x}, \,\mathrm{y}, \,\mathrm{z}\right\} \].

Initialize the Last Occurrence Table

Initialize an empty table that maps each character in the alphabet to an integer representing their last occurrence index in the pattern. Initially, set all values to -1, indicating that the character has not yet been encountered.

Populate the Last Occurrence Table

Scan through the pattern string from left to right. For each character, update its corresponding value in the table with its index (position) in the pattern. Overwrite the value if the character already has an entry, effectively keeping its most recent occurrence.

Construct the Table

After processing the pattern, construct the complete last occurrence table. For example, for the character 'a', find its last occurrence in the pattern string and note the index. Repeat this process for every character in the alphabet.

Key concepts

Boyer-Moore algorithm

The Boyer-Moore algorithm is a popular and efficient string searching algorithm. It is widely used for pattern matching in texts. This algorithm stands out due to its ability to skip sections of the text, thus minimizing unnecessary comparisons. It leverages two key insights: one about the rightmost occurrence of a mismatched character (using the last occurrence table) and the other about matching the suffix of the pattern. The focus here will be on the first insight, which uses the last occurrence table.

Pattern Matching

Pattern matching is the process of finding the occurrence of a pattern string within a larger text string. The Boyer-Moore algorithm is specifically designed to make this search efficient by skipping over sections of the text whenever possible. It ensures fewer comparisons between the pattern and the text, which is particularly useful when dealing with large texts. The idea is that instead of checking each position in the text sequentially, the algorithm uses information about the pattern to jump ahead by larger increments.

Last Occurrence Table

The last occurrence table is a crucial component of the Boyer-Moore algorithm. It records the most recent index of each character in the pattern. This table helps the algorithm to determine how many positions it can safely skip when a mismatch occurs. For example, if the character at a mismatched position in the text is not found at all in the last occurrence table, the pattern can be entirely skipped to the point after the mismatched character. If the character is present in the table, the algorithm skips to align the pattern with the last occurrence of that character.

Text Processing

Text processing involves analyzing and manipulating text to extract meaningful information or perform certain tasks, such as searching for patterns. The Boyer-Moore algorithm is instrumental in this field as it allows for efficient searching. Constructing the last occurrence table is a part of this process. For each character in the pattern, we determine its last appearance index and populate the table accordingly. This forms the basis for efficiently processing the text and searching for the pattern.

Substring Search

Substring search is the specific task of finding whether and where a smaller string (the 'pattern') exists within a larger string (the 'text'). The Boyer-Moore algorithm excels at this by using the last occurrence table and other heuristics to minimize the number of comparisons. For instance, when a mismatch happens, rather than moving the pattern one position to the right, the algorithm uses the last occurrence table to shift the pattern by several positions, reducing the overall search time.