Boyer-Moore Algorithm

Preview

1. How does the Boyer-Moore algorithm differ from other string matching algorithms, such as the naive approach or the Knuth-Morris-Pratt algorithm?
2. What is the significance of the “bad character rule” in the Boyer-Moore algorithm, and how does it help speed up the pattern matching process?
3. How does the Boyer-Moore algorithm handle patterns that contain repeating characters or substrings?
4. Can the Boyer-Moore algorithm be applied to search for multiple patterns at once, or is it strictly a single pattern matching algorithm?
5. Under what circumstances might the Boyer-Moore algorithm not be the best choice for string matching, and what other approaches might be preferable?

Explain

The Boyer-Moore algorithm is a pattern-matching algorithm that searches for a pattern in a given string. It relies on two main ideas: the use of a “bad character” rule and a “good suffix” rule to skip unnecessary comparisons.

The “bad character” rule states that if a mismatch occurs at a given position in the pattern, we can shift the pattern to align the last occurrence of that character in the pattern with the mismatched character in the text. This allows us to skip unnecessary comparisons if we know that a certain character cannot be part of the pattern at a given position.

The “good suffix” rule takes advantage of the fact that if a suffix of the pattern matches a substring of the text, we can shift the pattern to align that suffix with that substring, instead of starting from scratch. This helps reduce the number of comparisons needed.

For example, let’s say we want to search for the pattern “abc” in the text “abababc”.

1. Start at the end of the pattern and compare it to the last character of the text: “c” matches “c”.
2. Then we move one character to the left in the pattern and compare it to the character immediately to the left of the one we previously compared in the text: “b” does not match “b”, so we use the “bad character” rule and shift the pattern to align the last occurrence of “b” in the pattern with the mismatched character in the text.
3. We can now compare the “a” at position 0 of the pattern with the “a” at position 3 of the text. It matches, so we can move one position to the left in the pattern and repeat step 2.
4. We compare the “b” at position 1 of the pattern with the “b” at position 2 of the text. It matches, so we move one position to the left and repeat step 2.
5. We compare the “a” at position 0 of the pattern with the “a” at position 1 of the text. It matches, so we move one position to the left and repeat step 2.
6. We compare the “a” at position 0 of the pattern with the “a” at position 0 of the text. It matches, and we have found a match of the pattern in the text starting at position 0.

The algorithm use these rules to search the pattern in less than linear time complexity.

Keypoint

1. Boyer-Moore Algorithm is used for pattern matching in a given text string.

2. This algorithm is based on two major ideas: the “bad character rule” and the “good suffix rule”.

3. The bad character rule states that if a mismatch is found while comparing the pattern with the text, then we shift the pattern until the rightmost occurrence of the character in the pattern that doesn’t match the corresponding character in the text.

4. The good suffix rule is used when we find a match between the pattern and the text. It states that we can shift the pattern to the right by the difference between the length of the pattern and the position of the last occurrence of the matching suffix.

5. Boyer-Moore Algorithm is efficient for large pattern matching as it reduces the number of comparisons required.

6. The worst-case time complexity of this algorithm is O(mn), where m is the length of the pattern and n is the length of the text. However, in practice, it performs much faster than the other pattern matching algorithms.

Review

1. What is the main advantage of the Boyer-Moore algorithm over other string-searching algorithms?
   Answer: The Boyer-Moore algorithm has an average-case complexity of O(n/m) where n is the length of the string being searched and m is the length of the pattern being searched for. This makes it significantly faster than other algorithms in most cases.

2. How does the Boyer-Moore algorithm handle cases where the pattern being searched for has repeated characters?
   Answer: The algorithm uses two arrays (bad character and good suffix) to determine how far the pattern can be shifted to align with the current position in the text. If the pattern has repeated characters, the algorithm will use the array values corresponding to the farthest occurrence of that character.

3. What is the difference between the bad character and good suffix arrays used in the Boyer-Moore algorithm?
   Answer: The bad character array uses the last occurrence of a character in the pattern to determine how far to shift the pattern when a mismatch occurs. The good suffix array uses the length of the longest suffix (a sequence of characters at the end of the pattern) that matches a prefix (a sequence of characters at the beginning of the pattern).

4. Can the Boyer-Moore algorithm be used for searching in multiple patterns?
   Answer: Yes, the Boyer-Moore algorithm can be extended to search for multiple patterns at once. This is done by creating a trie data structure from the patterns and using it to guide the search.

5. Under what conditions does the Boyer-Moore algorithm have a worst-case complexity of O(nm)?
   Answer: The Boyer-Moore algorithm has a worst-case complexity of O(nm) when the pattern being searched has many repeated characters and there are many occurrences of the pattern within the text. This can cause the algorithm to have to shift the pattern back by a large amount for each occurrence, resulting in significant time complexity.