Question

Suppose the pattern-matching problem is changed to require locating only the first instance, if any, of the pattern within the text. a. Describe the worst case, give an example, and give the exact number of comparisons (of a pattern character with a text character) required. b. Describe the best case, give an example, and give the exact number of comparisons required.

Short answer

Worst case: "aaaaaaaaab" vs "aaaab" require 24 comparisons. Best case: "abcde" vs "abc" need 3 comparisons.

Step-by-step solution

Identifying the Worst-Case Scenario

The worst-case scenario in a pattern-matching problem occurs when the pattern and the text have few similarities in the substring arrangement, yet a portion of the beginning of the pattern mismatches often after beginning to match. This results in a maximal number of comparisons before realizing the pattern doesn't match or appears much later.

Constructing the Worst-Case Example

Consider the text 'aaaaaaaaab' and the pattern 'aaaab'. The pattern first seems to match, requiring backtracking upon each mismatch. In such a case, collateral backtracking could lead to repeated comparisons of characters in both pattern and text.

Calculating Comparisons in Worst-Case

In worst-case situations, like our example above, the number of comparisons can approach (but typically will not exceed) \((n - m + 1) \times m\), where \(n\) is the length of the text and \(m\) is the length of the pattern. If \(n=10\) and \(m=5\), then this results in 24 comparisons.

Identifying the Best-Case Scenario

The best-case scenario occurs when the pattern matches immediately and exactly from its starting position within the text, without needing to backtrack or slide further.

Constructing the Best-Case Example

For example, consider text 'abcde' and pattern 'abc'. The pattern immediately matches the start of the text.

Calculating Comparisons in Best-Case

In the best-case situation, the number of comparisons required is simply the length of the pattern \(m\), since no extra checking is done. For the example pattern 'abc', just 3 comparisons are necessary.

Key concepts

Worst-case scenario in Pattern Matching Algorithms

In the context of pattern matching algorithms, the worst-case scenario is a situation where the alignment of the pattern with the text leads to a maximal number of character comparisons. This happens when the text and pattern have certain sequences that overlap but don't ultimately match, requiring many retries. Essentially, when sections of the pattern start matching with the text but then a mismatch occurs, a lot of backtracking and rechecking is needed.
For instance, consider the text 'aaaaaaaaab' and the pattern 'aaaab'. Here, the first few characters of the pattern ('aaa') will appear to match repeatedly with the text, only for the final character of the pattern to cause failure. This mismatch demands resetting and retrying from various starting points in the text. This creates a high number of comparisons before concluding that either the text doesn't fully contain the pattern or discovering the match at a distant end.
These occurrences in a worst-case scenario lead to comparisons approaching \( (n - m + 1) \times m \), with \( n \) being the text’s length and \( m \) being the pattern's length. As demonstrated in our example with \( n=10 \) and \( m=5 \), this results in 24 character comparisons, though the count may vary slightly due to implementation specifics.

Best-case scenario in Pattern Matching Algorithms

Conversely, the best-case scenario for pattern matching algorithms is much simpler and faster. It occurs when the pattern matches the text immediately from the start. In this ideal situation, no extra movements or retries are necessary since the pattern aligns perfectly with the beginning of the text.
A practical example is when the text is 'abcde' and the pattern is 'abc'. Here, the pattern 'abc' directly aligns with the start of the text 'abcde', ensuring the algorithm only requires examining each of the pattern's characters once, without further adjustments.
This efficient process only requires a number of comparisons equal to the length of the pattern, which is represented mathematically as \( m \). For our specific example, with a pattern length of 3, just 3 comparisons are needed, showcasing the striking simplicity and speed of the best-case scenario.

Algorithm comparison steps in Pattern Matching

When comparing different pattern matching algorithms, it's essential to evaluate them based on various scenarios. Key aspects include their performance during the worst-case and best-case scenarios.

• Worst-case performance: Understand how many character comparisons occur when mismatches frequent every alignment attempt. This allows you to gauge the algorithm’s efficiency under stress.


• Best-case performance: Observe the number of comparisons when the pattern fits perfectly from the start of the text. This helps identify the algorithm's capability to leverage favorable conditions.


• Efficiency considerations: Overall efficiency is not only about the speed but also includes the memory usage and additional resources required. This is crucial for implementing pattern matching in resource-constrained environments.

By analyzing these steps meticulously, one can understand which algorithm suits specific requirements, providing better insight into their utility under varying conditions. This structured approach aids developers in choosing an algorithm that balances speed, accuracy, and resource use effectively.