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Pattern Matching Answered
For this assignment you will be coding 3 different pattern matching algorithms: Knuth-Morris-Pratt (KMP), Boyer-Moore, and Rabin-Karp. For all three algorithms, you should find all occurrences of the pattern in the text, not just the first match. The occurrences are returned as a list of integers; the list should contain the indices of occurrences in ascending order. There is information about all three algorithms in the javadocs with additional implementation details below. If you implement any of the three algorithms in an unexpected manner (i.e. contrary to what the Javadocs and PDF specify), you may receive a 0.
For all of the algorithms, make sure you check the simple failure cases as soon as possible. For ex-ample, if the pattern is longer than the text, don’t do any preprocessing on the pattern/text and just return an empty list since there cannot be any occurrences of the pattern in the text.
Note that for pattern matching, we refer to the text length as n and the pattern length as m.
CharacterComparator
CharacterComparator is a comparator that takes in two characters and compares them. This allows you to see how many times you have called compare(); besides this functionality, its return values are what you’d expect a properly implemented compare() method to return. You must use this comparator as the number of times you call compare() with it will be used when testing your assignment.
If you do not use the passed in comparator, this will cause tests to fail and will significantly lower your grade on this assignment. You must implement the algorithms as they were taught in class. We are expecting exact comparison counts for this homework. If you are getting fewer com-parison counts than expected, it means one of two things: either you implemented the algorithm wrong (most likely) or you are using an optimization not taught in the class (unlikely).
Knuth-Morris-Pratt
Failure Table
The Knuth-Morris-Pratt (KMP) algorithm relies on using the prefix of the pattern to determine how much to shift the pattern by. The algorithm itself uses what is known as the failure table (also called failure function). Before actually searching, the algorithm generates a failure table. This is an array of length m where each index will correspond to the substring in the pattern up to that index. Each index i of the failure table should contain the length of the longest proper prefix that matches a proper suffix of pattern[0, ..., i]. A proper prefix/suffix does not equal the string itself. There are different ways of calculating the failure table, but we are expecting the specific format described below.
For any string pattern, have a pointer i starting at the first letter, a pointer j starting at the sec-ond letter, and an array called table that is the length of the pattern. First, set index 0 of table to 0. Then, while j is still a valid index within pattern:
If the characters pointed to by i and j match, then write i + 1 to index j of the table and increment i and j.
If the characters pointed to by i and j do not match:
– If i is not at 0, then change i to table[i - 1]. Do not increment j or write any value to the table.
– If i is at 0, then write i to index j of the table. Increment only j.
Homework 9: PatternMatching Due: See Canvas
For example, for the string abacab, the failure table will be:
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For the string ababac, the failure table will be:
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0
For the string abaababa, the failure table will be:
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For the string aaaaaa, the failure table will be:
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Searching Algorithm
For the main searching algorithm, the search acts like a standard brute-force search for the most part, but in the case of a mismatch:
If the mismatch occurs at index 0 of the pattern, then shift the pattern by 1.
If the mismatch occurs at index j of the pattern and index i of the text, then shift the pattern such that index failure[j-1] of the pattern lines up with index i of the text, where failure is the failure table. Then, continue the comparisons at index i of the text (or index failure[j-1] of the pattern). Do not restart at index 0 of the pattern.
In addition, if the whole pattern is ever matched, instead of shifting the pattern over by 1 to continue searching for more matches, the pattern should be shifted so that the pattern at index failure[j-1], where j is at pattern.length, aligns with the index after the match in the text. KMP treats a match as a “mismatch” on the character immediately following the match.
Rabin-Karp
The Rabin-Karp algorithm relies on hashing to perform pattern matching. This algorithm, instead of using a sophisticated shift / skip through the text, uses a hash function to compare the given pattern with substrings of the text. This algorithm exploits the fact that if two strings are equal, their hash values must also be equal. The algorithm essentially reduces down to computing the hash value of the pattern and then looking for substrings of the text with the same hash value. Once a substring of the text with the same hash as the pattern is found, the substring is compared character by character with the pattern to ensure equality (as two strings with the same hash may not actually be equal).
Note: You must use the exact rolling hash function specified in the javadocs. You are not allowed to use Math.pow() for the intial hash calculation, nor are you allowed to use it for updating the text hash. This is because exponentiating a number is not an O(1) operation, so creating your own custom power method is also inefficient.
Homework 9: PatternMatching Due: See Canvas
Boyer-Moore
Last Occurrence Table
The Boyer-Moore algorithm, similar to KMP, relies on preprocessing the pattern. Before actually search-ing, the algorithm generates a last occurrence table. The table allows the algorithm to skip sections of the text, resulting in more efficient string searching. The last occurrence table should be a mapping from each character in the alphabet (the set of all characters that may be in the pattern or the text) to the last index the character appears in the pattern. If the character is not in the pattern, then -1 is used as the value, though you should not explicitly add all characters that are not in the pattern into the table. The getOrDefault() method from Java’s Map will be useful for this.
Searching Algorithm
Key properties of Boyer-Moore include matching characters starting at the end of the pattern, rather than the beginning and skipping along the text in jumps of multiple characters rather than searching every single character in the text.
The shifting rule considers the character in the text at which the comparison process failed (assum-ing that a failure occurred). If the last occurrence of that character is to the left in the pattern, shift so that the pattern occurrence aligns with the mismatched text occurrence. If the last occurrence of the mismatched character does not occur to the left in the pattern, shift the pattern over by one (to prevent the pattern from moving backwards). In addition, if the mismatched character does not exist in the pattern at all (no value in last table) then pattern shifts completely past this point in the text.
For finding multiple occurrences, if you find a match, shift the pattern over by one and continue searching.
Extra Credit: Galil Rule
The Galil Rule is an addition to Boyer-Moore that allows it to approach linear time in certain cases. Recall that Boyer-Moore shifts the pattern by one after finding a full match. The Galil Rule optimizes on this case of a full match by exploiting the periodicity of the pattern to shift the pattern intelligently.
Periodicity
The period, k, of a pattern is defined as the length of the shortest prefix of the pattern that when repeated contains the pattern itself at the beginning. The period of the pattern can be computed using the failure table of the pattern: k = m - ft[m - 1] where m is the length of the pattern and ft is the failure table of the pattern.
For the pattern string abacab, its period is 6 - ft[5] = 6 - 2 = 4.
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k = 4 corresponds to the prefix abac. If we repeat this prefix, we will create a string in the form abacabacabac.... Notice how the original pattern abacab is contained at the beginning of this repeated string: abacabacabac.... This is the shortest prefix of the pattern that satisfies these conditions.
After a full match, the Galil Rule shifts the pattern by its period, rather than by one. Thus, when a text has many occurrences of the pattern, the Galil Rule allows the algorithm to approach linear time.
For up to 50 points of extra credit, you may implmement the galilRule method on this homework.
This method is optional.
Homework 9: PatternMatching Due: See Canvas
Grading
Here is the grading breakdown for the assignment. There are various deductions not listed that are incurred when breaking the rules listed in this PDF and in other various circumstances.
This assignment includes 50 points of extra credit for implementing the boyerMooreGalilRule method. This means it is possible to earn up to 150 points on this assignment for total of 150%. Implementing the Galil Rule is optional.
Methods:
buildFailureTable
10pts
kmp
15pts
buildLastTable
10pts
boyerMoore
15pts
rabinKarp
25pts
Other:
Checkstyle
10pts
Efficiency
15pts
Total:
100pts
Extra Credit:
boyerMooreGalilRule
50pts
Provided
The following file(s) have been provided to you. There are several, but we’ve noted the ones to edit.
PatternMatching.java
This is the class in which you will implement the different pattern matching algorithms. Feel free to add private static helper methods but do not add any new public methods, new classes, instance variables, or static variables.
CharacterComparator.java
This is a comparator that will be used to count the number of comparisons used. You must use this comparator. Do not modify this file.
PatternMatchingStudentTests.java
This is the test class that contains a set of tests covering the basic algorithms in the PatternMatching class. It is not intended to be exhaustive and does not guarantee any type of grade. Write your own tests to ensure you cover all edge cases.
Deliverables
You must submit all of the following file(s) to the course Gradescope. Make sure all file(s) listed below are in each submission, as only the last submission will be graded. Make sure the filename(s) matches the filename(s) below, and thatonly the following file(s) are present. If you resubmit, be sure only one copy of each file is present in the submission. If there are multiple files, do not zip up the files before submitting; submit them all as separate files.
Once submitted, double check that it has uploaded properly on Gradescope. To do this, download
Homework 9: PatternMatching Due: See Canvas
your uploaded file(s) to a new folder, copy over the support file(s), recompile, and run. It is your sole responsibility to re-test your submission and discover editing oddities, upload issues, etc.
PatternMatching.java
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