1. Lecture 27. String Matching Algorithms 1

2. Floyd algorithm help to find the shortest path between every pair of vertices of a graph. Floyd graph may contain negative edges but no negative cycles A representation of weight matrix where W(i,j)=0 if i=j. W(i,j)=¥ if there is no edge between i and j. W(i,j)=“weight of edge” Recap 2

3. Definitions

7. The Concatenator

8. Definitions

12. Naïve String Matching Algorithm

13. Basic Explanation

14. Algorithm Pseudo Code

15. Algorithm Time Analysis

17. Boyer Moore Algorithm A String Matching Algorithm Preprocess a Pattern P (|P| = n) For a text T (| T| = m), find all of the occurrences of P in T Time complexity: O(n + m), but usually sub- linear

18. Right to Left Matching the pattern from right to left For a pattern abc: ↓ T: bbacdcbaabcddcdaddaaabcbcb P: abc Worst case is still O(n m)

19. The Bad Character Rule (BCR) On a mismatch between the pattern and the text, we can shift the pattern by more than one place. Sublinearity! ddbbacdcbaabcddcdaddaaabcbcb acabc ↑

20. BCR Preprocessing A table, for each position in the pattern and a character, the size of the shift. O(n |Σ|) space. O(1) access time. a b a c b: 1 2 3 4 5 A list of positions for each character. O(n + |Σ|) space. O(n) access time, But in total O(m). 12345 a11333 b2225 c44

21. BCR - Summary On a mismatch, shift the pattern to the right until the first occurrence of the mismatched char in P. Still O(n m) worst case running time: T: aaaaaaaaaaaaaaaaaaaaaaaaa P: abaaaa

22. The Good Suffix Rule (GSR) We want to use the knowledge of the matched characters in the pattern’s suffix. If we matched S characters in T, what is (if exists) the smallest shift in P that will align a sub-string of P of the same S characters ?

23. GSR (Cont…) Example 1 – how much to move: ↓ T: bbacdcbaabcddcdaddaaabcbcb P: cabbabdbab cabbabdbab

24. GSR (Cont…) Example 2 – what if there is no alignment: ↓ T: bbacdcbaabcbbabdbabcaabcbcb P: bcbbabdbabc bcbbabdbabc

25. GSR - Detailed We mark the matched sub-string in T with t and the mismatched char with x  In case of a mismatch: shift right until the first occurrence of t in P such that the next char y in P holds y≠x  Otherwise, shift right to the largest prefix of P that aligns with a suffix of t.

26. Boyer Moore Algorithm Preprocess(P) k := n while (k ≤ m) do – Match P and T from right to left starting at k – If a mismatch occurs: shift P right (advance k) by max(good suffix rule, bad char rule). – else, print the occurrence and shift P right (advance k) by the good suffix rule.

27. Algorithm Correctness The bad character rule shift never misses a match The good suffix rule shift never misses a match

28. Boyer Moore Worst Case Analysis Assume P consists of n copies of a single char and T consists of m copies of the same char: T: aaaaaaaaaaaaaaaaaaaaaaaaa P: aaaaaa Boyer Moore Algorithm runs in Θ(m n) when finding all the matches

29. String is combination of characters ends with a special character known as Null(in computer languages such as C/C++) A String comes with a prefix and suffex. One character or a string can be match with given string. Two important algorithm of string are Navii String matcher and Boyer Moore Algorithm which help to match a pattern of string over given string Summary 29

30. In next lecturer we will discuss Amortized analysis of different algorithms In Next Lecturer 30