1. Pattern Matching Rhys Price Jones Anne R. Haake

2. Pattern matching algorithms - Review Finding all occurrences of pattern p in text t P has length m, t has length n Naïve algorithm, Rabin-Karp both have worst- case O(mn) and expected case O(m+n) behavior Automaton approach preprocesses p to yield a O(n) algorithm

3. Suffix Tree Preprocess t To yield a O(m) algorithm Useful if t is fixed and there are lots of p’s that you want to search for.

4. Tries Are often used for retrieval of keywords to provide efficient indexing. Suppose you want to index: –PATTERN –MONKEY –PATAPAN –PROBOSCIS –PATHETIC

5. Build a TRIE For PATTERN PATTERN

6. Build a TRIE For PATTERN MONKEY PATTERNMONKEY

7. Build a TRIE For PATTERN MONKEY PATAPAN PAT MONKEY APAN TERN

8. Build a TRIE For PATTERN MONKEY PATAPAN PROBOSCIS MONKEY APAN P TERN ROBOSCIS AT

9. Build a TRIE For PATTERN MONKEY PATAPAN PROBOSCIS PATHETIC P M HETIC APAN TERN ROBOSCIS AT Each keyword can be located in O(k) steps where k is the length of the keyword

10. Applications of a Trie Dictionary –Just need to check if you get to a leaf to know the word exists –Or store a link to the word’s definition at the leaf Index for a book –Store a list of all pages where the keyword appears at the leaf Finding reserved words or filtering unwanted words …

11. Suffix Tree For a text t Is a Trie for the set of suffixes of t BIOINFORMATICS IOINFORMATICS OINFORMATICS INFORMATICS … ICS CS S Build it on the board

12. Suffix Tree For ACACTACT ACACTACT CACTACT ACTACT CTACT TACT ACT CT T ACACTACT 0

13. Suffix Tree For ACACTACT ACACTACT ACACTACT CACTACT ACTACT CTACT TACT ACT CT T ACACTACT CACTACT 0 1

14. Suffix Tree For ACACTACT ACACTACT CACTACT ACTACT CTACT TACT ACT CT T CACTACT AC ACTACTTACT 1 0 2

15. Suffix Tree For ACACTACT ACACTACT CACTACT ACTACT CTACT TACT ACT CT T AC ACTACTTACT C ACTACT 0 23 1

16. Suffix Tree For ACACTACT ACACTACT CACTACT ACTACT CTACT TACT ACT CT T AC ACTACTTACT C ACTACT 0 23 1 4 TACT

17. Suffix Tree For ACACTACT ACACTACT CACTACT ACTACT CTACT TACT ACT CT T AC ACTACTTACT C ACTACT 0 23 1 4 TACT

18. What’s the Problem ACACTACT CACTACT ACTACT CTACT TACT ACT CT T AC ACTACTTACT C ACTACT 0 23 1 4 TACT This suffix is a prefix of another suffix

19. What’s the Fix? Add a new symbol to the end of the string A symbol $ that does not appear elsewhere

20. Suffix Tree For ACACTACT$ ACACTACT$ CACTACT$ ACTACT$ CTACT$ TACT$ ACT$ CT$ T$ $ ACACTACT$ 0

21. Suffix Tree For ACACTACT$ ACACTACT$ CACTACT$ ACTACT$ CTACT$ TACT$ ACT$ CT$ T$ $ ACACTACT$ CACTACT$ 0 1

22. Suffix Tree For ACACTACT$ ACACTACT$ CACTACT$ ACTACT$ CTACT$ TACT$ ACT$ CT$ T$ $ CACTACT$ AC ACTACT$TACT$ 1 0 2

23. Suffix Tree For ACACTACT$ ACACTACT$ CACTACT$ ACTACT$ CTACT$ TACT$ ACT$ CT$ T$ $ AC ACTACT$TACT$ C ACTACT$ 0 23 1

24. Suffix Tree For ACACTACT$ ACACTACT$ CACTACT$ ACTACT$ CTACT$ TACT$ ACT$ CT$ T$ $ AC ACTACT$TACT$ C ACTACT$ 0 23 1 4 TACT

25. Suffix Tree For ACACTACT$ ACACTACT$ CACTACT$ ACTACT$ CTACT$ TACT$ ACT$ CT$ T$ $ AC ACTACT$T C TACT$ ACTACT$ 0 3 1 4 TACT ACT$ $ 2 5 Suffix is prefix problem went away

26. Suffix Tree For ACACTACT$ ACACTACT$ CACTACT$ ACTACT$ CTACT$ TACT$ ACT$ CT$ T$ $ AC ACTACT$T C ACT$ ACTACT$ 0 3 1 4 TACT ACT$ $ 2 5 $ T 6

27. Suffix Tree For ACACTACT$ ACACTACT$ CACTACT$ ACTACT$ CTACT$ TACT$ ACT$ CT$ T$ $ AC ACTACT$T C ACT$ ACTACT$ 0 3 1 T ACT$ $ 2 5 $ T 6 TACT $ 47

28. Suffix Tree For ACACTACT$ ACACTACT$ CACTACT$ ACTACT$ CTACT$ TACT$ ACT$ CT$ T$ $ AC ACTACT$T C ACT$ ACTACT$ 0 3 1 T ACT$ $ 2 5 $ T 6 TACT $ 47 $ 8

29. Use for Testing Substrings ACT follow links, get 2,5 CAC follow links, get 1 AC ACTACT$T C ACT$ ACTACT$ 0 3 1 T ACT$ $ 2 5 $ T 6 TACT $ 47 $ 8

30. Reprise To review the procedure, let’s build a suffix tree for MISSISSIPPI On the board Don’t forget the $

31. Code (define suffix-tree ; input string output suffix tree (lambda (t) (trie (suffixes-of t) (string-length t)))) (define suffixes-of ; input string output list of all its suffixes (lambda (t) (cond ((zero? (string-length t)) '()) (else (cons t (suffixes-of (substring t 1 (string-length t)))))))) (define trie ; input list of strings and n ; output suffix-tree style trie with ; n-(length of keyword) at the leaves (lambda (l n) ; list of keywords to put in a trie (tries (sort (lambda (x y) (string<=? x y)) l) n)))

32. More code (define tries ; builds a trie from sorted list l. Leaves as above (lambda (l n) (cond ((null? l) (make-empty-trie)) ((singleton? (samestarts l)) (make-internal-node (make-edge (car l) (make-leaf (- n (string-length (car l))))) (tries (cdr l) n))) (else (let ((childstrings (samestarts l))) (let ((label (commonprefix childstrings))) (let ((childnode (trie (map (chop (string-length label)) childstrings) (- n (string-length label))))) (make-internal-node (make-edge label childnode) (tries (nthcdr l (length childstrings)) n)))))))))

33. Analysis Building suffix tree: O(n 2 ) Searching for p: O(m+k) –Where p appears k times

34. Improvement possible Suffix tree for text length n can be built in time O(n) Thereafter all searches are O(m)

35. Applications in Biology Suffix Trees in Computational Biology Link doesn’t work