1. 1 Reverse Factor Algorithm Advisor: Prof. R. C. T. Lee Speaker: L. C. Chen Speeding up on two string matching algorithms, Algorithmica, Vol.12, 1994, pp.247-267 CROCHEMORE, M., CZUMAJ, A., GASIENIEC, L., JAROMINEK, S., LECROQ, T., PLANDOWSKI, W. and RYTTER, W.

2. 2 Rule 1: The Suffix to Prefix Rule For a window to have any chance to match a pattern, in some way, there must be a suffix of the window which is equal to a prefix of the pattern.

3. 3 Basic Ideas Open a window W with size |P| in the text. T |P||P| W p Find the longest suffix of W is also the prefix of pattern. T |P||P| p W Match! Case 1:

4. 4 T |P||P| W p Case 2: T |P||P| W p T |P||P| W p Case 3: |P||P| If there is no such suffix, we move W with length |P|.

5. 5 Preprocessing phase T=GCATCGGCGAGAGTATACAGTACG P=GCAGAGAG L(S): a set contains all prefixes of the pattern. 087654321 G A G A G GA C C C CA We construct the suffix automaton of P. Suffix Automaton

6. 6 Preprocessing: Construct a Suffix Tree P R : the reversal string of P. 1 86 47 5 3 2

7. 7 GCATCGCAGAGAGTATACAGTACG GCAGAGAG When there is a match, how do we move the window? T P

8. 8 GCATCGCAGAGAGTATACAGTACG GCAGAGAG T P

9. 9 GCATCGCAGGCAGTATACAGTACG GCAGAGAG T P Find the longest suffix of W is also the prefix of pattern.

10. 10 GCATCGCAGGCAGTATACAGTACG GCAGAGAG T P

11. 11 A Whole Example T=GCATCGCAGAGA GTATACAGTACG P=GCAGAGAG First attempt : GCATCGCAGAGAGTATACAGTACG GCAGAGAG Shift by: 5 (8 - 3) T P

12. 12 GCATCGCAGAGAGTATACAGTACG GCAGAGAG Second attempt : Shift by: 7 (8 - 1) T P

13. 13 Third attempt: GCATCGCAGAGAGTATACAGTACG GCAGAGAG Shift by: 7 (8 - 1) T P

14. 14 Third attempt: GCATCGCAGAGAGTATACAGTACG GCAGAGAG T P

15. 15 Conclusion Preprocessing phase is O (m). Searching phase is O (mn).

16. 16 Reference [A90]Algorithms for finding patterns in strings, A. V. Aho, Handbook of Theoretical Computer Science, Vol. A, Elsevier, Amsterdam, 1990, pp.255-300. [A85]The myriad virtues of suffix trees, Apostolico, A., Combinatorial Algorithms on words, NATO Advanced Science Institutes, Series F, Vol. 12, 1985, pp.85-96 [AG86]The Boyer-Moore-Galil string searching strategies revisited, Apostolico, A. and Giancarlo, R., SIAM, Comput. 15, 1986, pp98-105. [BR92]Average running time of the Boyer-Moore-Horspool algorithm, Baeza-Yates, R. A. and Regnier, M. Theoret. Comput. Sci., 1992, pp.19-31. [BKR91]Analysis of algorithms and Data Structures, Banachowski, L., Kreczmar, A. and Rytter, W., Addison- Wesley. Reading, MA,1991.