Reverse Factor Algorithm

1
Reverse Factor Algorithm
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.

• Advisor Prof. R. C. T. Lee
• Speaker L. C. Chen

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
Basic Ideas

• Open a window W with size P in the text.

W
T
P
p

• Find the longest suffix of W is also the prefix
  of pattern.

Case 1
W
T
P
p
Match!
4
Case 2
W
T
P
p
W
T
P
p
Case 3
If there is no such suffix, we move W with length
P.
W
T
P
P
p
5
Preprocessing phase

• TGCATCGGCGAGAGTATACAGTACG 
• PGCAGAGAG
• L(S) a set contains all prefixes of the pattern.

We construct the suffix automaton of P.
C
Suffix Automaton
A
G
G
A
C
G
G
A
0
8
7
6
5
4
3
2
1
C
A
C
6
Preprocessing Construct a Suffix Tree
PR the reversal string of P.
1
2
4
7
3
8
6
5
7
When there is a match, how do we move the window?
T
P
8
T
P
9
Find the longest suffix of W is also the prefix
of pattern.
T
P
10
T
P
11
A Whole Example

• TGCATCGCAGAGAGTATACAGTACG 
• PGCAGAGAG
• First attempt  

T
P
Shift by 5 (8 - 3)
12
Second attempt
T
P
Shift by 7 (8 - 1)
13
Third attempt
T
P
Shift by 7 (8 - 1)
14
Third attempt
T
P
15
Conclusion

• Preprocessing phase is O(m).
• Searching phase is O(mn).

16
Reference

• A90Algorithms for finding patterns in strings,
  A. V. Aho, Handbook of Theoretical Computer
  Science, Vol. A, Elsevier, Amsterdam, 1990,
  pp.255-300.
• A85The myriad virtues of suffix trees,
  Apostolico, A., Combinatorial Algorithms on
  words, NATO Advanced Science Institutes, Series
  F, Vol. 12, 1985, pp.85-96
• AG86The Boyer-Moore-Galil string searching
  strategies revisited, Apostolico, A. and
  Giancarlo, R., SIAM, Comput. 15, 1986, pp98-105.
• BR92Average running time of the
  Boyer-Moore-Horspool algorithm, Baeza-Yates, R.
  A. and Regnier, M. Theoret. Comput. Sci., 1992,
  pp.19-31.
• BKR91Analysis of algorithms and Data
  Structures, Banachowski, L., Kreczmar, A. and
  Rytter, W., Addison-Wesley. Reading, MA,1991.