# Factor Oracle Suffix Oracle by lqh68203

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```									      Factor Oracle
Suffix Oracle

Factor Oracle, Suffix Oracle   1
Outline
 Factor oracle definition
 Construction methods
 Suffix oracle
 Factor oracle for a set of words
 Applications in string matching

Factor Oracle, Suffix Oracle   2
Data Structures that Represent the
Factors of a String

   Suffix      trie   –     tree
representing all the suffixes
of the string.
Figure 1. Suffix Trie of the string abbc
   Suffix automaton (DAWG)
– the minimal automaton
recognizing all the suffixes
of the string.
Figure 2. Suffix Automaton of the string abbc

   Both the suffix automaton
and the factor oracle can
be obtained from the suffix
trie.                                     Figure 3. Factor Oracle for the string abbc

Factor Oracle, Suffix Oracle                                       3
Factor Oracle – Basic Ideas
 The factor oracle is a data structure used
for indexing all the factors of a given word.
 An automaton built on a string p that acts
like an oracle on the factors of the string.
 If a string is accepted by the automaton it
may be a factor of p – weak factor
recognition.
 All the correct factors are accepted.

Factor Oracle, Suffix Oracle   4
Factor Oracle – Example

Figure 4. Factor oracle for abbbaab
ba is a factor of baab so a transition from 2 to 5 by a is added

   Factor oracle for the string abbbaab.
   All states are considered final.
   The word abba is accepted although it is not a
factor of abbbaab.

Factor Oracle, Suffix Oracle                  5
Factor Oracle – Formal Definition

Figure 5. High level construction algorithm of Oracle(p). The algorithm
Definition 1. The factor oracle of a string p  p1 p2  pm
is the automaton built by the algorithm Build_Oracle,
where all the states are terminal.
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Factor Oracle – Properties

1. Acyclic homogenous deterministic automaton.
2. Recognizes at least the factors of p, the string
that it was built for.
3. Has the fewest states possible (for a string p of
length m there are precisely m+1 states).
4. Has a linear number of transitions (the total
number ranges between m and 2m-1).

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Factor Oracle – Construction

   In the sequential construction the letters of the word
are read from left to right and the automaton is

   We denote repet p (i ) the longest suffix of
pref p (i )  p1 p2  pi that appears at least twice in it.

   We define S p a function on the states of the automaton
called supply function that maps each state i of
Oracle(p) to the state j where the reading of
repet p (i ) ends.

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Factor Oracle – Construction Algorithm

Buid_Oracle_Sequential( p  p1 p2  pm )
1.     create initial state 0, set S p (0)  1
2.      for i=1 to m do
3.        create new state i
4.        add a new transition from i-1 to i by p i
5.        set k  S p (i  1)
6.        while      k  1 and there is no transition from k by   pi do
7.                   add new transition from k to i by pi
8.                   set k  S p (k )
9.        endwhile
10.       if k = -1 then set S p (i )  0
11.       else set S p (i )   ( k , pi )
12.    endfor

Factor Oracle, Suffix Oracle               9
Construction of the Factor Oracle for the string abbbaab

S p (0)  1, S p (1)  0                              S p ( 2)  0
Add a new transition from 0 to 2 by b

S p (3)  2                                         S p ( 4)  3
No new transition is needed

Factor Oracle, Suffix Oracle                           10
Construction of the Factor Oracle for the string abbbaab

S p (5)  1                                          S p ( 6)  1

Add new transitions from 3 and 2 to 5 by a               Add new transition from 1 to 6 by a

S p (7 )  2
No new transition is needed

Factor Oracle, Suffix Oracle                              11
Suffix Oracle - Definition

   We mark some states in the factor oracle for the string
p as final in order to recognize suffixes of p. The new
structure is called suffix oracle.

   A state q of the suffix oracle is terminal if and only if
there is a path labeled by a suffix of p from the initial

   Terminal states are determined by following the
supply function from state m of Oracle( p  p1 p2  pm ).

Factor Oracle, Suffix Oracle           12
Suffix Oracle – Example

Figure 6. Suffix oracle for the string abbbaab.
Double circled states are terminal.

   The suffix oracle is a little more complicated to
implement than the factor oracle. Also, it
requires more memory space.
Factor Oracle, Suffix Oracle            13
Factor Oracle for a Set of Words
   The factor oracle can be extended for a set of
words so that it contains at least all the factors of
the words from the set.
   We set an order on the words from the set, in
order to avoid the uniqueness problem.
   The oracle is built on a trie of all the words which
is updated similarly to the factor oracle for one
word.
   The supply function maps each state i of the
oracle to the state j where the reading of the
longest repeated suffix that appears in one of
the words ends.

Factor Oracle, Suffix Oracle     14
Factor Oracle for a Set of Words
Example

Figure 7. Trie for the set {abbba, baaa}              Figure 8. Intermediate phase in the
construction of the factor oracle for the
set {abbba, baaa}

Figure 9. Factor oracle for the set {abbba, baaa}

Factor Oracle, Suffix Oracle                                      15
Backward Oracle Matching Algorithm
   Version of the BDM algorithm using the factor
oracle instead of the suffix automaton.

    Fast in practice for very long patterns and small
alphabets.

   Preprocessing phase linear in time and space
complexity.

   Optimal on average (conjecture.)
Factor Oracle, Suffix Oracle   16
BOM – Main Idea

The search uses the oracle of the reversed pattern. The search
stops when the word is no longer recognized by the oracle (which
shows it is certainly not a factor of the reversed pattern).

The search window is shifted beyond the point the search
failed (safe shift).

Factor Oracle, Suffix Oracle                17
BOM – Facts
   The suffix oracle of the reversed pattern can be
used instead of the factor oracle. The shifts are
longer but there are more operations needed.
   Worst case complexity of BOM is O(mn), where
m is the length of the pattern, and n the total
length of the text.
   Because the factor oracle accepts some words
that are not really factors of the pattern in some
cases the total number of inspections is greater
than in BDM.
    TurboBOM combines BOM with KMP to obtain
an algorithm linear in the worst case.

Factor Oracle, Suffix Oracle   18
Factor Oracle – Applications

   Finding the repeats in a string

 Data   compression

 Bioinformatics

 Machine   improvisation

Factor Oracle, Suffix Oracle   19
Factor Oracle – Open Problems

Figure 10. The factor oracle for the string abbb accepts exactly all the
factors of the string.

   What     is    the    automaton-independent
characterization of the language recognized
by the oracle.

Factor Oracle, Suffix Oracle                     20
Factor Oracle – Open Problems

Figure 11. The factor oracle for the string abcacdace has 8 extra transitions

Figure 12. A similar automaton with 7 extra transitions

   The factor oracle is not the minimal
homogenous automaton which recognizes at
least the factors of the string.
Factor Oracle, Suffix Oracle                             21
References
1.   Cyril Allauzen, Maxime Crochemore, Mathieu Raffinot Efficient
Experimental String Matching by Weak Factor Recognition in
Proceedings of 12th conference on Combinatorial Pattern Matching, 2001

2.   Cyril Allauzen, Mathieu Raffinot Oracle des facteurs d’un ensemble de
mots Technical report 99-11, Institut Gaspard Monge Universite Marne la
Valee, 1999

3.   Loek Cleophas, Gerard Zwaan, Bruce Watson Constructing Factor
Oracles in Proceedings of the Prague Stringology Conference 2003,
2003

4.   Arnaud Levebvre, Thierry Lecroq Computing repeated factors with a
factor oracle in Proceedings of 11th Australian Workshop on
Combinatorial Algorithms, 2000

5.   G. Assayag, S. Dubnov Using Factor                    Oracles   for   Machine
Improvisation Soft Computing, 2004

Factor Oracle, Suffix Oracle                        22

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