Suffixes, Conjugates and Lyndon words

Transcript

Silvia Bonomo, Sabrina Mantaci, Antonio Restivo,
Giovanna Rosone and Marinella Sciortino
Dipartimento di Matematica e Informatica, University of Palermo
Palermo, ITALY
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Outline
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The problem
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SA, BWT and EBWT
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Lyndon words and sortings
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The algorithm for sorting conjugates
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Sorting suffixes by Lyndon factorization
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Applications
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The problem
The sorting problem in text processing tools
The sorting process is an important step for the construction of important
tools for text processing (applied in pattern matching, data compression,
sequences comparison, etc):
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The problem
The Suffix Array (SA);
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The problem
The Burrows Wheeler Transform (BWT)
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The problem
The Burrows Wheeler Transform (BWT)
The Extended Burrows-Wheeler Transform (EBWT)
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The problem
Different sorting relations
In spite of the closeness of these tools, the sorting processes involve
different sorting relations on different objects:
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The problem
SA → lexicographic order among suffixes of a single word;
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The problem
BWT → lexicographic order among conjugates of a single word;
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The problem
EBWT → ω order among conjugates of a set of words.
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The problem
EBWT → ω order among conjugates of a set of words.
The aim of this talk is to find the combinatorial connection among all
these sorting processes
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The problem
Preliminaries
Let Σ denote a non-empty finite alphabet.
A word w over an alphabet Σ is a finite sequence of letters of Σ. We
denote by Σ∗ the set of all words over Σ
Given a finite word w = a1 a2 · · · an , ai ∈ Σ, a factor of w is written
as w[i, j] = ai · · · aj . A factor w[1, j] is called a prefix, while a factor
w[i, n] is called a suffix.
Two words u, v ∈ Σ∗ are conjugate, if u = xy and v = yx for some
x, y ∈ Σ∗ .
Here, we denote by sufk (w) (resp. prefk (w)) the suffix (resp. prefix) of
w that has length k and by conjk (w) the conjugate of w starting at
position |w| − k + 1.
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SA, BWT and EBWT
Suffix array
Given a word w ∈ Σ∗ , with |w| = n , the suffix array (SA) of w (SA(w))
is the permutation of {1, 2, . . . , n} giving the starting positions of the
suffixes of w, sorted in lexicographic order.
Example
Given the word w = ababbaa its suffix array is SA(w) = [6, 7, 1, 3, 5, 2, 4]
In order to get the suffix array one need to lexicographically sort its
suffixes.
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SA, BWT and EBWT
The Burrows Wheeler Transform
Given a word w ∈ Σ∗ , the Burrows-Wheeler Transform of w, bwt(w) is a
permutation of the letters in w, obtained as concatenation of the last
letters of the lexicographically sorted list of its conjugates.
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SA, BWT and EBWT
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SA, BWT and EBWT
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The rows of M are conjugate of v sorted in
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SA, BWT and EBWT
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bwt(v) = L = nntltaoriiena and I = 4.
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bwt(v) = L = nntltaoriiena and I = 4.
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DLT 2013
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SA, BWT and EBWT
The Extended Burrows-Wheeler Transform
Given two words u, v ∈ Σ∗ we define the following order relation:
u ω v ⇐⇒ uω <lex v ω
Example
This order is different from lexicographic one. For instance ab <lex aba
but aba ω ab since abaω = abaabaaba · · · and abω = abababab · · · .
DLT 2013
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SA, BWT and EBWT
The Extended Burrows-Wheeler Transform
Given two words u, v ∈ Σ∗ we define the following order relation:
u ω v ⇐⇒ uω <lex v ω
Example
This order is different from lexicographic one. For instance ab <lex aba
but aba ω ab since abaω = abaabaaba · · · and abω = abababab · · · .
Given a set of words , W = {w1 , w2 , . . . wn } with w1 , w2 , . . . wn ∈ Σ∗ the
Extended Burrows-Wheeler Transform of W , denoted ebwt(W ), is a
transformation that produces a word as follows:
sort according to the ω order the conjugates of the words in W ;
take the concatenation of the last letters of the sorted list.
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SA, BWT and EBWT
Example
Consider the set W = {abac, bca, cbab, cba}.
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Figure : The output of E(S) is L = ccbbbcacaaabba
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Lyndon Words
A Lyndon word is a primitive word which is the minimum in its conjugacy
class, with respect to the lexicographic order relation.
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Lyndon Words
There exist linear algorithms that for any word w ∈ Σ∗ compute the
Lyndon word of its conjugacy class. We call it the Lyndon word of w and
we denote it by Tw .
DLT 2013
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Lyndon Words
Theorem (Chen, Fox, Lyndon 1958)
Every word w ∈ Σ+ has a unique factorization w = w1 · · · ws such that
w1 ≥lex · · · ≥lex ws is a non-increasing sequence of Lyndon words.
DLT 2013
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Lyndon Words
Theorem (Chen, Fox, Lyndon 1958)
Every word w ∈ Σ+ has a unique factorization w = w1 · · · ws such that
w1 ≥lex · · · ≥lex ws is a non-increasing sequence of Lyndon words.
The Lyndon factorization of a given word can be computed in linear time
[Duval 1983].
DLT 2013
10 / 22
One Word
Theorem (Giancarlo, Restivo, Sciortino 2007)
Let T be a Lyndon word. For any integers h, k with 1 ≤ h, k ≤ |T |, the
following statements are equivalent:
i. conjh (T ) <lex conjk (T );
ii. conjh (T ) ω conjk (T );
iii. sufh (T ) <lex sufk (T ).
This means that for a Lyndon word lexicographic sorting of the suffixes,
lexicographic sorting of the conjugates and ω sorting of the conjugates
are equivalent
DLT 2013
11 / 22
Two words
Theorem
Let T1 and T2 be two Lyndon words, then T1 ≤lex T2 if and only if
T1 ω T2 .
DLT 2013
12 / 22
Two words
Theorem
T1 ω T2 .
Lemma
Let T1 and T2 be two distinct Lyndon words. If T1 <lex T2 and h ≤ k then
conjh (T1 ) ≺ω conjk (T2 ) ⇔ sufh (T1 ) ≤lex sufk (T2 ).
DLT 2013
12 / 22
Two words
Theorem
T1 ω T2 .
Lemma
T1 =
T2 =
DLT 2013
12 / 22
Two words
Theorem
T1 ω T2 .
Lemma
T1 =
T2 =
DLT 2013
12 / 22
Lemma
Let T1 and T2 be two distinct Lyndon words. If T1 <lex T2 and h > k, then
conjh (T1 ) ≺ω conjk (T2 ) ⇔ sufh(T1 ) ≤lex sufk (T2 )prefh−k (T2ω ).
DLT 2013
13 / 22
Lemma
T1 =
T2 =
DLT 2013
13 / 22
Lemma
T1 =
T2 =
DLT 2013
13 / 22
A more general formulation of previuos lemmas is the following:
Theorem
Let u and v be primitive words, let Tu and Tv be their corresponding
Lyndon words. Let suppose Tu <lex Tv and let r be the integer such that
u = conjr (Tu ). Then
u ≺ω v ⇔ prefr (uω ) ≤lex prefr (v ω ).
DLT 2013
14 / 22
An algorithm for sorting the conjugates of a multiset of
Lyndon words
As an application of the previous results, given a set T = {T1 , T2 , . . . , Tm }
of lexicographically sorted multiset of Lyndon words, we provide an
algorithm that produces the ω sorted list of the conjugates of words in T .
DLT 2013
15 / 22
An algorithm for sorting the conjugates of a multiset of
Lyndon words
As an application of the previous results, given a set T = {T1 , T2 , . . . , Tm }
of lexicographically sorted multiset of Lyndon words, we provide an
algorithm that produces the ω sorted list of the conjugates of words in T .
Our strategy consists in inserting a new conjugate in an already sorted list
of other conjugates, analyzing them from right to left and proceeding from
the greatest Lyndon word down to the smallest one.
Actually our algorithm implicitely uses the above results, and do not use
any comparison.
More details in the proceedings.
DLT 2013
15 / 22
Let w = a1 · · · an ∈ Σ∗ and let w1 w2 · · · ws be its Lyndon factorization.
Let j1 , j2 , . . . , js be the positions in w of the last characters of factors
w1 , w2 , . . . , ws , respectively. Obviously js = n.
DLT 2013
16 / 22
For any position p in w, we define
L(p) = min{jk | 1 ≤ jk ≤ s and p ≤ jk }
and
l(p) = L(p) − p + 1.
DLT 2013
16 / 22
For any position p in w, we define
L(p) = min{jk | 1 ≤ jk ≤ s and p ≤ jk }
and
l(p) = L(p) − p + 1.
Theorem
Let w ∈ Σ∗ and let p and q be two positions in w, p < q. Then
w[q, n] < w[p, n] ⇔ prefl(q) (w[q, n]) ≤ prefl(q) (w[p, n]).
DLT 2013
16 / 22
Applications
Applications
REMARK 1:
If p and q are two positions in w with p < q, lcp(p, q) denotes the length
of the longest common prefix between the suffixes w[p, n] and w[q, n].
DLT 2013
17 / 22
Applications
Applications
REMARK 1:
A consequence of the previous theorem is that in order to get the mutual
order between w[p, n] and w[q, n] it is sufficient to execute
min(lcp(p, q) + 1, l(q)) symbol comparisons.
DLT 2013
17 / 22
Applications
Applications
REMARK 1:
A consequence of the previous theorem is that in order to get the mutual
order between w[p, n] and w[q, n] it is sufficient to execute
min(lcp(p, q) + 1, l(q)) symbol comparisons.
This would allow the decreasing of the number of comparisons in
algorithms for SA based on symbol comparisons.
The following example shows that l(q) can be much smaller than
lcp(p, q) + 1.
DLT 2013
17 / 22
Applications
Example
Example
Let w = abaaaabaaaaabaaaabaaaaaab. The Lyndon factorization of w is
ab|aaaab|aaaaabaaaab|aaaaaab
Consider the suffixes w[2, 25] = baaaabaaaaabaaaabaaaaaab and
w[13, 25] = baaaabaaaaaab. We have lcp(2, 13) = 11 and l(13) = 6.
l (13)
lcp(2, 13) + 1
↓
↓
w[2, 25] = baaaa b aaaaa b aaaabaaaaaab
w[13, 25] = baaaa b aaaaa a b
DLT 2013
18 / 22
Applications
Applications
REMARK 2:
If p and q are in the same Lyndon factor, i.e. L(p) = L(q), then the order
of the two suffixes is the same as the order of the suffixes inside their
common Lyndon factor. This property can be extended to blocks of
consecutive Lyndon factors (Lyndon blocks).
DLT 2013
19 / 22
Applications
Applications
REMARK 2:
Application: new strategy for the construction of the suffix array of a word
[MRRS, to be presented in PSC2013]:
1
2
3
find the Lyndon factorization of a word;
find separately the suffix array of every Lyndon factor;
merge the suffix arrays of pairs of adjacent Lyndon blocks into one.
DLT 2013
19 / 22
Applications
Applications
REMARK 2:
Application: new strategy for the construction of the suffix array of a word
[MRRS, to be presented in PSC2013]:
1
2
3
find the Lyndon factorization of a word;
find separately the suffix array of every Lyndon factor;
merge the suffix arrays of pairs of adjacent Lyndon blocks into one.
This strategy allows:
Parallelization;
Online sorting.
DLT 2013
19 / 22
Applications
Applications
The suffix permutation of a word w = a1 a2 · · · an is the permutation πw
over {1, . . . , n}, where πw (i) is the rank of the suffix w[i, n] in the set of
the lexicographically sorted suffixes of w (i.e. the inverse of the
permutation defined by the suffix array SAw ).
Given a permutation π over {1, . . . , n}, an integer i (1 ≤ i ≤ n) is a
left-to-right minimum of π if π(j) > π(i), for all j < i.
DLT 2013
20 / 22
Applications
Applications
The suffix permutation of a word w = a1 a2 · · · an is the permutation πw
over {1, . . . , n}, where πw (i) is the rank of the suffix w[i, n] in the set of
the lexicographically sorted suffixes of w (i.e. the inverse of the
permutation defined by the suffix array SAw ).
Given a permutation π over {1, . . . , n}, an integer i (1 ≤ i ≤ n) is a
left-to-right minimum of π if π(j) > π(i), for all j < i.
Theorem (Hohlweg, Reutenauer 2003)
Let w be a word, let i1 = 1, i2 , . . . , ik be the start positions of the factors
in its Lyndon factorization and let πw be its suffix permutation. Then the
values i1 , i2 , . . . , ik correspond to the positions of the left to right minima
of πw .
This theorem can be deduced from the results in the present paper.
DLT 2013
20 / 22
Applications
Example
The suffix array of the word w = abaaaabaaaaabaaaabaaaaaab is
[19, 20, 8, 21, 14, 3, 9, 22, 15, 4, 10, 23, 16, 5, 11, 24, 17, 6, 12, 1, 25, 18, 7, 13, 2].
The suffix permutation is given by:
[20, 25, 6, 10, 14, 18, 23, 3, 7, 11, 15, 19, 24, 5, 9, 13, 17, 22, 1, 2, 4, 8, 12, 16, 21].
Since there exist linear algorithms for the construction of SA, this give an
linear alternative to Duval’s algorithm for finding Lyndon Factorization.
DLT 2013
21 / 22
Applications
Thanks for your attention!
DLT 2013
22 / 22

Suffixes, Conjugates and Lyndon words

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