Combinatorial aspects of the Burrows-Wheeler transform

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Combinatorial aspects of the Burrows-Wheeler
transform
Sabrina Mantaci Antonio Restivo Marinella
Sciortino
University of Palermo
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Burrows-Wheeler Transform

• In 1994 M. Burrows and D. Wheeler introduced a
  new data compression method based on a
  preprocessing on the input string. Such a
  preprocessing, called after them the
  Burrows-Wheeler Transform (BWT), produces a
  permutation of the letters in the input string
  such that
• the transformed string is easier to compress
  than the original one.
• the original string can be recovered
• The use of this preprocessing allowed to define a
  class of lossless data compression algorithms
  that
• achieve speed comparable to the algorithms based
  on the techniques by Lempel and Ziv
• obtains a compression ratio close to the best
  statistical modelling techniques.

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How does BWT work ?

• INPUT w abraca

• Lexicographically sort the cyclic rotations of w

• The following properties hold
• the character Li is followed in w by Fi
• for each character ch, the i-th occurrence of ch
  in F corresponds to the i-th occurrence of ch in
  L.

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Reversibility
The Burrows-Wheeler transform is reversible, in
the sense that given BWT(w) and an index I, it is
possible to recover w.

• Given LBWT(w)caraab and I1
• Construct F by alphabetically sorting the
  letters in L

• Define a permutation ? on 0,1,,n-1,
  establishing a correspondence between the
  positions of the same letters in F and in L

• Starting from position I, we can recover ww0
  wn as follows
• wi F?i(I), where ?0(x)x, ?i1(x) ?(?i(x))

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We can deduce that
Therefore we can study combinatorial properties
of the BWT by studying the conjugacy classes of
primitive words.
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Standard Words
d1, d2,,dn, a sequence of natural numbers d1?0,
gt0 i 2,,n Consider the sequence snn ?0
defined as

• s is a characteristic
  Sturmian word
• sn ?0 is called approximating sequence of s
• (d1, d2,,dn, ) is the directive sequence of s
• Each finite word sn is a standard word

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Characterization of standard words

• A word w is standard if and only if it is a
  letter or wvab (or equivalently wvba) and v has
  periods p,q such that gcd(p,q)1 and
  vpq-2.(extremal case of Fine and Wilf
  theorem)
• A word w is standard if and only if it is a
  letter or there exist palindrome words P,Q,R,
  such that w QR Pxy where x,ya,b.
• Standard words correspond to an extremal case of
  Knuth-Morris-Pratt algorithm.

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Rotations
Standard words can also be generated by
rotations. Let p,q?2 such that gcd(p,q)1 and
npq. ?p0,1,,n-1?0,1,,n-1 defined as
?p(z)zp (mod n)
If n8, p3, q5, wabaababa
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A new characterization of standard words
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Idea of the proof
The permutation ? giving the correspondence
between the positions of characters in F and L is
?(z)zp(mod n). Starting, for example, from the
position Ip we can recover the word u,
uiF(?i(p)).
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Further Research
Further Research

• Study extremal case of the BWT for k-letters
  alphabets with kgt2.
• For instance for k3, characterize the words w
  such that BWT(w) belongs to cab or bca.
• This property does work neither with 3-Standard
  words nor with balanced words.

• Does a relation between the complexity function
  of a word w and the structure of BWT(w) exist?

• Given a language L, one can define
  BWT(L)BWT(w) w in L. One can ask whether BWT
  preserves some properties of a language L, such
  as belonging to a certain family of languages in
  the Chomsky Hierarchy.
• We found negative results

L1(ab), BWT(L1)bnan n0 a context free
language
L2(abc), BWT(L2)cnanbn n0 a context
sensitive language
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Further Research

• Is it possible to characterize interesting
  families of words in terms of their BWT?