Constructing Antidictionaries of Long Texts in Output-Sensitive Space
A word x that is absent from a word y is called minimal if all its proper factors occur in y . Given a collection of k words y 1 , … , y k over an alphabet Σ , we are asked to compute the set M { y 1 , … , y k } ℓ of minimal absent words of length at most ℓ of the collection { y 1 , … , y k }. The s...
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| Published in: | Theory of computing systems Vol. 65; no. 5; pp. 777 - 797 |
|---|---|
| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
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Springer US
01.07.2021
Springer Nature B.V |
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| ISSN: | 1432-4350, 1433-0490 |
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| Abstract | A word
x
that is absent from a word
y
is called
minimal
if all its proper factors occur in
y
. Given a collection of
k
words
y
1
, … ,
y
k
over an alphabet
Σ
, we are asked to compute the set
M
{
y
1
,
…
,
y
k
}
ℓ
of minimal absent words of length at most
ℓ
of the collection {
y
1
, … ,
y
k
}. The set
M
{
y
1
,
…
,
y
k
}
ℓ
contains all the words
x
such that
x
is absent from all the words of the collection while there exist
i
,
j
, such that the maximal proper suffix of
x
is a factor of
y
i
and the maximal proper prefix of
x
is a factor of
y
j
. In data compression, this corresponds to computing the antidictionary of
k
documents. In bioinformatics, it corresponds to computing words that are absent from a genome of
k
chromosomes. Indeed, the set
M
y
ℓ
of minimal absent words of a word
y
is equal to
M
{
y
1
,
…
,
y
k
}
ℓ
for any decomposition of
y
into a collection of words
y
1
, … ,
y
k
such that there is an overlap of length at least
ℓ
− 1 between any two consecutive words in the collection. This computation generally requires
Ω
(
n
) space for
n
= |
y
| using any of the plenty available
O
(
n
)
-time algorithms. This is because an
Ω
(
n
)-sized text index is constructed over
y
which can be impractical for large
n
. We do the identical computation incrementally using output-sensitive space. This goal is reasonable when
∥
M
{
y
1
,
…
,
y
N
}
ℓ
∥
=
o
(
n
)
, for all
N
∈ [1,
k
], where ∥
S
∥ denotes the sum of the lengths of words in set
S
. For instance, in the human genome,
n
≈ 3 × 10
9
but
∥
M
{
y
1
,
…
,
y
k
}
12
∥
≈
1
0
6
. We consider a constant-sized alphabet for stating our results. We show that
all
M
y
1
ℓ
,
…
,
M
{
y
1
,
…
,
y
k
}
ℓ
can be computed in
O
(
k
n
+
∑
N
=
1
k
∥
M
{
y
1
,
…
,
y
N
}
ℓ
∥
)
total time using
O
(
MaxIn
+
MaxOut
)
space, where MaxIn is the length of the longest word in {
y
1
, … ,
y
k
} and
MaxOut
=
max
{
∥
M
{
y
1
,
…
,
y
N
}
ℓ
∥
:
N
∈
[
1
,
k
]
}
. Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution. |
|---|---|
| AbstractList | A word x that is absent from a word y is called minimal if all its proper factors occur in y. Given a collection of k words y1, … , yk over an alphabet Σ, we are asked to compute the set M{y1,…,yk}ℓ of minimal absent words of length at most ℓ of the collection {y1, … , yk}. The set M{y1,…,yk}ℓ contains all the words x such that x is absent from all the words of the collection while there exist i,j, such that the maximal proper suffix of x is a factor of yi and the maximal proper prefix of x is a factor of yj. In data compression, this corresponds to computing the antidictionary of k documents. In bioinformatics, it corresponds to computing words that are absent from a genome of k chromosomes. Indeed, the set Myℓ of minimal absent words of a word y is equal to M{y1,…,yk}ℓ for any decomposition of y into a collection of words y1, … , yk such that there is an overlap of length at least ℓ − 1 between any two consecutive words in the collection. This computation generally requires Ω(n) space for n = |y| using any of the plenty available O(n)-time algorithms. This is because an Ω(n)-sized text index is constructed over y which can be impractical for large n. We do the identical computation incrementally using output-sensitive space. This goal is reasonable when ∥M{y1,…,yN}ℓ∥=o(n), for all N ∈ [1,k], where ∥S∥ denotes the sum of the lengths of words in set S. For instance, in the human genome, n ≈ 3 × 109 but ∥M{y1,…,yk}12∥≈106. We consider a constant-sized alphabet for stating our results. We show that allMy1ℓ,…,M{y1,…,yk}ℓ can be computed in O(kn+∑N=1k∥M{y1,…,yN}ℓ∥) total time using O(MaxIn+MaxOut) space, where MaxIn is the length of the longest word in {y1, … , yk} and MaxOut=max{∥M{y1,…,yN}ℓ∥:N∈[1,k]}. Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution. A word x that is absent from a word y is called minimal if all its proper factors occur in y . Given a collection of k words y 1 , … , y k over an alphabet Σ , we are asked to compute the set M { y 1 , … , y k } ℓ of minimal absent words of length at most ℓ of the collection { y 1 , … , y k }. The set M { y 1 , … , y k } ℓ contains all the words x such that x is absent from all the words of the collection while there exist i , j , such that the maximal proper suffix of x is a factor of y i and the maximal proper prefix of x is a factor of y j . In data compression, this corresponds to computing the antidictionary of k documents. In bioinformatics, it corresponds to computing words that are absent from a genome of k chromosomes. Indeed, the set M y ℓ of minimal absent words of a word y is equal to M { y 1 , … , y k } ℓ for any decomposition of y into a collection of words y 1 , … , y k such that there is an overlap of length at least ℓ − 1 between any two consecutive words in the collection. This computation generally requires Ω ( n ) space for n = | y | using any of the plenty available O ( n ) -time algorithms. This is because an Ω ( n )-sized text index is constructed over y which can be impractical for large n . We do the identical computation incrementally using output-sensitive space. This goal is reasonable when ∥ M { y 1 , … , y N } ℓ ∥ = o ( n ) , for all N ∈ [1, k ], where ∥ S ∥ denotes the sum of the lengths of words in set S . For instance, in the human genome, n ≈ 3 × 10 9 but ∥ M { y 1 , … , y k } 12 ∥ ≈ 1 0 6 . We consider a constant-sized alphabet for stating our results. We show that all M y 1 ℓ , … , M { y 1 , … , y k } ℓ can be computed in O ( k n + ∑ N = 1 k ∥ M { y 1 , … , y N } ℓ ∥ ) total time using O ( MaxIn + MaxOut ) space, where MaxIn is the length of the longest word in { y 1 , … , y k } and MaxOut = max { ∥ M { y 1 , … , y N } ℓ ∥ : N ∈ [ 1 , k ] } . Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution. A word x that is absent from a word y is called minimal if all its proper factors occur in y . Given a collection of k words y 1 , … , y k over an alphabet Σ , we are asked to compute the set $\mathrm {M}^{\ell }_{\{y_1,\ldots ,y_k\}}$ M { y 1 , … , y k } ℓ of minimal absent words of length at most ℓ of the collection { y 1 , … , y k }. The set $\mathrm {M}^{\ell }_{\{y_1,\ldots ,y_k\}}$ M { y 1 , … , y k } ℓ contains all the words x such that x is absent from all the words of the collection while there exist i , j , such that the maximal proper suffix of x is a factor of y i and the maximal proper prefix of x is a factor of y j . In data compression, this corresponds to computing the antidictionary of k documents. In bioinformatics, it corresponds to computing words that are absent from a genome of k chromosomes. Indeed, the set $\mathrm {M}^{\ell }_{y}$ M y ℓ of minimal absent words of a word y is equal to $\mathrm {M}^{\ell }_{\{y_1,\ldots ,y_k\}}$ M { y 1 , … , y k } ℓ for any decomposition of y into a collection of words y 1 , … , y k such that there is an overlap of length at least ℓ − 1 between any two consecutive words in the collection. This computation generally requires Ω ( n ) space for n = | y | using any of the plenty available $\mathcal {O}(n)$ O ( n ) -time algorithms. This is because an Ω ( n )-sized text index is constructed over y which can be impractical for large n . We do the identical computation incrementally using output-sensitive space. This goal is reasonable when $\| \mathrm {M}^{\ell }_{\{y_1,\ldots ,y_N\}}\| =o(n)$ ∥ M { y 1 , … , y N } ℓ ∥ = o ( n ) , for all N ∈ [1, k ], where ∥ S ∥ denotes the sum of the lengths of words in set S . For instance, in the human genome, n ≈ 3 × 10 9 but $\| \mathrm {M}^{12}_{\{y_1,\ldots ,y_k\}}\| \approx 10^{6}$ ∥ M { y 1 , … , y k } 12 ∥ ≈ 1 0 6 . We consider a constant-sized alphabet for stating our results. We show that all $\mathrm {M}^{\ell }_{y_{1}},\ldots ,\mathrm {M}^{\ell }_{\{y_1,\ldots ,y_k\}}$ M y 1 ℓ , … , M { y 1 , … , y k } ℓ can be computed in $\mathcal {O}(kn+{\sum }^{k}_{N=1}\| \mathrm {M}^{\ell }_{\{y_1,\ldots ,y_N\}}\| )$ O ( k n + ∑ N = 1 k ∥ M { y 1 , … , y N } ℓ ∥ ) total time using $\mathcal {O}(\textsc {MaxIn}+\textsc {MaxOut})$ O ( MaxIn + MaxOut ) space, where MaxIn is the length of the longest word in { y 1 , … , y k } and $\textsc {MaxOut}=\max \limits \{\| \mathrm {M}^{\ell }_{\{y_1,\ldots ,y_N\}}\| :N\in [1,k]\}$ MaxOut = max { ∥ M { y 1 , … , y N } ℓ ∥ : N ∈ [ 1 , k ] } . Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution. |
| Author | Pissis, Solon P. Badkobeh, Golnaz Héliou, Alice Fici, Gabriele Ayad, Lorraine A.K. |
| Author_xml | – sequence: 1 givenname: Lorraine A.K. surname: Ayad fullname: Ayad, Lorraine A.K. organization: Department of Informatics, King’s College London – sequence: 2 givenname: Golnaz surname: Badkobeh fullname: Badkobeh, Golnaz organization: Department of Computing, Goldsmiths University of London – sequence: 3 givenname: Gabriele orcidid: 0000-0002-3536-327X surname: Fici fullname: Fici, Gabriele email: gabriele.fici@unipa.it organization: Dipartimento di Matematica e Informatica, Università di Palermo – sequence: 4 givenname: Alice surname: Héliou fullname: Héliou, Alice organization: Laboratoire d’Informatique de l’École Polytechique, École Polytechnique – sequence: 5 givenname: Solon P. surname: Pissis fullname: Pissis, Solon P. organization: CWI, Vrije Universiteit |
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| Cites_doi | 10.1101/gr.229102 10.1109/DCC.2008.95 10.1007/978-3-030-00479-8_12 10.1093/bioinformatics/btx209 10.1017/CBO9780511574931 10.1109/5.892711 10.1007/978-3-319-32152-3_23 10.1007/978-3-030-00479-8_11 10.1007/978-3-030-00479-8 10.1145/3386369 10.1109/ISITA.2010.5649621 10.1016/j.ic.2018.06.002 10.1007/3-540-61258-0_11 10.1017/CBO9780511546853 10.1093/bioinformatics/btv189 10.1007/978-3-642-40450-4_12 10.1186/s13015-017-0094-z 10.1007/s00453-017-0286-4 10.1007/978-3-662-55751-8_14 10.1186/s12859-014-0388-9 10.1109/DCC.2019.00062 10.1007/978-3-319-19929-0_28 10.1016/S0020-0190(98)00104-5 10.1109/ISIT.2012.6283021 10.1007/978-3-030-32686-9_11 |
| ContentType | Journal Article |
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| Keywords | String algorithm Output sensitive algorithm Absent word Antidictionary Data compression |
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IEEE Computer Society, Snowbird (2008) Fujishige, Y, Takagi, T, Hendrian, D: Truncated DAWGs and their application to minimal absent word problem. In: Gagie, T., Moffat, A., Navarro, G., Cuadros-Vargas, E. (eds.) String Processing and Information Retrieval - 25th International Symposium, SPIRE 2018, Lima, Proceedings, Lecture Notes in Computer Science, vol. 11147, pp 139–152. Springer (2018) CrochemoreMMignosiFRestivoASalemiSData compression using antidictionariesProc. IEEE200088111756176810.1109/5.892711 Ota, T, Morita, H: On the adaptive antidictionary code using minimal forbidden words with constant lengths. In: Proceedings of the International Symposium on Information Theory and its Applications, ISITA 2010, pp. 72–77. IEEE, Taichung (2010) HéliouAPissisSPPuglisiSJemMAW: computing minimal absent words in external memoryBioinformatics201733172746274910.1093/bioinformatics/btx209 SilvaRMPratasDCastroLPinhoAJFerreiraPJSGThree minimal sequences found in Ebola virus genomes and absent from human DNABioinformatics201531152421242510.1093/bioinformatics/btv189 Farach, M, Muthukrishnan, S: Perfect hashing for strings: Formalization and algorithms. In: Hirschberg, D.S., Myers, E.W. (eds.) Combinatorial Pattern Matching, 7th Annual Symposium, CPM 96, Laguna Beach, Proceedings, Lecture Notes in Computer Science, vol. 1075, pp 130–140. Springer (1996) KociumakaTKubicaMRadoszewskiJRytterWWalenTA linear-time algorithm for seeds computationACM Trans. Algorithm.202016227:127:23408776910.1145/3386369 CharalampopoulosPCrochemoreMFiciGMercasRPissisSPAlignment-free sequence comparison using absent wordsInf. 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(eds.): String processing and information retrieval - 25th international symposium, SPIRE 2018, Lima, proceedings, Lecture Notes in Computer Science, vol. 11147 Springer (2018) BartonCHéliouAMouchardLPissisSPLinear-time computation of minimal absent words using suffix arrayBMC Bioinform.20141538810.1186/s12859-014-0388-9 AlmirantisYCharalampopoulosPGaoJIliopoulosCSMohamedMPissisSPPolychronopoulosDOn avoided words, absent words, and their application to biological sequence analysisAlgorithm. Mol. Biol.20171215:15:1210.1186/s13015-017-0094-z BelazzouguiDCunialFA framework for space-efficient string kernelsAlgorithmica2017793857883369610010.1007/s00453-017-0286-4 CrochemoreMMignosiFRestivoAAutomata and forbidden wordsInf. Process. Lett.1998673111117163817810.1016/S0020-0190(98)00104-5 Crochemore, M, Navarro, G: Improved antidictionary based compression. 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| Snippet | A word
x
that is absent from a word
y
is called
minimal
if all its proper factors occur in
y
. Given a collection of
k
words
y
1
, … ,
y
k
over an alphabet
Σ
,... A word x that is absent from a word y is called minimal if all its proper factors occur in y. Given a collection of k words y1, … , yk over an alphabet Σ, we... |
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| Title | Constructing Antidictionaries of Long Texts in Output-Sensitive Space |
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