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Superstring Graph in compact space
Bastien Cazaux and Eric Rivals∗
∗ LIRMM & IBC, Montpellier
February 5, 2020 (DSB 2020)
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Superstring Problems
Bastien Cazaux and Eric Rivals∗ 2 / 1
Superstring Graph in compact space Bastien Cazaux and Eric Rivals - - PowerPoint PPT Presentation
Superstring Graph in compact space Bastien Cazaux and Eric Rivals - - PowerPoint PPT Presentation
Superstring Graph in compact space Bastien Cazaux and Eric Rivals LIRMM & IBC, Montpellier February 5, 2020 (DSB 2020) Bastien Cazaux and Eric Rivals 1 / 1 Superstring Problems Bastien Cazaux and Eric Rivals 2 / 1 Linear and
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Linear and Cyclic words
a b b a b b b a b b a b
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Notation
Definition [Gusfield 1997] Let w a string.
◮ a substring of w is a string included in w, ◮ a prefix of w is a substring which begins w and ◮ a suffix is a substring which ends w. ◮ an overlap from w over v is a suffix of w that is also a prefix of v.
w a b a b b a b a a a
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Notation
Definition [Gusfield 1997] Let w a string.
◮ a substring of w is a string included in w, ◮ a prefix of w is a substring which begins w and ◮ a suffix is a substring which ends w. ◮ an overlap from w over v is a suffix of w that is also a prefix of v.
w a b a b b a b a a a
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Notation
Definition [Gusfield 1997] Let w a string.
◮ a substring of w is a string included in w, ◮ a prefix of w is a substring which begins w and ◮ a suffix is a substring which ends w. ◮ an overlap from w over v is a suffix of w that is also a prefix of v.
w a b a b b a b a a a
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Notation
Definition [Gusfield 1997] Let w a string.
◮ a substring of w is a string included in w, ◮ a prefix of w is a substring which begins w and ◮ a suffix is a substring which ends w. ◮ an overlap from w over v is a suffix of w that is also a prefix of v.
w a b a b b a b a a a
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Notation
Definition [Gusfield 1997] Let w a string.
◮ a substring of w is a string included in w, ◮ a prefix of w is a substring which begins w and ◮ a suffix is a substring which ends w. ◮ an overlap from w over v is a suffix of w that is also a prefix of v.
w a b a b b a b a a a v a b a a a b b b b
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Notation
Definition [Gusfield 1997] Let w a string.
◮ a substring of w is a string included in w, ◮ a prefix of w is a substring which begins w and ◮ a suffix is a substring which ends w. ◮ an overlap from w over v is a suffix of w that is also a prefix of v.
w a b a b b a b a a a v a b a a a b b b b
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Notation
Definition [Gusfield 1997] Let w a string.
◮ a substring of w is a string included in w, ◮ a prefix of w is a substring which begins w and ◮ a suffix is a substring which ends w. ◮ an overlap from w over v is a suffix of w that is also a prefix of v.
w a b a b b a b a a a v a b a a a b b b b u a b a a a
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Notation
Definition [Gusfield 1997] Let w a string.
◮ a substring of w is a string included in w, ◮ a prefix of w is a substring which begins w and ◮ a suffix is a substring which ends w. ◮ an overlap from w over v is a suffix of w that is also a prefix of v.
w a b a b b a b a a a v a b a a a b b b b u a b a a a a b a b b b b b b
- v(w,v)
pr(w,v) su(w,v)
M
w v
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Superstring
Definition Let P = {s1,s2,...,sp} be a set of strings. A superstring of P is a string w such that any si is a substring of w. w : s1 : s2 : s3 : a b a a b a 1 2 3 4 5 6 a a b a b a b a
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Shortest Superstrings problems
Input
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Shortest Superstrings problems
Input Shortest Linear Superstring
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Shortest Superstrings problems
Input Shortest Linear Superstring Shortest Cyclic Superstring
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Shortest Superstrings problems
Input Shortest Linear Superstring Shortest Cyclic Superstring Shortest Cyclic Cover of Strings
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Shortest Superstrings problems
Input Shortest Cyclic Cover of Strings Shortest Linear Superstring
◮ NP-hard [Gallant 1980]
◮ APX-hard [Blum 1991] ◮ Approximation 2+ 11
30
[Paluch 2015] ◮ NP-hard [Cazaux PhD 2016]
Shortest Cyclic Superstring
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Shortest Superstrings problems
Input Shortest Linear Superstring
◮ NP-hard [Gallant 1980]
◮ APX-hard [Blum 1991] ◮ Approximation 2+ 11
30
[Paluch 2015] ◮ NP-hard [Cazaux PhD 2016]
Shortest Cyclic Superstring
◮ In O(|P|3 +||P||) time [Papadimitriou 1982]
◮ Linear time in ||P|| [Cazaux, Rivals JDA 2016]
Shortest Cyclic Cover of Strings
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Superstring Graph
One graph to rule them all
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Greedy Algorithm for SCCS
a a b a b b a a b a a a b a b b
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Greedy Algorithm for SCCS
a a b a b b a a b a a a b a b b
|ov(ababb,abba)| = |abb| = 3
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Greedy Algorithm for SCCS
a a b a b b a a b a a a b a b b
|ov(ababb,abba)| = |abb| = 3
a b a b b a a b b a a b a b b
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Greedy Algorithm for SCCS
a a b a b a a
|ov(ababb,abba)| = |abb| = 3
a b a b b a a b a b b a
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Greedy Algorithm for SCCS
a a b a b a a a b a b b a
|ov(abaa,aab)| = |aa| = 2
a b a a a a b a b a a b
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Greedy Algorithm for SCCS
a b a b b a
|ov(abaa,aab)| = |aa| = 2
a b a a b a b a a b
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Greedy Algorithm for SCCS
a b a b b a a b a a b
|ov(abaab,abaab)| = |ab| = 2
a b a a b a b a a b a b a
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Greedy Algorithm for SCCS
a b a b b a
|ov(abaab,abaab)| = |ab| = 2
a b a a b a
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Greedy Algorithm for SCCS
a b a a b a b b An other solution a b a a b a b b Theorem [Cazaux et al. 2014] The greedy algorithm solves exactly the Shortest Cyclic Cover
- f Strings problem.
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Extended Hierarchical Overlap Graph (EHOG)
ababb aab abba abaa
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Extended Hierarchical Overlap Graph (EHOG)
ababb aab abba abaa abb aa ab a
ε
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Extended Hierarchical Overlap Graph (EHOG)
ababb aab abba abaa abb aa ab a
ε
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Extended Hierarchical Overlap Graph (EHOG)
ababb aab abba abaa abb aa ab a
ε
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Extended Hierarchical Overlap Graph (EHOG)
ababb aab abba abaa abb aa ab a
ε
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Permutation of words on the EHOG
ababb abba aab abaa ababb aab abba abaa abb aa ab a
ε
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Results on the Superstring Graph
Definition All the solutions of the greedy algorithm for SCCS give the same graph
- n the EHOG, and it is called Superstring Graph.
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Results on the Superstring Graph
Definition All the solutions of the greedy algorithm for SCCS give the same graph
- n the EHOG, and it is called Superstring Graph.
Propositions [Cazaux et al. 2015]
◮ The size of the Superstring Graph is linear in the size of the input. ◮ We can build the Superstring Graph in liner time in the size of the
input.
◮ A labeled eulerian cycle of the Superstring Graph is a solution of
the greedy algorithm for the Shortest Cyclic Cover of Strings problem.
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Superstring Graph on the EHOG
ababb aab abba abaa abb aa ab a
ε
b a abb aa b
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Why do we want to compute the Superstring Graph?
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Application 1: Compute bounds for optimal solution of SLS
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Application 1: Compute bounds for optimal solution of SLS
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Application 1: Compute bounds for optimal solution of SLS
A B C
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Application 1: Compute bounds for optimal solution of SLS
A B C a b c
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Application 1: Compute bounds for optimal solution of SLS
A B C a b c With a ≥ b and a ≥ c, A + a + B + C ≤ | Optimal solution of SLS | ≤ A + a + B + b + C + c
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Application 1: Compute bounds for optimal solution of SLS
A B C a b c With a ≥ b and a ≥ c, A + a + B + C ≤ | Optimal solution of SLS | ≤ A + a + B + b + C + c Results on real data [Cazaux et al. 2018] Input: E. coli genome of 50x: 4 503 422 reads (454 845 622 symbols) Result: length of optimal solutions between 187 250 434 and 187 250 672 A difference of 710 symbols (0,00038%)
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Application 2: Use SG as a genome assembly graph
aa da ae ac cb bb bd eb bf fa a b c d e f ε
Results [Cazaux et al. 2016] We can build a mixed cover that includes unitigs of the dBG (or Variable
- rder dBG) in time in O(||P||), and in linear space in the size of the de
Bruijn Graph.
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How can we store the Superstring Graph in compact space?
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Difficulty
The Superstring Graph is a graph with integer value (each value in
log||P|| for a set of strings P)
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Relation between ST and EHOG
4 4 3 1 b a 2 1 a 1 bb a 3 1 a b b a 5 3 3 2 a 2 bb a 4 2 a b b Propositions Each node of EHOG(P) is an explicit node of the Suffix Tree of P.
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Use the BWT-AC decomposition
S = {aaa$,aac$,baa$,bbc$,bbb$} a a a c b a a b c b
$ $ $ $ $
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Use the BWT-AC decomposition
S = {aaa$,aac$,baa$,bbc$,bbb$} 1 6 2 3 9 5 7 4 10 8 a a a c b a a b c b
$ $ $ $ $
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Use the BWT-AC decomposition
S = {aaa$,aac$,baa$,bbc$,bbb$} 1 6 2 3 9 5 7 4 10 8 a a a c b a a b c b
$ $ $ $ $
b a b a b a a c a
$ $
a b a b c b
$ $ $
BWT(mP)
mP = aaa$caa$aab$bbb$cbb$ 1 2 3 4 5 6 7 8 9 10
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Use the BWT-AC decomposition
S = {aaa$,aac$,baa$,bbc$,bbb$} 1 6 2 3 9 5 7 4 10 8 a a a c b a a b c b
$ $ $ $ $
b a b a b a a c a
$ $
a b a b c b
$ $ $
BWT(mP)
mP = aaa$caa$aab$bbb$cbb$ 1 2 3 4 5 6 7 8 9 10 1 1 1 1 1 1 1 1 1 1 1
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Use the BWT-AC decomposition
S = {aaa$,aac$,baa$,bbc$,bbb$} 1 6 2 3 9 5 7 4 10 8 a a a c b a a b c b
$ $ $ $ $
b a b a b a a c a
$ $
a b a b c b
$ $ $
BWT(mP)
mP = aaa$caa$aab$bbb$cbb$ 1 2 3 4 5 6 7 8 9 10 1 1 1 1 1 1 1 1 1 1 1
1 1 1 01 01 01 01 1 01 01 1 1 1 1 01 01 1 1 1 01 01 01 Blue(mP) Red(mP)
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Use the BWT-AC decomposition
Result For a set of strings P,the size of the tables Blue et Red are bounded by
||P|| (= ∑s∈P |s|). → By using BWT and BP
, we can store the Superstring Graph in O(nlogσ) bits.
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How can we build the Superstring Graph in compact space?
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Difficulty
If we want just to compute the tables Blue and Red.
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Difficulty
If we want just to compute the tables Blue and Red. t w v1 v2 u1 u2
n(v) n(u1) n(u2) d( v) d(v1) d(v2)
v For now, algorihtm in O(||P||) linear time and in O(||P||log||P||) bits space.
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Difficulty
If we want just to compute the tables Blue and Red. t w v1 v2 u1 u2
n(v) n(u1) n(u2) d( v) d(v1) d(v2)
v For now, algorihtm in O(||P||) linear time and in O(||P||log||P||) bits space. Can we find an algorithm in O(||P||) linear time and in O(||P||) bits space?
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Thank you for your attention
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