succinct - ucf computer sciencecs.ucf.edu/stringbio2018/talks/talk9.pdf · 10/26/18 diego díaz...

Succinct de Bruijn graphsDiego Díaz DomínguezDepartment of Computer Science

University of Chile

de Bruijn graphs

Given a parameter k and a set ! of strings, a de Bruijn graph

(dBG) is a directed graph that encodes the k-1 overlaps

between all the k-substrings (k-mers) of !.

ATGC TGCT

GCTA

CTAT

TATG

TGCG GCGT

k=5

T

A

G

T

G

C

T

ATGCTATGCGTATGCTGCT

k-mer = ATGCT

!={ }

10/25/18 2Diego Díaz Domínguez

de Bruijn graphs

Different values for k yield different graph topologies


!=

C

ATGCTACTATGC

ATGCGT

ATG

TGC GCT

CTATAT

GCG CGT

k=4

T

A

TG

G

T

ATGC TGCT

GCTA

CTAT

TATG

TGCG GCGT

k=5

T

A

G

T

G

C

de Bruijn graphsAdvantages:• They are easy to construct• They catch the underlying structure of the string set• They can be used to solve several problems in Bioinformatics

Disadvantages:• Choosing the right value for k is always a major concern.• The size of the graph grows with k.• Having several dBG, with different orders, for the same

dataset is expensive.


Encoding a dBG


Things we have to do:• Represent the nodes and the edges succinctly• Traverse the graph efficiently• Combine the information of several graph orders• Colour the nodes

Operations in a dBG


Navigational operations we are interested in:

• outdegree(v):number of outgoing edges of node v• indegree(v):number of incoming edges of v• outgoing(v,a):follow outgoing edge of v labeled with a• incoming(v,a):follow an incoming node of v whose label

starts with a

Succinct dBG: BOSS


C

$$$ATGCTA$$$$$$CTATGC$$$$$$ATGCGT$$$

ATG

TGC

GCTCTA

TATGCG

CGT

k=4

T

A

T

G

G

T

$AT$$A

$$$

$$C $CT

TA$A$$

GT$

T$$

GC$

C$$

A

TG

C

TA

$

$$

$$

$

$$

$

$$$ A1 0

C1 1A$$ $ 1C$$ $ 1T$$ $ 1TA$ $ 1GC$ $ 1GT$ $ 1$$A T1 1CTA $ 0

T2 1$$C T3 1TGC $ 0

G1 0

T4 1GCG T5 1ATG C2 1$AT G2 1TAT *G2 1$CT A1 1GCT *A1 1CGT $ 1

BOSSMatrix

EdgeBWT

NodeMarks

C[$] = 0C[A] = 7C[C] = 9

C[G] = 11 C[T] = 13

2

2

BOSS:outdegree


2

2

select(NodeMarks,11)

select(NodeMarks,10)+1

BOSS: forward


$$$ A1 0


T2 1$$C T3 1TGC $ 0

G1 0


BOSSMatrix

EdgeBWT

NodeMarks

C

ATG

TGC

GCTCTA

TATGCG

CGT

T

A

T

G

G

T

$AT$$A

$$$

$$C $CT

TA$A$$

GT$

T$$

GC$

C$$

A

TG

C

TA

$

$$

$$

$

$$

$

11

18

C[$]=0C[A]=7C[C]=9C[G]=11C[T]=13

forward(11,T)= 17= C[C] + rank(EdgeBWT,T,15)= 13 + 4 = 17

2

217

11

15

$$$ A1 0


T2 1$$C T3 1TGC $ 0

G1 0


BOSSMatrix

EdgeBWT

NodeMarks

C

ATG

TGC

GCTCTA

TATGCG

CGT

T

A

T

G

G

T

$AT$$A

$$$

$$C $CT

TA$A$$

GT$

T$$

GC$

C$$

A

TG

C

TA

$

$$

$$

$

$$

$

C[$]=0C[A]=7C[C]=9C[G]=11C[T]=13

a

b

BOSS: indegree


select(EdgeBWT,G,3)=b

select(EdgeBWT,G,2)=aindegree(13)= 2= 1 + rank(EdgeBWT,*G,b)–

rank(EdgeBWT,*G,a)= 2

13

2

2

Bidirectional BOSS


incoming is more expensive than outgoing because we don’t have accessto the first symbol of the nodes. By doubling the size of the index, we cancompute incoming fast.

C

ATG

TGC

GCTCTA

TATGCG

CGT

T

A

T

G

G

T

$AT$$A

$$$

$$C $CT

TA$A$$

GT$

T$$

GC$

C$$

A

TG

C

TA

$

$$

$$

$

$$

$

$$$ A 0

C 1A$$ $ 1C$$ $ 1T$$ $ 1TA$ $ 1GC$ $ 1GT$ $ 1$$A T 1CTA $ 0

T 1$$C T 1TGC $ 0

G 0

T 1GCG T 1ATG C 1$AT G 1TAT *G 1$CT A 1GCT *A 1CGT $ 1

$$$ $ 1A$$ *$ 1C$$ *$ 1TA$ $ 1TC$ $ 1$$A T 1GTA $ 0

T 1$$C G 1TGC G 1ATC $ 0

G 1$CG T 1GCG *T 1TCG *T 1$TG C 1$$T G 1$AT C 1TAT *C 1CGT A 1

BOSSrBOSS

outgoingselect

incoming(13,G)= 21

13

21

Variable-order BOSS

10/26/18 12Diego Díaz Domínguezk=4 k=2

By augmenting BOSS with the LCP, we can represent all the dBG up to order k

i

i’->LCP[i’]<(k-1)

j->LCP[j]<(k-1)

Encoding EdgeBWT: Wavelet trees


Encoding EdgeBWT: RL-BWT


Closing remarks


• We can compress several dBG and still supporting fast operations

• It is still necessary to propose new algorithms that work ontop of BOSS

• Can we encode the patterns of the graph? (bubbles, tips cross-links)

• Can we combine BOSS with other compression schemes (LZ, grammars) to improve compression even further?


Thanks!

Encoding EdgeBWT: RL-BWT


1 $1 *$1 *$1 $1 $0 T1 $0 T0 G0 G1 $0 G0 T0 *T0 *T0 C0 G0 C0 *C0 A

0 T0 T0 G0 G0 G0 T1 *T1 *T0 C0 G0 C1 *C0 A

T T

T T

C G

G

G

T

C

G

C

A

D EdgeBWT

M

E* RL-E

succinct - ucf computer sciencecs.ucf.edu/stringbio2018/talks/talk9.pdf · 10/26/18 diego díaz...

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