From Indexing Data Structures to de Bruijn Graphs
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1 From Indexing Dt Structures to de Bruijn Grphs Bstien Czux, Thierry Lecroq, Eric Rivls LIRMM & IBC, Montpellier - LITIS Rouen June 1, 201 Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
2 Introduction Motivtion De Bruijn Grph is lrgely used in de novo ssemly. [Pevzner et l., 2001] One uilds suffix tree efore the ssemly for some pplictions, for instnce for the error correction. [Slmel, 2010] There exist lgorithms to uild directly the De Bruijn Grph. [Onoder et l., 2013] [Rødlnd, 2013] None is le to uild the Contrcted De Bruijn Grph directly. Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
3 Introduction Indexing dt structures Numerous dt structures: suffix tree, ffix tree, suffix rry, etc. to index one or severl texts (generlized index) functionlly equivlent to compressed versions (CSA, FM-index, CST, etc) Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
4 Introduction Reltion etween indexing structures nd ssemly grphs Generlised Suffix Tree (GST) indexes ll fctors of set of texts is functionlly equivlent to other indexes (SA, CSA, FM-index) Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, 201 / 3
5 Introduction Reltion etween indexing structures nd ssemly grphs Generlised Suffix Tree (GST) indexes ll fctors of set of texts is functionlly equivlent to other indexes (SA, CSA, FM-index) Question: How to directly uild, e.g., the ssemly De Bruijn grph? in clssicl or contrcted form? Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, 201 / 3
6 Generlized Suffix Tree (GST) 1 Generlized Suffix Tree (GST) 2 De Bruijn Grphs 3 From the Generlized Suffix Tree to the DBG Contrcted De Bruijn Grph Conclusion nd Future works Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, 201 / 3
7 Generlized Suffix Tree (GST) Generlized Suffix Tree (GST) Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, 201 / 3
8 Generlized Suffix Tree (GST) Exmple S = {c, cc, c, cc, cc} S = s i S s i S = = 31 Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
9 Generlized Suffix Tree (GST) Suffix Tree c$ $ c 6 $ 1$c c$ 2$c 3 c$ 7 c $ c 6 c 2c 3 c c Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
10 Generlized Suffix Tree (GST) $ Feture 1 tke in input set of words 2 no use n end mrking specil symol i.e. no hypothesis requiring tht suffix is not the prefix of nother suffix Theorem The GST of set of words S tkes liner spce in S. Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
11 3 Generlized Suffix Tree (GST) 2 Generlized Suffix Tree (GST) c c c 6 6 1c c c 3 3c 1c 2c 3 2c 2c c 1c S = {c, cc, c, cc, cc} Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
12 De Bruijn Grphs De Bruijn Grphs Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
13 De Bruijn Grphs De Bruijn Grph The ssemly De Bruijn Grph (DBG + k ) Let k e positive integer stisfying k 2 nd S = {s 1,..., s n } e set of n words. The De Bruijn grph of order k of S, denoted y DBG + k tht the nodes re the k-mers of S nd, is grph such n rc links two k-mers of S if there exists n integer i such tht these k-mers strt t successive positions in s i. Remrk: the rc definition implies tht the two k-mers overlp y k 1 positions. Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
14 De Bruijn Grphs De Bruijn Grph for ssemly c c c c S = {c, cc, c, cc, cc} Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
15 From the Generlized Suffix Tree to the DBG From the Generlized Suffix Tree to the DBG Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
16 From the Generlized Suffix Tree to the DBG One defines 3 specific types of nodes in the GST (in V S ): u < k v k v initil node v = k v initil exct node v = k 1 v suinitil node Some properties Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
17 From the Generlized Suffix Tree to the DBG 2 Generlized Suffix Tree (GST) c c c 6 6 1c c c 3 3c 1c 2c 3 2c 2c S = {c, cc, c, cc, cc} nd k = 2 1c 1c Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
18 From the Generlized Suffix Tree to the DBG 2 Generlized Suffix Tree (GST) c c c 3 c initil exct node c 6 6 c 1c 1c suinitil node initil node 2c 3 2c 2c S = {c, cc, c, cc, cc} nd k = 2 1c 1c Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
19 From the Generlized Suffix Tree to the DBG Nodes of the de Bruijn Grph Nottion: Init S Let Init S denote the set of initil nodes of the GST of S. Property: node correspondence The set of k-mers of DBG + k of S is isomorphic to Init S. Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
20 From the Generlized Suffix Tree to the DBG Arcs of the de Bruijn Grph Ide 1 Tke n initil node v 2 follow its suffix link to node z (lose the first letter of its k-mer) 3 if needed, go the children of z to find its extensions check whether the extensions re vlid Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
21 From the Generlized Suffix Tree to the DBG Let v e n initil node, u its fther, nd z the node pointed t y the suffix link of v. Property 2 Let v e node of suffix tree. If it exists, the suffix link of v elongs to the su-tree of the suffix link of pr(v). u SL(u) z = SL(v) v Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
22 From the Generlized Suffix Tree to the DBG Arcs of DBG + k : cse figure v strting initil node: green z := SL(v) node pointed t y its suffix link: ornge lck stright rrows : rcs of ST dotted rrows: suffix links colored plin rrows: creted rcs of the DBG + k Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
23 From the Generlized Suffix Tree to the DBG Arcs of DBG + k : Type 1 v is initil not exct & z is initil 6 6 c c c 1c 2 Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
24 From the Generlized Suffix Tree to the DBG Arcs of DBG + k : Type v is initil not exct & z is not initil i.e. z is deeper thn word depth k c 2c 2c 2 c 1 1 Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
25 From the Generlized Suffix Tree to the DBG 3 Arcs of DBG + k : Type 3 c 7 7 c v is initil exct with single child c c 1c Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
26 From the Generlized Suffix Tree to the DBG 3 Arcs of DBG + k : Type 7 7 c v is initil exct with severl children 3c 3 3c 2c 2c c Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
27 From the Generlized Suffix Tree to the DBG DBG construction Theorem Given the GST of set of words S. The construction of the De Bruijn Grph tkes liner time in S. Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
28 From the Generlized Suffix Tree to the DBG DBG construction Theorem Given the GST of set of words S. The construction of the De Bruijn Grph tkes liner time in S. Proof All four cses of the typology re processed in constnt time. Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
29 Contrcted De Bruijn Grph Contrcted De Bruijn Grph Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
30 Contrcted De Bruijn Grph Exmple not contrcted contrcted c c c c c c c c c c c c S = {c, cc, c, cc, cc} Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
31 Contrcted De Bruijn Grph Left nd right extensile nodes Left or right extensile node Let v e node of the De Bruijn grph; we sy tht v is left extensile if nd only if d in (v) = 1. v is right extensile if nd only if d out (v) = 1. d in (u) = 2 d out (u) = 1 u v d in (v) = 1 d out (v) = 3 Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
32 Contrcted De Bruijn Grph Contrcted De Bruijn Grph Contrcted De Bruijn Grph Let G := (V G, E G ) e the De Bruijn grph of order k of S. We cll K := (V K, E K ) the Contrcted De Bruijn Grph (CDBG + k ) of order k of S, the grph otined from G y contrcting itertively the rcs (u, v) where u is right extensile nd v is left extensile. u v w Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
33 Contrcted De Bruijn Grph Find right extensile nodes Property : right extensile Initil nodes of type 1, 2 or 3 re right-extensile. Initil nodes of type re not right-extensile. Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
34 Contrcted De Bruijn Grph Find right extensile nodes Property : right extensile Initil nodes of type 1, 2 or 3 re right-extensile. Initil nodes of type re not right-extensile. Proof Types 1, 2 or 3: word of node v hs only one extension in S. Types nodes hve severl extensions in S. Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
35 3 Contrcted De Bruijn Grph 2 Cse of right extensile nodes ex. for k = 2 c c c 6 6 1c c c 3 3c 1c 2c 3 2c 2c c 1c Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
36 Contrcted De Bruijn Grph 2 Find left extensile nodes Let v e right extensile k-mer/node nd (v, u) n rc of DBG + k. Question: is u left extensile? Property : left extensile u is left extensile if nd only if V suff (u) V ter (u) = V suff (v) V suff (u) = {, 2,, 2, 2, 1} V ter (u) = {1} u c 6 6 c 1 V suff (1) = {1} V ter (1) = {1} 2c 2c 1 Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
37 Contrcted De Bruijn Grph Find left extensile nodes Let v e right extensile k-mer/node nd (v, u) n rc of DBG + k. Question: is u left extensile? Property : left extensile u is left extensile if nd only if V suff (u) V ter (u) = V suff (v) Ide determine how mny nodes in su-tree of u hve Suffix Links pointing to them coming from the su-tree of v do not ccount for nodes representing the first suffix of words in S which we cll terminl nodes compre the crdinlities of non terminl suffixes in u su-tree with suffixes in v su-tree. Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
38 Contrcted De Bruijn Grph v u w Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
39 Contrcted De Bruijn Grph v u w Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
40 Contrcted De Bruijn Grph v u is left extensile u w Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
41 Contrcted De Bruijn Grph v u w Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
42 Contrcted De Bruijn Grph v u is not left extensile u w Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
43 3 Contrcted De Bruijn Grph 2 Cse of left extensile nodes ex. for k = 2 c c c 6 6 1c c c 3 3c 1c 2c 3 2c 2c c 1c Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
44 Contrcted De Bruijn Grph Contrcted DBG construction Theorem Given the GST of set of words S. The construction of the Contrcted DBG + k tkes liner time in S. Ide of proof. Detection of right extensile nodes is otined from the types. To get left nd right extensile nodes in constnt time preprocess: V suff (u) nd V ter (u) for ech node u of GST, nd for ech node w of ST such tht w k pointer to its ncestrl initil node. Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
45 Conclusion nd Future works Conclusion nd Future works Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
46 Conclusion nd Future works Conclusion A liner time lgorithm tht uilds the Contrcted De Bruijn Grph from Generlized Suffix Tree or Suffix Arry Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
47 Conclusion nd Future works Future works Use only slice of the suffix tree Updte the order of the DBG: dynmiclly chnging k Go for compressed indexes insted of Suffix Tree Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
48 Conclusion nd Future works Funding nd cknowledgments Thnks for your ttention Questions? Czux, Lecroq, Rivls (LIRMM) Generlized Suffix Tree & DBG June 1, / 3
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