Tries & Suffix Tries

Transcription

1 Tries & Suffix Tries Be Lgmed Deprme of Compuer Sciece Plese sig guesook ( o ell me riefly how you re usig he slides. For origil Keyoe files, emil me (e.lgmed@gmil.com).

2 Tries A rie ( ry ) is ree represeig collecio of srigs (keys): he smlles ree such h Ech edge is leled wih chrcer c Σ For give ode, mos oe child edge hs lel c, for y c Σ Ech key is spelled ou log some ph srig roo Helpful for implemeig se or mp whe he keys re srigs

3 Tries Keys: is, ierl, iere Key Vlue is 1 ierl 2 s i iere 3 e Smlles ree such h: Ech edge is leled wih chrcer c Σ For give ode, mos oe child edge hs lel c, for y c Σ Ech key is spelled ou log some ph srig roo 1 l r 2 3 e

4 Tries How do we check wheher ifer is i he rie? Sr roo d ry o mch successive chrcers of ifer o edges i rie s i oops, o f e r 1 l e 2 3

5 Tries Mchig ieresig i s e r oops, o e 1 l e 2 3

6 Tries Mchig is i s e No vlue ssocied wih ode, so is ws key r 1 l e 2 3

7 Tries Mchig is i s e r I s key! 1 l e 2 3

8 Tries s i Checkig for presece of key P, where P = rverses? edges If ol legh of ll keys is N, rie hs N? edges e r e 1 l 2 3

9 Tries i How o represe edges ewee ode d is childre? s Mp (from chrcers o child odes) Ide 1: Hsh le e r Ide 2: Sored liss Assumig hsh le, i s resole o sy queryig wih P, P =, is O() ime e 1 l 2 3

10 Tries Could use rie o represe k-mer idex. Mp k-mers o offses where hey occur c 4 g 8 14 cc 12 cc 2 c 6 g 18 g Idex roo: c g c g c , ,

11 Tries: implemeio hp://i.ly/cg_triemp

12 Tries: lerives Tries re he oly wy o ecode ses or mps over srigs usig ree. E.g. erry serch ree: Beley, Jo L., d Roer Sedgewick. "Fs lgorihms for sorig d serchig srigs." Proceedigs of he eighh ul ACM-SIAM symposium o Discree lgorihms. Sociey for Idusril d Applied Mhemics, 1997

13 Idexig wih suffixes We sudied idexes uil over susrigs of T Differe pproch is o idex suffixes of T. This yields surprisigly ecoomicl & prcicl d srucures: A ANA ANANA BANANA NA NANA B A N A N A A B A N A N A N A B A N A N A N A B B A N A N A N A B A N A N A N A B A Suffix Tree Suffix Arry FM Idex

14 Suffix rie Build rie coiig ll suffixes of ex T T: GTTATAGCTGATCGCGGCGTAGCGG GTTATAGCTGATCGCGGCGTAGCGG TTATAGCTGATCGCGGCGTAGCGG TATAGCTGATCGCGGCGTAGCGG ATAGCTGATCGCGGCGTAGCGG TAGCTGATCGCGGCGTAGCGG AGCTGATCGCGGCGTAGCGG GCTGATCGCGGCGTAGCGG CTGATCGCGGCGTAGCGG TGATCGCGGCGTAGCGG GATCGCGGCGTAGCGG ATCGCGGCGTAGCGG TCGCGGCGTAGCGG CGCGGCGTAGCGG GCGGCGTAGCGG CGGCGTAGCGG GGCGTAGCGG GCGTAGCGG CGTAGCGG GTAGCGG TAGCGG AGCGG GCGG CGG GG G m(m+1)/2 chrs

15 Suffix rie Firs dd specil ermil chrcer o he ed of T is chrcer h does o pper elsewhere i T, d we defie i o e less h oher chrcers ( < A < C < G < T) eforces fmilir rule: e.g. s comes efore sh i he diciory. lso gurees o suffix is prefix of y oher suffix. T: GTTATAGCTGATCGCGGCGTAGCGG GTTATAGCTGATCGCGGCGTAGCGG TTATAGCTGATCGCGGCGTAGCGG TATAGCTGATCGCGGCGTAGCGG ATAGCTGATCGCGGCGTAGCGG TAGCTGATCGCGGCGTAGCGG AGCTGATCGCGGCGTAGCGG GCTGATCGCGGCGTAGCGG CTGATCGCGGCGTAGCGG TGATCGCGGCGTAGCGG GATCGCGGCGTAGCGG ATCGCGGCGTAGCGG TCGCGGCGTAGCGG CGCGGCGTAGCGG GCGGCGTAGCGG

16 Suffix rie T: Wh s he suffix rie?

17 Suffix rie Excerp from SuffixTrie clss: hp://i.ly/cg_suffixtrie def ii (self, ): """ Mke suffix rie from """ += '' # dd ermior self.roo = {} for i i rge(le()): # for ech suffix cur = self.roo for c i [i:]: # for ech chrcer i i'h suffix if c o i cur: cur[c] = {} # dd ougoig edge if ecessry cur = cur[c] # follow he edge d coiue

18 Suffix rie T: Shores (o-empy) suffix Ech ph from roo o lef represes suffix; ech suffix is represeed y some ph from roo o lef Would his sill e he cse if we hd dded? Loges suffix

19 Suffix rie T: Ech ph from roo o lef represes suffix; ech suffix is represeed y some ph from roo o lef Would his sill e he cse if we hd dded? No

20 Suffix rie Thik of ech ode s hvig lel, spellig ou chrcers o ph from roo o ode

21 Suffix rie How do we check wheher srig S is susrig of T? Noe: Ech of T s susrigs is spelled ou log ph from he roo. Every susrig is prefix of some suffix of T. Sr he roo d follow he edges leled wih he chrcers of S If we fll off he rie -- i.e. here is o ougoig edge for ex chrcer of S, he S is o susrig of T S = Yes, i s susrig If we exhus S wihou fllig off, S is susrig of T

22 Suffix rie T = How do we check wheher srig S is susrig of T? Noe: Ech of T s susrigs is spelled ou log ph from he roo. Every susrig is prefix of some suffix of T. Sr he roo d follow he edges leled wih he chrcers of S If we fll off he rie -- i.e. here is o ougoig edge for ex chrcer of S, he S is o susrig of T If we exhus S wihou fllig off, S is susrig of T S = Yes, i s susrig

23 Suffix rie T = How do we check wheher srig S is susrig of T? Noe: Ech of T s susrigs is spelled ou log ph from he roo. Every susrig is prefix of some suffix of T. Sr he roo d follow he edges leled wih he chrcers of S If we fll off he rie -- i.e. here is o ougoig edge for ex chrcer of S, he S is o susrig of T x S = No, o susrig If we exhus S wihou fllig off, S is susrig of T

24 Suffix rie Excerp from SuffixTrie clss: hp://i.ly/cg_suffixtrie def follow_ph(self, s): """ Follow ph give y chrcers of s. Reur ode ed of ph, or Noe if we fll off. """ cur = self.roo for c i s: if c o i cur: reur Noe # o ougoig edge o ex chrcer cur = cur[c] # desced oe level reur cur def hs_susrig(self, s): """ Reur rue if s ppers s susrig of """ reur self.follow_ph(s) is o Noe

25 Suffix rie How do we check wheher srig S is suffix of T? Sme procedure s for susrig, u ddiiolly check ermil ode for child S = No suffix

26 Suffix rie How do we check wheher srig S is suffix of T? Sme procedure s for susrig, u ddiiolly check ermil ode for child S = Is suffix

27 Suffix rie Excerp from SuffixTrie clss: hp://i.ly/cg_suffixtrie def hs_suffix(self, s): """ Reur rue if s is suffix of """ ode = self.follow_ph(s) reur ode is o Noe d '' i ode

28 Suffix rie How do we cou he umer of imes srig S occurs s susrig of T? Follow ph leled wih S. If we fll off, swer is 0. If we ed up ode, swer equls # of leves i suree rooed. S = 2 occurreces Leves c e coued wih deph-firs rversl.

29 Suffix rie How do we fid he loges repeed susrig of T? Fid he deepes ode wih more h oe child

30 Suffix rie How does he suffix rie grow wih T = m?

31 Suffix rie T = How does he suffix rie grow wih T = m? Is here clss of srig where he umer of suffix rie odes grows lierly wih m? Yes: srig of m s i row ( m ) 1 Roo m odes wih "" edge o pre m + 1 odes wih icomig edge 2m + 2 odes

32 Suffix rie T = How does he suffix rie grow wih T = m? Is here clss of srig where he umer of suffix rie odes grows wih m 2? Yes: where 2 = m 1 roo odes log chi, righ odes log chi, middle chis of odes hgig off chi ( 2 ol) leves (o show) Figure & exmple y Crl Kigsford odes, where m = 2

33 Suffix rie: upper oud o size: ide 1 Recll our fucio for uild uildig suffix rie def ii (self, ): """ Mke suffix rie from """ += '' self.roo = {} for i i rge(le()): cur = self.roo for c i [i:]: if c o i cur: cur[c] = {} cur = cur[c] GTTATAGCTGATCGCGGCGTAGCGG GTTATAGCTGATCGCGGCGTAGCGG TTATAGCTGATCGCGGCGTAGCGG TATAGCTGATCGCGGCGTAGCGG Adds 1 ode, rus mos m(m+1)/2 imes ATAGCTGATCGCGGCGTAGCGG TAGCTGATCGCGGCGTAGCGG AGCTGATCGCGGCGTAGCGG GCTGATCGCGGCGTAGCGG CTGATCGCGGCGTAGCGG TGATCGCGGCGTAGCGG GATCGCGGCGTAGCGG ATCGCGGCGTAGCGG TCGCGGCGTAGCGG CGCGGCGTAGCGG GCGGCGTAGCGG CGGCGTAGCGG GGCGTAGCGG GCGTAGCGG CGTAGCGG GTAGCGG TAGCGG AGCGG GCGG CGG GG G

34 Suffix rie: cul growh Buil suffix ries for he firs 500 prefixes of he lmd phge virus geome Blck curve shows how # odes icreses wih prefix legh # suffix rie odes m(m+1)/2 cul 2m Legh prefix over which suffix rie ws uil

35 Suffix rie: cul growh Buil suffix ries for he firs 500 prefixes of he lmd phge virus geome Blck curve shows how # odes icreses wih prefix legh Acul growh much closer o wors cse h o es! # suffix rie odes m(m+1)/2 cul 2m Legh prefix over which suffix rie ws uil