Tissue quality and patient comfort 1 Prof. Jaak Ph. Janssens MD, PhD.
Lecture 14 Full Text - ut text indexing Jaak Vilo ... A suffix tree T for a string S ... • Each...
Transcript of Lecture 14 Full Text - ut text indexing Jaak Vilo ... A suffix tree T for a string S ... • Each...
Problem
• GivenPandS– findallexactorapproximateoccurrencesofPinS
• YouareallowedtopreprocessS(andP,ofcourse)
• Goal:tospeedupthesearches
E.g.Dictionaryproblem
• DoesPbelongtoadictionaryD={d1,…,dn}– BuildabinarysearchtreeofD– B-TreeofD– Hashing– Sorting+Binarysearch
• Buildakeywordtrie:searchinO(|P|)– Assumingalphabethasuptoaconstantsizec– SeeAho-Corasickalgorithm,Trieconstruction
Sortedarrayandbinarysearch
he
hershis
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indexhappy
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informal
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Sortedarrayandbinarysearch
he
hershis
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informal
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O( |P| log n )
S!=setofwords
• Soflengthn
• Howtoindex?
• Indexfromeverypositionofatext
• Prefixofeverypossiblesuffixisimportant
Suffixtree• Definition: Acompactrepresentationofatriecorrespondingtothe
suffixesofagivenstringwhereallnodeswithonechildaremergedwiththeirparents.
• Definition(suffixtree).AsuffixtreeTforastringS(withn=|S|)isarooted,labeledtreewithaleafforeachnon-emptysuffixofS.Furthermore,asuffixtreesatisfiesthefollowingproperties:
• Eachinternalnode,otherthantheroot,hasatleasttwochildren;• Eachedgeleavingaparticularnodeislabeledwithanon-emptysubstring
ofSofwhichthefirstsymbolisuniqueamongallfirstsymbolsoftheedgelabelsoftheedgesleavingthisparticularnode;
• Foranyleafinthetree,theconcatenationoftheedgelabelsonthepathfromtheroottothisleafexactlyspellsoutanon-emptysuffixofs.
• DanGusfield: AlgorithmsonStrings,Trees,andSequences:ComputerScienceandComputationalBiology.Hardcover- 534pages1stedition(January15,1997).CambridgeUnivPr(Short);ISBN:0521585198.
Literatureonsuffixtrees• http://en.wikipedia.org/wiki/Suffix_tree• DanGusfield: AlgorithmsonStrings,Trees,andSequences:Computer
ScienceandComputationalBiology.Hardcover- 534pages1stedition(January15,1997).CambridgeUnivPr(Short);ISBN:0521585198.(pages:89--208)
• E.Ukkonen.On-lineconstructionofsuffixtrees.Algorithmica,14:249-60,1995. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.751
• Ching-FungCheung,JeffreyXuYu,HongjunLu."ConstructingSuffixTreeforGigabyteSequenceswithMegabyteMemory,"IEEETransactionsonKnowledgeandDataEngineering,vol.17,no.1,pp.90-105,January,2005.http://www2.computer.org/portal/web/csdl/doi/10.1109/TKDE.2005.3
• CPMarticlesarchive:http://www.cs.ucr.edu/~stelo/cpm/
• MarkNelson.FastStringSearchingWithSuffixTreesDr.Dobb'sJournal,August,1996.http://www.dogma.net/markn/articles/suffixt/suffixt.htm
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ThesuffixtreeTree(T)ofT
• datastructuresuffixtree, Tree(T),iscompactedtrie thatrepresentsallthesuffixesofstringT
• linearsize:|Tree(T)|=O(|T|)• canbeconstructedinlineartimeO(|T|)• hasmyriadvirtues (A.Apostolico)• iswell-known:366000Googlehits
E. Ukkonen: http://www.cs.helsinki.fi/u/ukkonen/Erice2005.ppt
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Suffix tree andsuffix array techniques forpatternanalysis instringsEskoUkkonenUniv Helsinki
Erice School30Oct 2005E. Ukkonen: http://www.cs.helsinki.fi/u/ukkonen/Erice2005.ppt
Partlybasedon:
High-throughput genome-scale sequence analysis andmapping using compressed datastructures
VeliMäkinenDepartmentofComputerScience
University ofHelsinki
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ttttttttttttttgagacggagtctcgctctgtcgcccaggctggagtgcagtggcgggatctcggctcactgcaagctccgcctcccgggttcacgccattctcctgcctcagcctcccaagtagctgggactacaggcgcccgccactacgcccggctaattttttgtatttttagtagagacggggtttcaccgttttagccgggatggtctcgatctcctgacctcgtgatccgcccgcctcggcctcccaaagtgctgggattacaggcgt
E. Ukkonen: http://www.cs.helsinki.fi/u/ukkonen/Erice2005.ppt
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Analysisofastringofsymbols
• T=hattivatti’text’• P=att’pattern’
• FindtheoccurrencesofPinT:hattivatti
• Patternsynthesis:#(t)=4#(atti)=2#(t****t)=2
E. Ukkonen: http://www.cs.helsinki.fi/u/ukkonen/Erice2005.ppt
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 16
Solution:backtrackingwithsuffixtree
...ACACATTATCACAGGCATCGGCATTAGCGATCGAGTCG.....
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Patternfinding&synthesisproblems• T=t1t2 …tn,P=p1p2 …pn ,stringsofsymbolsinfinitealphabet
• Indexingproblem:PreprocessT(buildanindexstructure)suchthattheoccurrencesofdifferentpatternsPcanbefoundfast– statictext,anygivenpatternP
• Patternsynthesisproblem:LearnfromTnewpatternsthatoccursurprisinglyoften
• Whatisapattern?Exactsubstring,approximatesubstring,withgeneralizedsymbols,withgaps,…
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1. Suffix tree
2. Suffix array
3. Some applications
4. Finding motifs
E. Ukkonen: http://www.cs.helsinki.fi/u/ukkonen/Erice2005.ppt
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ThesuffixtreeTree(T)ofT
• datastructuresuffixtree, Tree(T),iscompactedtrie thatrepresentsallthesuffixesofstringT
• linearsize:|Tree(T)|=O(|T|)• canbeconstructedinlineartimeO(|T|)• hasmyriadvirtues (A.Apostolico)• iswell-known:366000Googlehits
E. Ukkonen: http://www.cs.helsinki.fi/u/ukkonen/Erice2005.ppt
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Suffixtrieandsuffixtree
a
b
b
aaa
aa
b
b
b
abaabbaabaababb
Trie(abaab)
E. Ukkonen: http://www.cs.helsinki.fi/u/ukkonen/Erice2005.ppt
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Trie(T)canbelarge
• |Trie(T)|=O(|T|2)• badexample:T=anbn
• Trie(T)canbeseenasaDFA:languageaccepted=thesuffixesofT
• minimizetheDFA=>directedcyclicwordgraph(’DAWG’)
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Tree(T)isoflinearsize
• onlytheinternalbranchingnodesandtheleavesrepresentedexplicitly
• edgeslabeledbysubstringsofT• v=node(α)ifthepathfromroottovspellsα• one-to-onecorrespondenceofleavesandsuffixes
• |T|leaves,hence<|T|internalnodes• |Tree(T)|=O(|T|+size(edgelabels))
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Tree(hattivatti)hattivatti
attivattittivatti
tivattiivatti
vattiatti
ttiti
i
hattivattiattivatti ttivatti
tivatti
ivatti
vatti
vattivatti
attiti
i
i
tti
ti
t
i
vatti
vatti
vatti
hattivatti
atti
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Tree(hattivatti)hattivatti
attivattittivatti
tivattiivatti
vattiatti
ttiti
i
hattivattiattivatti ttivatti
tivatti
ivatti
vatti
vattivatti
attiti
i
i
tti
ti
t
i
vatti
vatti
vatti
hattivatti
hattivatti
atti
substring labels of edges represented as pairs of pointers
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Tree(hattivatti)hattivatti
attivattittivatti
tivattiivatti
vattiatti
ttiti
i1 2 3
4
5
6
6,106,10
2,54,5
i
10
8
9
3,3
i
vatti
vatti
vatti
hattivatti
hattivatti
7
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Tree(T)isfull textindexTree(T)
P
31 8
P occurs in T at locations 8, 31, …
P occurs in T ó P is a prefix of some suffix of T ó Path for P exists in Tree(T)
All occurrences of P in time O(|P| + #occ)
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Findatt fromTree(hattivatti)hattivatti
attivattittivatti
tivattiivatti
vattiatti
ttiti
i
hattivattiattivatti ttivatti
tivatti
ivatti
vatti
vattivatti
attiti
2
i
tti
ti
t
i
vatti
vatti
vatti
hattivatti
atti7
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LineartimeconstructionofTree(T)
hattivatti
attivattittivatti
tivattiivatti
vattiatti
ttiti
i
Weiner (1973),
’algorithm of the year’
McCreight (1976)
’on-line’ algorithm (Ukkonen 1992)
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On-lineconstructionofTrie(T)
• T=t1t2 …tn$• Pi =t1t2 …ti i:th prefix ofT• on-lineidea:updateTrie(Pi) toTrie(Pi+1)• =>verysimpleconstruction
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Trie(abaab)
a a
b
b a
b
b
aa
Trie(a) Trie(ab) Trie(aba)
chain of links connects the end points of current suffixes
abaabaaaaεaε
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Trie(abaab)
a a
b
b a
b
b
aa
a
b
b
aaa
aa
Trie(abaa)
Add next symbol = b
From here on b-arc already exists
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WhathappensinTrie(Pi) =>Trie(Pi+1) ?
ai
ai
ai
ai
ai
ai
Before
After
New nodes
New suffix links
From here on the ai-arc exists already => stop updating here
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WhathappensinTrie(Pi) =>Trie(Pi+1) ?
• time:O(sizeofTrie(T))• suffixlinks:
slink(node(aα))=node(α)
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On-lineprocedureforsuffixtrie
1. Create Trie(t1): nodes root and v, an arc son(root, t1) = v, and suffix links slink(v) := root and slink(root) := root
2. for i := 2 to n do begin3. vi-1 := leaf of Trie(t1…ti-1) for string t1…ti-1 (i.e., the deepest leaf)
4. v := vi-1; v´ := 0
5. while node v has no outgoing arc for ti do begin
6. Create a new node v´´ and an arc son(v,ti) = v´´
7. if v´ ≠ 0 then slink(v) := v´´
8. v := slink(v); v´ := v´´ end9. for the node v´´ such that v´´= son(v,ti) do
if v´´ = v´ then slink(v’) := root else slink(v´) := v´´
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Suffixtreeson-line
• ’compactedversion’oftheon-linetrieconstruction:simulatetheconstructiononthelinearsizetreeinsteadofthetrie=>timeO(|T|)
• alltrienodesareconceptuallystillneeded=>implicit andreal nodes
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Implicitandrealnodes
• Pair(v,α)isanimplicitnode inTree(T)ifvisanodeofTreeandα isa(proper)prefixofthelabelofsomearcfrom v.Ifα istheemptystringthen (v,α)isa ’real’ node(=v).
• Let v=node(α´)in Tree(T). Then implicitnode(v,α)representsnode(α´α)ofTrie(T)
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Suffixlinksandopenarcs
…
v
aα
…
α
root
slink(v)
label [i,*] instead of [i,j] if w is a leaf and j is the scanned position of T
w
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Bigpicture
… … …
…
…
suffix link path traversed: total work O(n)
new arcs and nodes created: total work O(size(Tree(T))
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On-lineprocedureforsuffixtree
Input: string T = t1t2 … tn$
Output: Tree(T)
Notation: son(v,α) = w iff there is an arc from v to w with label α
son(v,ε) = v
Function Canonize(v, α):
while son(v, α´) ≠ 0 where α = α´ α´´, | α´| > 0 do
v := son(v, α´); α := α´´
return (v, α)
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Suffix-treeon-line:mainprocedure
Create Tree(t1); slink(root) := root(v, α) := (root, ε) /* (v, α) is the start node */for i := 2 to n+1 do
v´ := 0while there is no arc from v with label prefix αti do
if α ≠ ε then /* divide the arc w = son(v, αη) into two */son(v, α) := v´´; son(v´´,ti) := v´´´; son(v´´,η) := w
elseson(v,ti) := v´´´; v´´ := v
if v´ ≠ 0 then slink(v´) := v´´v´ := v´´; v := slink(v); (v, α) := Canonize(v, α)
if v´ ≠ 0 then slink(v´) := v(v, α) := Canonize(v, αti) /* (v, α) = start node of the next round */
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Theactualtimeandspace• |Tree(T)|isabout20|T|inpractice• brute-forceconstructionisO(|T|log|T|)forrandomstringsastheaveragedepthofinternalnodesisO(log|T|)
• differencebetweenlinearandbrute-forceconstructionsnotnecessarilylarge(Giegerich&Kurtz)
• truncatedsuffixtrees:ksymbolslongprefixofeachsuffixrepresented(Naetal.2003)
• alphabetindependentlineartime(Farach1997)
ApplicationsofSuffixTrees• DanGusfield: AlgorithmsonStrings,Trees,andSequences:
ComputerScienceandComputationalBiology.Hardcover-534pages1stedition(January15,1997).CambridgeUnivPr(Short);ISBN:0521585198.- book
• APL1:ExactStringMatchingSearchforPfromtextS.Solution1:buildSTree(S)- oneachievesthesameO(n+m)asKnuth-Morris-Pratt,forexample!
• SearchfromthesuffixtreeisO(|P|)• APL2:ExactsetmatchingSearchforasetofpatternsP
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 55
C
Backtobacktracking
AC
T
4 2 1 5 36
CT
TAT
TAC
T
AT C
T
A
ACA, 1 mismatch
Same idea can be used to many otherforms of approximate search, like Smith-Waterman, position-restricted scoringmatrices, regular expression search, etc.
ApplicationsofSuffixTrees
• APL3:substringproblemforadatabaseofpatternsGivenasetofstringsS=S1,...,Sn--- adatabaseFindallSithathavePasasubstring
• GeneralizedsuffixtreecontainsallsuffixesofallSi• QueryintimeO(|P|),andcanidentifytheLONGESTcommonprefixofPinallSi
ApplicationsofSuffixTrees
• APL4:Longestcommonsubstringoftwostrings• FindthelongestcommonsubstringofSandT.• OveralltherearepotentiallyO(n2 )suchsubstrings,ifnisthelengthofashorterofSandT
• Donald Knuthonce(1970)conjectured thatlinear-timealgorithmisimpossible.
• Solution:constructtheSTree(S+T)andfindthenodedeepestinthetreethathassuffixesfrombothSandTinsubtreeleaves.
• Ex:S=superiorcalifornialives T=sealiver havebothasubstringalive.
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 58
Simpleanalysistask:LCSS
• LetLCSSA(A,B) denotethelongestcommonsubstringtwosequencesA andB. E.g.:– LCSS(AGATCTATCT,CGCCTCTATG)=TCTAT.
• Agoodsolutionistobuildsuffixtreefortheshortersequenceandmakeadescendingsuffixwalk withtheothersequence.
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 60
Descendingsuffixwalk
suffix tree of A Read B left-to-right,always going down in thetree when possible.If the next symbol of B doesnot match any edge labelon current position, takesuffix link, and try again.(Suffix link in the root to itself emits a symbol).The node v encountered with largest string depthis the solution.v
ApplicationsofSuffixTrees
• APL5:RecognizingDNAcontaminationRelatedtoDNAsequencing,searchforlongeststrings(longerthanthreshold)thatarepresentintheDBofsequencesofothergenomes.
• APL6: CommonsubstringsofmorethantwostringsGeneralizationofAPL4,canbedoneinlinear(intotallengthofallstrings)time
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 62
Anothercommontool:Generalizedsuffixtree
ACCTTA....ACCT#CACATT..CAT#TGTCGT...GTA#TCACCACC...C$
AC
C
node info:subtree size 47813871sequence count 87
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 63
Generalizedsuffixtreeapplication
...ACC..#...ACC...#...ACC...ACC..ACC..#..ACC..ACC...#...ACC...#...
...#....#...#...#...ACC...#...#...#...#...#...#..#..ACC..ACC...#......#...
AC
C
node info:subtree size 4398blue sequences 12/15red sequences 2/62
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 64
Casestudycontinued
genome
regions with ChIP-seq matches
suffix tree of genome
5 blue1 red
TAC..........T
motif?
ApplicationsofSuffixTrees
• APL7:Buildingadirectedgraphforexactmatching: Suffixgraph - directedacyclicwordgraph(DAWG),asmallestfinitestateautomaton recognizingallsuffixesofastringS.Thisautomatoncanrecognize membership,butnottellwhichsuffixwasmatched.
• Construction:mergeisomorficsubtrees.• IsomorficinSuffixTreewhenexistssuffixlinkpath,andsubtreeshaveequalnr.ofleaves.
ApplicationsofSuffixTrees
• APL8:Areverseroleforsuffixtrees,andmajorspacereductionIndexthepattern,nottree...
• Matchingstatistics.• APL10:All-pairssuffix-prefixmatchingForallpairsSi, Sj, findthelongestmatchingsuffix-prefixpair.Usedinshortestcommonsuperstringgeneration(e.g.DNAsequenceassembly),ESTalignmentmetc.
ApplicationsofSuffixTrees
• APL11:Findingallmaximalrepetitivestructuresinlineartime
• APL12:Circularstringlinearizatione.g.circularchemicalmoleculesinthedatabase,onewantstolienarizetheminacanonicalway...
• APL13:Suffixarrays- morespacereductionwilltouchthatseparately
ApplicationsofSuffixTrees
• APL14:Suffixtreesingenome-scaleprojects• APL15:ABoyer-Mooreapproachtoexactsetmatching
• APL16:Ziv-Lempeldatacompression• APL17:MinimumlengthencodingofDNA
ApplicationsofSuffixTrees• AdditionalapplicationsMostlyexercises...• Extrafeature:CONSTANTtimelowestcommonancestorretrieval(LCA)
Andmestruktuurmisvõimaldableidakonstantseajagaalumistühistvanemat(seevastabpikimaleühiseleprefixile!)onvõimalikkoostadalineaarseajaga.
• APL:Longestcommonextension:abridgetoinexactmatching• APL:Findingallmaximalpalindromesinlineartime
Palindromereadsfromcentralpositionthesametoleftandright.E.g.:kirik,saippuakivikauppias.
• BuildthesuffixtreeofSandinvertedS(aabcbad=>aabcbad#dabcbaa)andusingtheLCAonecanaskforanypositionpair(i,2i-1),thelongestcommonprefixinconstanttime.
• ThewholeproblemcanbesolvedinO(n).
ApplicationsofSuffixTrees
• APL:Exactmatchingwithwildcards• APL:Thek-mismatchproblem• Approximatepalindromesandrepeats• Fastermethodsfortandemrepeats• Alinear-timesolutiontothemultiplecommonsubstringproblem
• Andmany-manymore...
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 71
Propertiesofsuffixtree
• Suffixtreehasn leavesandatmostn-1internalnodes,wheren isthetotallengthofallsequencesindexed.
• Eachnoderequiresconstantnumberofintegers(pointerstofirstchild,sibling,parent,textrangeofincomingedge,statisticscounters,etc.).
• Canbeconstructedinlineartime.
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 72
Propertiesofsuffixtree...inpractice
• Hugeoverheadduetopointerstructure:– Standardimplementationofsuffixtreeforhumangenomerequiresover200GB memory!
– Acarefulimplementation(usinglogn -bitfieldsforeachvalueandarraylayoutforthetree)stillrequiresover40GB.
– Humangenomeitselftakeslessthan1GB using2-bitsperbp.
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Suffixes- sorted
• Sortallsuffixes.Allowstoperformbinarysearch!
hattivattiattivattittivattitivattiivattivattiattittitiiε
εattiattivattihattivattiiivattititivattittittivattivatti
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Suffixarray:example
• suffixarray=lexicographicorderofthesuffixes
hattivattiattivattittivattitivattiivattivattiattittitiiε
εattiattivattihattivattiiivattititivattittittivattivatti
1172110594836
1234567891011
1172110594836
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Suffixarrayconstruction:sort!
• suffixarray=lexicographicorderofthesuffixes
hattivattiattivattittivattitivattiivattivattiattittitiiε
1172110594836
1234567891011
1172110594836
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Suffixarray
• suffixarray SA(T)=anarraygivingthelexicographicorderofthesuffixesofT
• spacerequirement:5|T|• practitionerslikesuffixarrays(simplicity,spaceefficiency)
• theoreticianslikesuffixtrees(explicitstructure)
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 78
Reducingspace:suffixarray
AC
T
4 2 1 5 36
C A T A C T1 2 3 4 5 6
=[3,3]=[3,3]=[2,2]
suffix array
=[4,6]=[6,6]=[2,6]
=[3,6]=[5,6]
CC
TTA
T
TAC
T CT
A
T
A
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 79
Suffixarray
• Manyalgorithmsonsuffixtreecanbesimulatedusingsuffixarray...– ...andcoupleofadditionalarrays...– ...formingso-calledenhancedsuffixarray...– ...leadingtothesimilarspacerequirementascarefulimplementationofsuffixtree
• Notasatisfactorysolutiontothespaceissue.
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Patternsearchfromsuffixarrayhattivattiattivattittivattitivattiivattivattiattittitiiε
εattiattivattihattivattiiivattititivattittittivattivatti
1172110594836
att binary search
ISMB2009Tutorial VeliMäkinen:"...analysisandmapping..." 81
Whatwelearntoday?
• Welearnthatitispossibletoreplacesuffixtreeswithcompressedsuffixtrees thattake8.8GB forthehumangenome.
• Welearnthatbacktracking canbedoneusingcompressedsuffixarrays requiringonly2.1GBforthehumangenome.
• Welearnthatdiscovering interestingmotifseedsfromthehumangenometakes40hoursandrequires9.3GB space.
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Recentsuffixarrayconstructions
• Manber&Myers(1990):O(|T|log|T|)• lineartimeviasuffixtree• January/June2003:directlineartimeconstructionofsuffixarray- Kim,Sim,Park,Park(CPM03)- Kärkkäinen&Sanders(ICALP03)- Ko&Aluru(CPM03)
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Kärkkäinen-Sandersalgorithm
1.Construct the suffix array of the suffixes starting at positions i mod 3 ≠ 0. This is done by reduction to the suffix array construction of a string of two thirds the length, which is solved recursively.
2.Construct the suffix array of the remaining suffixes using the result of the first step.
3.Merge the two suffix arrays into one.
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Notation
• stringT=T[0,n)=t0t1 …tn-1• suffixSi =T[i,0)=titi+1 …tn-1• forC\subset[0,n]:SC ={Si|iinC}
• suffixarray SA[0,n]ofTisapermutationof[0,n]satisfyingSSA[0] <SSA[1] <…<SSA[n]
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Runningexample
• T[0,n)=yabbadabbado00…
• SA=(12,1,6,4,9,3,8,2,7,5,10,11,0)
0 1 2 3 4 5 6 7 8 9 10 11
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Step0:Constructasample
• fork=0,1,2Bk={iє [0,n]|imod3=k}
• C=B1UB2samplepositions• SC samplesuffixes
• Example:B1={1,4,7,10},B2={2,5,8,11},C={1,4,7,10,2,5,8,11}
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Step1:Sortsamplesuffixes• fork=1,2,construct
Rk=[tktk+1tk+2][tk+3tk+4tk+5]…[tmaxBktmaxBk+1tmaxBk+2]
R=R1^R2concatenationofR1andR2
SuffixesofRcorrespondtoSC:suffix[titi+1ti+2]…correspondstoSi;correspondenceisorderpreserving.
SortthesuffixesofR:radixsortthecharactersandrenamewithrankstoobtainR´.Ifallcharactersdifferent,theirorderdirectlygivestheorderofsuffixes.Otherwise,sortthesuffixesofR´ usingKärkkäinen-Sanders.Note:|R´|=2n/3.
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Step1(cont.)
• oncethesamplesuffixesaresorted,assignaranktoeach:rank(Si)=therankofSiinSC;rank(Sn+1)=rank(Sn+2)=0
• Example:R=[abb][ada][bba][do0][bba][dab][bad][o00]R´ =(1,2,4,6,4,5,3,7)SAR´ =(8,0,1,6,4,2,5,3,7)rank(Si)- 14- 26- 53– 78– 00
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Step2:Sortnonsamplesuffixes
• foreachnon-sampleSi є SB0 (notethatrank(Si+1)isalwaysdefinedforiє B0):
Si ≤Sj ↔(ti,rank(Si+1))≤(tj,rank(Sj+1))• radixsortthepairs(ti,rank(Si+1)).
• Example:S12 <S6 <S9 <S3 <S0because(0,0)<(a,5)<(a,7)<(b,2)<(y,1)
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Step3:Merge• mergethetwosortedsetsofsuffixesusingastandardcomparison-basedmerging:
• tocompareSi є SC withSj є SB0,distinguishtwocases:
• iє B1:Si ≤Sj ↔(ti,rank(Si+1))≤(tj,rank(Sj+1))• iє B2:Si ≤Sj ↔(ti,ti+1,rank(Si+2))≤(tj,tj+1,rank(Sj+2))
• notethattheranksaredefinedinallcases!• S1 <S6 as(a,4)<(a,5)andS3 <S8 as(b,a,6)<(b,a,7)
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RunningtimeO(n)
• excludingtherecursivecall,everythingcanbedoneinlineartime
• therecursionisonastringoflength2n/3• thusthetimeisgivenbyrecurrence
T(n)=T(2n/3)+O(n)• henceT(n)=O(n)
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LCPtable
• LongestCommonPrefixofsuccessiveelementsofsuffixarray:
• LCP[i]=lengthofthelongestcommonprefixofsuffixesSSA[i] andSSA[i+1]
• buildinversearraySA-1 fromSAinlineartime• thenLCPtablefromSA-1 inlineartime(Kasaietal,CPM2001)
• OxfordEnglishDisctionary http://www.oed.com/• Example- WordoftheDay,Fourth
http://biit.cs.ut.ee/~vilo/edu/2005-06/Text_Algorithms/L7_SuffixTrees/wotd_fourth.htmlhttp://www.oed.com/cgi/display/wotd
• PATindex- byGastonGonnet(taonsamutiMapletarkvaraüksloojatestninghiljemmolekulaarbioloogiatarkvarapaketiväljatöötajaid)
• PATindexisessentiallyasuffixarray.Tosavespace,indexedonlyfromfirstcharacterofeveryword
• XML-tagging(orSGML,atthattime!)alsoindexed• TomarkcertainfieldsofXML,thebitvectorswereused.• Mainconcern- improvethespeedofsearchontheCD- minimizerandom
accesses.• Forslowmediumeven15-20accessesistooslow...• G.H.Gonnet,R.A.Baeza-Yates,andT.Snider,Lexicographicalindicesfor
text:Invertedfilesvs.PATtrees,TechnicalReportOED-91-01,CentrefortheNewOED,UniversityofWaterloo,1991.
97
SubstringmotifsofstringT
• stringT =t1 …tninalphabetA.• Problem:whatarethefrequentlyoccurring(ungapped)substringsofT?Longestsubstringthatoccursatleastq times?
• Thm:SuffixtreeTree(T) givescompleteoccurrencecountsofallsubstringmotifsofTinO(n) time(althoughT mayhaveO(n2)substrings!)
98
Countingthesubstringmotifs
• internalnodesofTree(T)↔repeatingsubstringsofT
• numberofleavesofthesubtreeofanodeforstringP=numberofoccurrencesofPinT
99
Substringmotifsofhattivatti
hattivattiattivatti ttivatti
tivatti
ivatti
vatti
vattivatti
attiti
i
i
tti
ti
t
i
vatti
vatti
vatti
hattivatti
atti
2
2 2
24
Counts for the O(n) maximal motifs shown
100
FindingrepeatsinDNA
• humanchromosome3• thefirst48999930bases• 31mincputime(8processors,4GB)
• Humangenome:3x109 bases• Tree(HumanGenome)feasible
101
Longestrepeat?
Occurrences at: 28395980, 28401554r Length: 2559
ttagggtacatgtgcacaacgtgcaggtttgttacatatgtatacacgtgccatgatggtgtgctgcacccattaactcgtcatttagcgttaggtatatctccgaatgctatccctcccccctccccccaccccacaacagtccccggtgtgtgatgttccccttcctgtgtccatgtgttctcattgttcaattcccacctatgagtgagaacatgcggtgtttggttttttgtccttgcgaaagtttgctgagaatgatggtttccagcttcatccatatccctacaaaggacatgaactcatcatttttttatggctgcatagtattccatggtgtatatgtgccacattttcttaacccagtctacccttgttggacatctgggttggttccaagtctttgctattgtgaatagtgccgcaataaacatacgtgtgcatgtgtctttatagcagcatgatttataatcctttgggtatatacccagtaatgggatggctgggtcaaatggtatttctagttctagatccctgaggaatcaccacactgacttccacaatggttgaactagtttacagtcccagcaacagttcctatttctccacatcctctccagcacctgttgtttcctgactttttaatgatcgccattctaactggtgtgagatggtatctcattgtggttttgatttgcatttctctgatggccagtgatgatgagcattttttcatgtgttttttggctgcataaatgtcttcttttgagaagtgtctgttcatatccttcgcccacttttgatggggttgtttgtttttttcttgtaaatttgttggagttcattgtagattctgggtattagccctttgtcagatgagtaggttgcaaaaattttctcccattctgtaggttgcctgttcactctgatggtggtttcttctgctgtgcagaagctctttagtttaattagatcccatttgtcaattttggcttttgttgccatagcttttggtgttttagacatgaagtccttgcccatgcctatgtcctgaatggtattgcctaggttttcttctagggtttttatggttttaggtctaacatgtaagtctttaatccatcttgaattaattataaggtgtatattataaggtgtaattataaggtgtataattatatattaattataaggtgtatattaattataaggtgtaaggaagggatccagtttcagctttctacatatggctagccagttttccctgcaccatttattaaatagggaatcctttccccattgcttgtttttgtcaggtttgtcaaagatcagatagttgtagatatgcggcattatttctgagggctctgttctgttccattggtctatatctctgttttggtaccagtaccatgctgttttggttactgtagccttgtagtatagtttgaagtcaggtagcgtgatggttccagctttgttcttttggcttaggattgacttggcaatgtgggctcttttttggttccatatgaactttaaagtagttttttccaattctgtgaagaaattcattggtagcttgatggggatggcattgaatctataaattaccctgggcagtatggccattttcacaatattgaatcttcctacccatgagcgtgtactgttcttccatttgtttgtatcctcttttatttcattgagcagtggtttgtagttctccttgaagaggtccttcacatcccttgtaagttggattcctaggtattttattctctttgaagcaattgtgaatgggagttcactcatgatttgactctctgtttgtctgttattggtgtataagaatgcttgtgatttttgcacattgattttgtatcctgagactttgctgaagttgcttatcagcttaaggagattttgggctgagacgatggggttttctagatatacaatcatgtcatctgcaaacagggacaatttgacttcctcttttcctaattgaatacccgttatttccctctcctgcctgattgccctggccagaacttccaacactatgttgaataggagtggtgagagagggcatccctgtcttgtgccagttttcaaagggaatgcttccagtttttgtccattcagtatgatattggctgtgggtttgtcatagatagctcttattattttgagatacatcccatcaatacctaatttattgagagtttttagcatgaagagttcttgaattttgtcaaaggccttttctgcatcttttgagataatcatgtggtttctgtctttggttctgtttatatgctggagtacgtttattgattttcgtatgttgaaccagccttgcatcccagggatgaagcccacttgatcatggtggataagctttttgatgtgctgctggattcggtttgccagtattttattgaggatttctgcatcgatgttcatcaaggatattggtctaaaattctctttttttgttgtgtctctgtcaggctttggtatcaggatgatgctggcctcataaaatgagttagg
102
Tenoccurrences?
ttttttttttttttgagacggagtctcgctctgtcgcccaggctggagtgcagtggcgggatctcggctcactgcaagctccgcctcccgggttcacgccattctcctgcctcagcctcccaagtagctgggactacaggcgcccgccactacgcccggctaattttttgtatttttagtagagacggggtttcaccgttttagccgggatggtctcgatctcctgacctcgtgatccgcccgcctcggcctcccaaagtgctgggattacaggcgt
Length: 277
Occurrences at: 10130003, 11421803, 18695837, 26652515, 42971130, 47398125In the reversed complement at: 17858493, 41463059, 42431718, 42580925
103
Usingsuffixtrees:plagiarism
• findlongestcommonsubstringofstringsXandY
• buildTree(X$Y)andfindthedeepestnodewhichhasaleafpointingtoXandanotherpointingtoY
104
Usingsuffixtrees:approximatematching
• editdistance:insertions,deletions,changes
• STOCKHOLMvsTUKHOLMA
105
Stringdistance/similarityfunctions
STOCKHOLM vs TUKHOLMA
STOCKHOLM__TU_ KHOLMA
=> 2 deletions, 1 insertion, 1 change
106
Dynamicprogrammingdi,j = min(if ai=bj then di-1,j-1 else ¥,
di-1,j + 1, di,j-1 + 1)
= distance between i-prefix of A and j-prefix of B(substitution excluded)
di,j
di-1,j-1
di,j-1
di-1,j
dm,n
mxn table d
A
B
ai
bj
+1
+1
107
A\B s t o c k h o l m0 1 2 3 4 5 6 7 8 9
t 1 2 1 2 3 4 5 6 7 8u 2 3 2 3 4 5 6 7 8 9k 3 4 3 4 5 4 5 6 7 8h 4 5 4 5 6 5 4 5 6 7o 5 6 5 4 5 6 5 4 5 6l 6 7 6 5 6 7 6 5 4 5m 7 8 7 6 7 8 7 6 5 4a 8 9 8 7 8 9 8 7 6 5
di,j = min(if ai=bj then di-1,j-1 else ¥, di-1,j + 1, di,j-1 + 1)
dID(A,B)optimal alignment by trace-back
108
Searchproblem
• findapproximateoccurrencesofpatternPintextT:substringsP’ofTsuchthatd(P,P’)small
• dynprogrwithsmallmodification:O(mn)• lotsof(practical)improvementtricks
P
T P’
Burrows-Wheeler• SeeFAQ http://www.faqs.org/faqs/compression-faq/part2/section-9.html• Themethoddescribedintheoriginalpaperisreallyacompositeofthreedifferent
algorithms:– theblocksortingmainengine(alossless,veryslightlyexpansivepreprocessor),– themove-to-frontcoder(abyte-for-bytesimple,fast,locallyadaptivenoncompressivecoder)and– asimplestatisticalcompressor(firstorderHuffmanismentionedasacandidate)eventuallydoing
thecompression.
• Ofthesethreemethodsonlythefirsttwoarediscussedhereastheyarewhatconstitutestheheartofthealgorithm.Thesetwoalgorithmscombinedformacompletelyreversible(lossless)transformationthat- withtypicalinput- skewsthefirstordersymboldistributionstomakethedatamorecompressiblewithsimplemethods.Intuitivelyspeaking,themethodtransformsslackinthehigherorderprobabilitiesoftheinputblock(thusmakingthemmoreeven,whiteningthem)toslackinthelowerorderstatistics.Thiseffectiswhatisseeninthehistogramoftheresultingsymboldata.
• Please,readthearticlebyMarkNelson:• DataCompressionwiththeBurrows-WheelerTransformMarkNelson,Dr.Dobb'sJournal
September,1996.http://marknelson.us/1996/09/01/bwt/
CODE:t: hat acts like this:<13><10><1t: hat buffer to the constructort: hat corrupted the heap, or woW: hat goes up must come down<13t: hat happens, it isn't likelyw: hat if you want to dynamicallt: hat indicates an error.<13><1t: hat it removes arguments fromt: hat looks like this:<13><10><t: hat looks something like thist: hat looks something like thist: hat once I detect the mangled