Introduction to Triangulated Graphstandy.cs.illinois.edu/581-2018-chordal.pdf · • Every chordal graph has at most n maximal cliques, and these can be found in polynomial time:

IntroductiontoTriangulatedGraphs

TandyWarnow

Topicsfortoday

•  Triangulatedgraphs:theoremsandalgorithms(Chapters11.3and11.9)

•  Examplesoftriangulatedgraphsinphylogenyestimation(Chapters4.8,11.3-11.5)

Triangulated(i.e.,Chordal)Graphs

•  Definition:Agraphistriangulatedifithasnosimplecyclesofsizefourormore.

DCMsareDivide-and-Conquerstrategies!

DCMsforphylogenyreconstruction

•  Defineatriangulatedgraphsothatitsverticescorrespondto

theinputtaxa(orsequences)•  Decomposethegraphintooverlappingsubgraphs,thus

decomposingthetaxaintooverlappingsubsets.•  Applythe“basemethod” toeachsubsetoftaxa,toconstruct

asubsettree•  Applyasupertreemethodtothesubsettreestoobtaina

singletreeonthefullsetoftaxa.

DCMs(Disk-CoveringMethods)

•  DCMsforpolynomialtimemethodsimprovetopologicalaccuracy(empiricalobservation)andhaveprovabletheoreticalguaranteesunderMarkovmodelsofevolution.

•  DCMsforhardoptimizationproblemsreducerunningtimeneededtoachievegoodlevelsofaccuracy(empiricalobservation)

DecomposingTriangulatedGraphs

Max Clique Decomposition Separator-component Decomposition

Given:TriangulatedgraphG=(V,E)Output:DecompositionoftheverticesintooverlappingsubsetsRequire:Polynomialtime!Technique:Usespecialpropertiesabouttriangulatedgraphs

SimplicialVertices

Definition:LetG=(V,E)beagraph,andletvbeavertexinV.Thenvissimplicialifitssetofneighbors(i.e.,Γ(v))isaclique.Todo:•  Giveexampleofagraphthathasnosimplicialvertices.

•  Giveexampleofagraphwhereeveryvertexissimplicial.

PerfectEliminationOrdering

Definition:LetG=(V,E)beagraphonnvertices.Aperfecteliminationorderingisanorderingoftheverticesv1,v2,…,vnofGsothateachvertexviissimplicialinthegraphinducedon{vi+1,vi+2,…,vn}.Theorems:•  Everytriangulatedgraphhasasimplicialvertex.•  Infact,everytriangulatedgraphthatisnotacliquehastwonon-adjacentsimplicialvertices.

PerfectEliminationOrdering

Definition:LetG=(V,E)beagraphonnvertices.Aperfecteliminationorderingisanorderingoftheverticesv1,v2,…,vnofGsothateachvertexviissimplicialinthegraphinducedon{vi+1,vi+2,…,vn}.Theorems(Rose1970):AgraphGistriangulatedifandonlyifithasaperfecteliminationordering.Furthermore,givenatriangulatedgraph,aperfecteliminationorderingcanbefoundinpolynomialtime.

Somepropertiesofchordalgraphs

•  Theorem:EverychordalgraphG=(V,E)hasatmost|V|maximalcliques,andthesecanbefoundinpolynomialtime:–  Maxcliquedecomposition.

•  Provethisusingtheexistenceofaperfecteliminationordering.


•  Theorem:EverychordalgraphG=(V,E)hasatmost|V|maximalcliques,andthesecanbefoundinpolynomialtime:–  Maxcliquedecomposition.

•  Provethisusingtheexistenceofaperfecteliminationordering.


•  Everychordalgraphthatisnotacliquehasavertexseparatorthatisamaximalclique,anditcanbefoundinpolynomialtime:–  Separator-componentdecomposition.


•  Everychordalgraphhasatmostnmaximalcliques,andthesecanbefoundinpolynomialtime:Maxcliquedecomposition.

•  Everychordalgraphthatisnotacliquehasavertexseparatorthatisamaximalclique,anditcanbefoundinpolynomialtime:Separator-componentdecomposition.

DecomposingTriangulatedGraphs

Max Clique Decomposition Separator-component Decomposition

Given:TriangulatedgraphG=(V,E)Output:DecompositionoftheverticesintooverlappingsubsetsRequire:Polynomialtime!Technique:Usespecialpropertiesabouttriangulatedgraphs

DCMsareDivide-and-Conquerstrategies!

Howtocombinesubsettrees?

•  Everytriangulatedgraphhasaperfecteliminationordering:–  enablesustomergecorrectsubtreesandgetacorrectsupertreeback,ifsubtreesarebigenough(sothattheycontainalltheshortquartettrees).

TriangulatedGraphsandTrees

Theorem(Gravil1974,Buneman1974):AgraphGistriangulatedifandonlyifGistheintersectiongraphofasetofsubtreesofatree.Proof:Onedirectioniseasy,andtheotherisnot…

ExamplesofTriangulatedGraphs

•  ThresholdgraphsTG(D,q):Disadditiveandqisanyrealnumber,and(x,y)isanedgeifandonlyifD[x,y]<=q.

•  ShortSubtreeGraphsSSG(T,w):Tisatreewithedge-weightingw,andeveryshortquartetcontributesa4-clique.

•  Character-stateintersectiongraphsfromperfectphylogenies.


•  ThresholdgraphsTG(D,q):Disadditiveandqisanyrealnumber,and(x,y)isanedgeifandonlyifD[x,y]<=q.

•  Theorem:ForalladditivematricesDandthresholdsq,TG(D,q)istriangulated.

•  Proof:Usethefactthatallsubtreeintersectiongraphsaretriangulated.

DCM1-boostingdistance-basedmethods[Nakhlehetal.ISMB2001]

• Theorem(Warnowetal.,SODA2001):DCM1-NJconvergestothetruetreefrompolynomiallengthsequences

NJ DCM1-NJ

0 400 800 1600 1200 No. Taxa

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Erro

r Rat

e


•  ShortSubtreeGraphsSSG(T,w):Tisatreewithedge-weightingw,andeveryshortquartetcontributesa4-clique.

•  Theorem:ForalltreesTwithedgeweightingw,SSG(T,w)isadditive.


Rec-I-DCM3 significantly improves performance

Comparison of TNT to Rec-I-DCM3(TNT) on one large dataset

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Average MP score above

optimal, shown as a percentage of

the optimal

Current best techniques (TNT)

Rec-I-DCM3(TNT)


•  Character-stateintersectiongraphsfromperfectphylogenies.

•  Theorem:Forallperfectphylogenies,thecharacterstateintersectiongraph(wherenodescorrespondtocharacterstatesandedgescorrespondtoanytwostatesatanynodeinthetree–internalandleaf)istriangulated.


“Homoplasy-Free”Evolution(perfectphylogenies)

YESNO

PerfectPhylogeny

•  AphylogenyTforasetSoftaxaisaperfectphylogenyifeachstateofeachcharacteroccupiesasubtree(nocharacterhasback-mutationsorparallelevolution)

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Perfectphylogenies,cont.

•  A=(0,0),B=(0,1),C=(1,3),D=(1,2)hasaperfectphylogeny!

•  A=(0,0),B=(0,1),C=(1,0),D=(1,1)doesnothaveaperfectphylogeny!

Aperfectphylogeny

•  A=00•  B=01•  C=13•  D=12•  E=03•  F=13

A

B

C

D

Aperfectphylogeny

•  A=00•  B=01•  C=13•  D=12•  E=03•  F=13

A

B

C

D

E F

ThePerfectPhylogenyProblem

•  GivenasetSoftaxa(species,languages,etc.)determineifaperfectphylogenyTexistsforS.

•  TheproblemofdeterminingwhetheraperfectphylogenyexistsisNP-hard(McMorrisetal.1994,Steel1991).

TriangulatedGraphs

•  Agraphistriangulatedifithasnosimplecyclesofsizefourormore.

TriangulatedGraphs

Theorem(Gravil1974,Buneman1974):AgraphGistriangulatedifandonlyifGistheintersectiongraphofasetofsubtreesofatree.Proof:Onedirectioniseasy,andtheotherisnot…

PerfectPhylogeniesandTriangulatedColoredGraphs

•  SupposeMisacharactermatrixandTisaperfectphylogenyforM.

•  ThenletM’betheextensionofMtoincludetheadditional“species”addedattheinternalnodes.

•  CharacterStateIntersectionGraphGbasedonM’:–  ForeachcharacteralphaandstateiinM’,giveavertexv(alpha,i)andcolorthevertexwiththecolorforalpha.

–  Putedgesbetweentwoverticesiftheyshareanyspecies.–  Gistriangulatedandproperlycolored.

PerfectPhylogeniesandTriangulatedColoredGraphs

•  SupposeMisacharactermatrixandTisaperfectphylogenyforM.

•  ThenletM’betheextensionofMtoincludetheadditional“species”addedattheinternalnodes.

•  ThecharacterstateintersectiongraphGbasedonM’istriangulatedandproperlycolored.(Why?)

•  ButifwehadbaseditonMitmightnothavebeentriangulated.(Why?)

Aperfectphylogeny

•  A=00•  B=01•  C=13•  D=12•  E=03•  F=13Drawthecharacterstateintersectiongraph.

A

B

C

D

E F

Matrixwithaperfectphylogeny

c1c2c3s1321s2122s3113s4211

Draw the perfect phylogeny and compute the sequences at the internal nodes.

Matrixwithaperfectphylogeny

c1c2c3s1321s2122s3113s4211

Draw the character state intersection graph for the extended matrix (including the sequences at the internal nodes).

Thepartitionintersectiongraph

“Yes”InstanceofPP:c1c2c3s1321s2122s3113s4211

Triangulatingcoloredgraphs

•  LetG=(V,E)beagraphandcbeavertexcoloringofG.ThenGcanbec-triangulatedifasupergraphG’=(V,E’)existsthatistriangulatedandwherethecoloringcisproper.

•  Inotherwords,Gcanbec-triangulatedifandonlyifwecanaddedgestoGtomakeittriangulatedwithoutaddingedgesbetweenverticesofthesamecolor.

Agraphthatcanbec-triangulated

Agraphthatcanbec-triangulated

A vertex-colored graph G=(V,E) can be c-triangulated if a supergraph G’=(V,E’) exists that is triangulated and where the coloring is proper.

Agraphthatcannotbec-triangulated

TriangulatingColoredGraphs(TCG)

TriangulatingColoredGraphs:givenavertex-coloredgraphG,determineifGcanbec-triangulated.

ThePPandTCGProblems

•  Buneman’sTheorem:AperfectphylogenyexistsforasetSifandonlyiftheassociatedcharacterstateintersectiongraphcanbec-triangulated.

•  ThePPandTCGproblemsarepolynomiallyequivalentandNP-hard.

Ano-instanceofPerfectPhylogeny

•  A=00•  B=01•  C=10•  D=11

0 1

0

1

Aninputtoperfectphylogeny(left)offoursequencesdescribedbytwocharacters,anditscharacterstateintersectiongraph.Notethatthecharacterstateintersectiongraphis2-colored.

SolvingthePPProblemUsingBuneman’sTheorem


SolvingthePPProblemUsingBuneman’sTheorem


Somespecialcasesareeasy

•  Binarycharacterperfectphylogenysolvableinlineartime

•  r-statecharacterssolvableinpolynomialtimeforeachr(combinatorialalgorithm)

•  Twocharacterperfectphylogenysolvableinpolynomialtime(produces2-coloredgraph)

•  k-characterperfectphylogenysolvableinpolynomialtimeforeachk(producesk-coloredgraphs--connectionstoRobertson-Seymourgraphminortheory)

EarlyHistory•  LeQuesne(1969,1972,1974,1977):initialformulationofperfectphylogenies•  Estabrook(1972),Estabrooketal.(1975):mathematicalfoundationsofperfect

phylogenies•  McMorris(1997):binarycharactercompatibility•  Felsenstein(1984):reviewpaper•  Estabrook&Landrum,Fitch1975:compatibilityoftwocharacters•  Steel(1992)andBodlaenderetal.(1992):NP-hardness•  Buneman(1974):reducedperfectphylogenytotriangulatingcoloredgraphs•  KannanandWarnow(1992):establishedequivalenceofTCGandPP

SeeT.Warnow,1993.Constructingphylogenetictreesefficientlyusingcompatibilitycriteria.NewZealandJournalofBotany,31:3,pp.239-248(linkedoffmyhomepage)forasurveyoftheearlyliteratureandthefullcitations.

Literaturesample•  R.AgarwalaandD.Fernandez-Baca,1994.Apolynomial-timealgorithmfortheperfectphylogenyproblem

whenthenumberofcharacterstatesisfixed.SIAMJournalonComputing,23,1216–1224.•  R.AgarwalaandD.Fernandez-Baca,1996.Simplealgorithmsforperfectphylogenyandtriangulating

coloredgraphs.InternationalJournalofFoundationsofComputerScience,7,11–21.•  H.L.Bodlaender,M.R.Fellows,MichaelT.Hallett,H.ToddWareham,andT.Warnow.2000.Thehardnessof

perfectphylogeny,feasibleregisterassignmentandotherproblemsonthincoloredgraphs,TheoreticalComputerScience244(2000)167-188

•  H.L.BodlaenderandT.KIoks,1993:Asimplelineartimealgorithmfortriangulatingthree-coloredgraphs.JournalofalgorithmsJ5:160-172.

•  D.Fernandez-Baca,2000.Theperfectphylogenyproblem.Pages203–234of:Du,D.-Z.,andCheng,X.(eds),SteinerTreesinIndustries.KluwerAcademicPublishers.

•  D.Gusfield,1991.Efficientalgorithmsforinferringevolutionarytrees.Networks21,19–28.•  R.IduryandA.Schaffer.1993:Triangulatingthree-coloredgraphsinlineartimeandlinearspace.SIAM

journalondiscretemathematics6:289-294.•  S.KannanandT.Warnow,1992.Triangulating3-coloredgraphs.SIAMJ.onDiscreteMathematics,Vol.5

No.2,pp.249-258(alsoSODA1991)•  S.KannanandT.Warnow,1997.Afastalgorithmforthecomputationandenumerationofperfect

phylogenieswhenthenumberofcharacterstatesisfixed.SIAMJ.Computing,Vol.26,No.6,pp.1749-1763(alsoSODA1995)

•  F.R.McMorris,T.Warnow,andT.Wimer,1994.TriangulatingVertexColoredGraphs.SIAMJ.onDiscreteMathematics,Vol.7,No.2,pp.296-306(alsoSODA1993).

ApplicationsofPerfectPhylogeny

•  TumorPhylogenetics(MohammedEl-KebirwilltalkthisonApril10-17,2018)

•  HistoricalLinguistics(IwilltalkaboutthisonMarch15,2018)

•  PopulationgeneticsandHaplotypeinference

Introduction to Triangulated Graphstandy.cs.illinois.edu/581-2018-chordal.pdf · • Every chordal graph has at most n maximal cliques, and these can be found in polynomial time:

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