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Efficient Bounds for 3d Cayley Configuration Space of Partial 2-Trees

Permanent Link: http://ufdc.ufl.edu/UFE0042297/00001

Material Information

Title: Efficient Bounds for 3d Cayley Configuration Space of Partial 2-Trees
Physical Description: 1 online resource (41 p.)
Language: english
Creator: Chittamuru, Ugandharreddy
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: 2tree, 3tree, cayley, configuration, euclidean, graph, linear, partial, quadratic, realizabilty, sampling
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: An Euclidean Distance Constraint System (EDCS) is defined as a graph, G(V,E), with fixed or interval euclidean distance assignment to its edges. A non-edge is a vertex pair that is not connected by an edge, and hence is not assigned a distance. Each non-edge can take a single or range of values that is continuous or discontinuous. The set of (squares of) values the non-edges can take is called the Cayley configuration space of the EDCS. The set of (squares of) values the non-edges in a set F can take is the Cayley configuration space of the EDCS projected on F. If G union F is minimally rigid, the parameterized Cayley configuration space taken on F is called complete Cayley configuration space. A graph G is d-realizable if for every realization in some finite euclidean dimension there is a realization with the same edge lengths in d-dimension. Graphs sharing a complete-k-vertex graph are said to be in a k-sum. A k-tree is defined recursively: a complete k + 1-vertex graph is a k-tree; and any two k-trees in a k-sum together form a k-tree. The graph formed by any subset of edges of a k-tree is called a partial k-tree. From a result by Sitharam & Gao we know, if a graph G together with a set of non-edges F is d-realizable, then for any EDCS based on G, the d-dimensional Cayley configuration space on F is convex. Partial 3-trees are a special subset of 3-realizable graphs. A 3-tree is minimally rigid in 3D. There are known methods for computing bounds in 3D for non-edges that extend partial 3-trees into complete 3-trees. These methods can involve solving system of inequalities consisting of polynomials of higher degree ( > 2) with many terms and variables. From a result by Sitharam & Gao, the inequalities that express the range of a non-edge can be classified into either a linear or non-linear class. A sampling algorithm by nature adds edges to the graph it is sampling. Sometimes a non-linear class non-edge after addition of certain edges fall into a linear class. Bounding box, a box enclosing the configuration space and touching its boundaries, is a nice approximation for a configuration space. An EDCS can have different complete Cayley configuration spaces varying in parameter set. Some of these complete Cayley configuration spaces might have a bounding box expressible by linear inequalities. In this thesis we (a) give an iterative algorithm for computing bounds for complete Cayley configuration spaces of partial 2-trees (a subset of partial 3-trees) in 3D, where complexity of computing each bound is either solving linear inequalities or a quadratic in single variable. (b) prove a necessary but not sufficient condition to identify partial 2-trees that have a bounding box expressible by linear inequalities. (c) prove an exact characteristic of non-edges, in a partial 3-tree, that fall into a linear class on addition of edges, such that the graph along with new edges is also a partial 3-tree. Problems in applications like molecular assembly have associated EDCS representations and desired solution for these problems are usually a set or series of configurations. Sampling or searching the configuration space of the underlying EDCS is one way to find these configurations. In this thesis, we take macromolecular assembly as an example and show how sampling or searching can be improved using the above algorithm.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ugandharreddy Chittamuru.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Sitharam, Meera.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042297:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042297/00001

Material Information

Title: Efficient Bounds for 3d Cayley Configuration Space of Partial 2-Trees
Physical Description: 1 online resource (41 p.)
Language: english
Creator: Chittamuru, Ugandharreddy
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: 2tree, 3tree, cayley, configuration, euclidean, graph, linear, partial, quadratic, realizabilty, sampling
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: An Euclidean Distance Constraint System (EDCS) is defined as a graph, G(V,E), with fixed or interval euclidean distance assignment to its edges. A non-edge is a vertex pair that is not connected by an edge, and hence is not assigned a distance. Each non-edge can take a single or range of values that is continuous or discontinuous. The set of (squares of) values the non-edges can take is called the Cayley configuration space of the EDCS. The set of (squares of) values the non-edges in a set F can take is the Cayley configuration space of the EDCS projected on F. If G union F is minimally rigid, the parameterized Cayley configuration space taken on F is called complete Cayley configuration space. A graph G is d-realizable if for every realization in some finite euclidean dimension there is a realization with the same edge lengths in d-dimension. Graphs sharing a complete-k-vertex graph are said to be in a k-sum. A k-tree is defined recursively: a complete k + 1-vertex graph is a k-tree; and any two k-trees in a k-sum together form a k-tree. The graph formed by any subset of edges of a k-tree is called a partial k-tree. From a result by Sitharam & Gao we know, if a graph G together with a set of non-edges F is d-realizable, then for any EDCS based on G, the d-dimensional Cayley configuration space on F is convex. Partial 3-trees are a special subset of 3-realizable graphs. A 3-tree is minimally rigid in 3D. There are known methods for computing bounds in 3D for non-edges that extend partial 3-trees into complete 3-trees. These methods can involve solving system of inequalities consisting of polynomials of higher degree ( > 2) with many terms and variables. From a result by Sitharam & Gao, the inequalities that express the range of a non-edge can be classified into either a linear or non-linear class. A sampling algorithm by nature adds edges to the graph it is sampling. Sometimes a non-linear class non-edge after addition of certain edges fall into a linear class. Bounding box, a box enclosing the configuration space and touching its boundaries, is a nice approximation for a configuration space. An EDCS can have different complete Cayley configuration spaces varying in parameter set. Some of these complete Cayley configuration spaces might have a bounding box expressible by linear inequalities. In this thesis we (a) give an iterative algorithm for computing bounds for complete Cayley configuration spaces of partial 2-trees (a subset of partial 3-trees) in 3D, where complexity of computing each bound is either solving linear inequalities or a quadratic in single variable. (b) prove a necessary but not sufficient condition to identify partial 2-trees that have a bounding box expressible by linear inequalities. (c) prove an exact characteristic of non-edges, in a partial 3-tree, that fall into a linear class on addition of edges, such that the graph along with new edges is also a partial 3-tree. Problems in applications like molecular assembly have associated EDCS representations and desired solution for these problems are usually a set or series of configurations. Sampling or searching the configuration space of the underlying EDCS is one way to find these configurations. In this thesis, we take macromolecular assembly as an example and show how sampling or searching can be improved using the above algorithm.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ugandharreddy Chittamuru.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Sitharam, Meera.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042297:00001


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Beforestartingthiswork,Iwasholdingontoassumptionsthatweredetrimentalforresearch.IhavemyutmostrespecttowardsProfessorMeeraSitharamforherknowledgeacquisitionandapplicationskills,forshowingremarkablepatienceandcomposureinhandlingmyinexperienceandforstimulatingmyself-reectionintomyassumptions.Heradviceandquestionsmadethisstrugglefruitful.Exposuretoherdeductiveprocess,employedduringproblem-solvingnurturedmythinking.KnowledgeobtainedwhileresolvingconictingassumptionsduringdiscussionsonvariedtopicswithChristopherGoddard,RaviTejaChinta,SivaKumarBalaga,VenkatakrishnanRamaswamy,JamesPence,onseparateoccasions,hasinuencedmylearningandthinkingprocess.Teammates,AysegulOzkanandJamesPence,wereveryhelpfulinsharingtheirknowledgeonthissubjectandespeciallyansweringmyrepetitivequestions.AysegulOzkanintegratedtheimplementationofalgorithmbasedonthisworkintoavisualizingtoolformacromolecularassembly,co-developedbyher.TwoOPSjobs,heldatdierentpointsoftime,oeredbyDr.NicoCellinese,Dr.AmrAbd-Elrahma,providedthenecessarynancialsupport.InteractionwithDr.AlperUngorinuencedmydecisiontopursuethiseld.Dr.JorgPeter'squote,simplifyknowledgebyabstractingiti.egivingitstructure,hasopenedupnewperspectives.Dr.OscarBoykinwashelpfulwhenneededmost.Myuncles,ManoharandNarendranath,havealwaysassistedmeinmycareerchoices.Iamgratefultoallthesepeoplefortheirinuences,advices,andhelp. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 10 1.1Introduction ................................... 10 1.2Questions .................................... 13 1.3OrganizationofThesis ............................. 13 2BACKGROUND .................................. 14 3MACROMOLECULARASSEMBLY ........................ 16 3.1MacromolecularAssembly ........................... 16 3.2GeometricRepresentationofMacromolecularAssembly ........... 16 3.2.1RigidAssembly ............................. 17 3.2.2ChoosingCayleyParameters ...................... 17 3.3TheThesisProblem:BoundingCayleyCongurationSpace ........ 18 3.3.1GeneralStep-wiseSamplingAlgorithm ................ 19 3.3.2OtherRelatedQuestions ........................ 19 4GRADATIONINCOMPUTINGBOUNDSFORCAYLEYPARAMETERSIN3-TREES ....................................... 21 4.1Non-edgeBoundsbyPolynomialRepresentation ............... 21 4.1.1Linear .................................. 21 4.1.2Quadratic ................................ 21 4.1.2.1Nonsharingnon-edge .................... 21 4.1.2.2Singlenon-edgesharedbetweenmultipletetrahedra .... 21 4.1.3Cubic ................................... 22 4.1.3.1Twovariables ......................... 22 4.1.3.2Threevariables ........................ 23 4.1.3.3Four,ve,sixvariables .................... 23 5EFFICIENTMETHODSFORCOMPUTINGBOUNDS ............. 27 5.1ChoosingNon-sharingNon-edgesforaPartial2-treetoExtendtoa3-tree 27 5.1.1Theorems ................................ 27 5.1.2ImprovedSamplingAlgorithm ..................... 29 5.1.2.1Choosingnon-edgestoconstruct3-tree ........... 29 5.1.2.2Orderofpickingnon-edges .................. 29 5

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.................................... 29 5.2.1Theorems ................................ 30 5.2.2ApplicationtoSampling ........................ 33 5.3LinearBoundingBoxforPartial2-trees:ANecessarybutNotSucientCondition .................................... 33 5.3.1Theorems ................................ 34 5.3.2PracticalApplication .......................... 37 6CONCLUSIONANDOPENPROBLEM ..................... 38 REFERENCES ....................................... 40 BIOGRAPHICALSKETCH ................................ 41 6

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Figure page 1-1Non-edgefcomputablebytriangularinequalities ................. 12 1-2Dottededges(F)areexpressiblebylinearinequalitiesandG[Fisa3-tree .. 12 3-1Bi-Tetherbetweenmiandmj 17 3-23x3moleculeactiveconstraintgraphby'JamesPence' ............. 18 4-1Onenon-edgesharedbetweentetrahedrons ..................... 25 4-2Fourpathsbetweentwovertices ........................... 25 4-3Cubicsystemwithtwovariables(a)sharingtwovariables(b)sharingonevariable 25 4-4Cubicsystemwiththreevariables(a)sharingthreevariable(b)sharingtwovariables(c)sharingonevariable ................................ 25 4-5Cubicsystemwithfourvariables(a)sharingthreevariable(b)sharingtwovariables(c)sharingonevariable ................................ 26 4-6Cubicsystemwithvevariables(a)sharingthreevariables(b)sharingtwovariable 26 4-7Cubicsystemwithsixvariables,sharingatmostthreevariables ......... 26 5-1Non-sharingnon-edgef 28 5-2ComponentofS1 32 5-3ThereexistsanorderwhereStep1solveslinearinequalitieswhilesamplingtheCayleycongurationspaceconsistingofthedottededges ............ 33 5-4Foreveryorder,everyiterationofStep1solvesnon-linearinequalitieswhilesamplingtheCayleycongurationspaceconsistingofthedottededges ..... 34 5-5Non-edgefbetweenPi;Pj 36 7

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AnEuclideanDistanceConstraintSystem(EDCS)isdenedasagraph,G(V;E),withxedorintervaleuclideandistanceassignmenttoitsedges.Anon-edgeisavertexpairthatisnotconnectedbyanedge,andhenceisnotassignedadistance.Eachnon-edgecantakeasingleorrangeofvaluesthatiscontinuousordiscontinuous.Thesetof(squaresof)valuesthenon-edgescantakeiscalledtheCayleycongurationspaceoftheEDCS.Thesetof(squaresof)valuesthenon-edgesinasetFcantakeistheCayleycongurationspaceoftheEDCSprojectedonF.IfG[Fisminimallyrigid,theparameterizedCayleycongurationspacetakenonFiscalledcompleteCay-leycongurationspace.AgraphGisd-realizableifforeveryrealizationinsomeniteeuclideandimensionthereisarealizationwiththesameedgelengthsind-dimension.Graphssharingacomplete-k-vertexgrapharesaidtobeinak-sum.Ak-treeisdenedrecursively:acompletek+1-vertexgraphisak-tree;andanytwok-treesinak-sumtogetherformak-tree.Thegraphformedbyanysubsetofedgesofak-treeiscalledapartialk-tree. FromaresultbySitharam&Gao[ 1 ]weknow,ifagraphGtogetherwithasetofnon-edgesFisd-realizable,thenforanyEDCSbasedonG,thed-dimensionalCayleycongurationspaceonFisconvex.Partial3-treesareaspecialsubsetof3-realizablegraphs.A3-treeisminimallyrigidin3D.Thereareknownmethodsforcomputingboundsin3Dfornon-edgesthatextendpartial3-treesintocomplete3-trees.These 8

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From[ 1 ],theinequalitiesthatexpresstherangeofanon-edgecanbeclassiedintoeitheralinearornon-linearclass.Asamplingalgorithmbynatureaddsedgestothegraphitissampling.Sometimesanon-linearclassnon-edgeafteradditionofcertainedgesfallintoalinearclass. Boundingbox,aboxenclosingthecongurationspaceandtouchingitsboundaries,isaniceapproximationforacongurationspace.AnEDCScanhavedierentcompleteCayleycongurationspacesvaryinginparameterset.SomeofthesecompleteCayleycongurationspacesmighthaveaboundingboxexpressiblebylinearinequalities. Inthisthesiswe ProblemsinapplicationslikemolecularassemblyhaveassociatedEDCSrepresentationsanddesiredsolutionfortheseproblemsareusuallyasetorseriesofcongurations.SamplingorsearchingthecongurationspaceoftheunderlyingEDCSisonewaytondthesecongurations.Inthisthesis,wetakemacromolecularassemblyasanexampleandshowhowsamplingorsearchingcanbeimprovedusingtheabovealgorithm. 9

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Problemslikemolecularassembly,proteinfoldinghavearepresentationusingEDCS.UsuallythesolutionfortheseproblemsareasetorseriesofcongurationsthatarepartofthecompletecongurationspaceoftheEDCS.OnewayistosearchforthedesiredcongurationsbysamplingthroughthecompleteCayleycongurationspaceofthegivenEDCS.Thesetofrealizabledistanceassignmentstothechosenparameters(non-edges)yieldsaparameterizedCayleyCongurationSpace.IfG[Fisminimallyrigid,theparameterizedCayleycongurationspacetakenonFisaCompleteCayleyCongurationSpace.Resultfrom[ 1 ]promisesanecientstructureforCayleycongurationspace,i.esquaredconvex,whenEDCSalongwithnon-edgesresultsinad-realizablegraphind-dimension.From[ 1 ],thesamplingcomplexityisameasureofeciencyforwalkingthroughacongurationspace.Itdependsonthecomplexityof(a)choosingthesetofnon-edges,Fand(b)thedescriptionofcongurationspaceassemi-algebraicset. TheproblemsmentionedabovehaveEDCSrepresentationsin3D.Incaseofassembly,majorityofthegraphsthatoccurareinfact3-realizable.Thecongurationsofinterestinmolecularassemblyproblemsarepointsinthecongurationspacethatsatisestheconstraints.Thesecongurationsarecrucialforagivenassemblyastheyhelpto 10

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determinethecongurationalentropythatinturndeterminesfreeenergy; (b) toisolatethoseintermolecularinteractionsthatarecrucialfordrivingassemblyofacollectionofmolecularunits. Solvingtheseproblemswouldinturnshedlightonrobustandspontaneous-butpoorlyunderstood-supramolecularandmacromolecularself-assemblyprocessessuchashelixpacking,viralself-assembly,proteincrystallization,prionaggregation,ligandanddrugdockingetc. Anyecientalgorithmsamplingaconvexspacewillnotsteponpointsoutsideofthespace.Boundsonparametersarecrucialinavoidingtheun-populatedspace.Computingdescriptionforapartial3-treesisrelativelyeasy.Fromthedescriptionofconvexspace,ndingboundsforparametersisanon-trivialproblem.Findingboundsusingthisdescriptionfornon-edgesina3-treeisofvaryingdicultyrangingfromsolvingsystemoflinearinequalitiestosystemofcubicinequalities.Inthisthesis,forpartial2-treesweshowaparticularchoiceandorderfornon-edgescanguaranteeecientsamplingwherendingboundsforeachnon-edgeiseithersolvingalinearinequalityorsolvingasinglequadraticofonevariable. From[ 1 ],inequalitiesthatexpresstherangeofanon-edgecanbeclassiedintoeitheralinearornon-linearclass.Bynature,asamplingalgorithmaddsedgestothegraphitissampling.Insomegraphs,thoughsomenon-edgesfallintonon-linearclass,onadditionofedges(edgesthatwithholdpartial3-treeproperty),thesamenon-edgecanfallintolinearclass.Forinstance,addinge1tothegraphintheFigure 1-1 resultsinanewpartial3-treeandalsomakesffallintolinearclass.Acharacterizationforsuchnon-edgescanimprovesamplingperformance. Boundingbox,aboxenclosingthecongurationspaceandtouchingitsboundaries,isaniceapproximationforthecongurationspace.Boundingboxisusedtoapproximatelyvalidateagivenconguration.Computingsuchaboxisequivalenttondingtherangeofallthenon-edgesthatconstitutetheCayleycongurationspace.Hence,complexityof 11

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Non-edgefcomputablebytriangularinequalities ndingaboundingboxisrelatedtothecomplexityofndingtherangefornon-edges.From[ 1 ]weknowacharacteristicoflinearclassofnon-edges,i.enon-edgesexpressiblebyacollectionoflinearinequalities.Graphswithaboundingboxexpressiblebylinearinequalitieshaveafastalgorithmtocomputetheboundingbox.Soacharacterizationthatcanidentifysuchgraphsiscrucial.ForthegraphintheFigure 1-2 ,foreachdotted-edgealltheminimum2-sumcomponentscontainingthatdotted-edgearepartial2-trees.Thisprovestheexistenceofaboundingboxexpressiblebylinearinequalitiesforaspecicpartial2-treein3D. Dottededges(F)areexpressiblebylinearinequalitiesandG[Fisa3-tree 12

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1. Whatarethevariouslevelsofdicultyincomputingboundsofnon-edgesinpartial3-trees,usingknownmethods? 2. Givenapartial2-treeG,whatchoiceandorderofnon-edgesexistsforG,sothatndingboundsiterativelyisecient? 3. Whatarethecharacteristicsofpartial2-treesthathaveaboundingboxexpressiblebylinearinequalities? 4. Whatarethecharacteristicsofnon-edges,inapartial3-tree,thatswitchintolinearclassonadditionofedges,suchthatthegraphalongwithnewedgesisalsoapartial3-tree? Chapter6concludeswithanopenproblem. 13

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Thischapterprovidesdenitionsnecessarytounderstandtheissuesaddressedinthisthesis. { (b) Eucledianspaceisametricspacewitheucledianmetric. Eucledianmetricisthedistancerelationbetweentwopoints.Ifp(p1;p2);q(q1;q2)arethepointsinEuclidean2-spacetheneucledianmetricisbythefollowingformula.p 2 ] 2 ] 3 ]

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4 ] CongurationSpaceofEDCSissetofcongurationswhereeachcongurationrepresentsauniquesetofdistancevaluescorrespondingtoallvertexpairsorsetofallvalidrealizationsforagiven(G,). 1 ] [ 1 ]

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Thischapterintroducesmacromolecularassemblyandgivesageometricalinterpretationusingcongurationspaces.Weuserecentadvancesincharacterizinggraphswithspecialtypesofcongurationspacesonthequestionofsampling,searchingandvisualizationofmacromolecularassemblycongurationspaces. 3-1 .Molecularunitstogetherwithconstraintsisinputtoanassemblyproblem.Anyvalidrealizationsatisfyingalltheconstraintsiscalledanassembly.Acompleteunderstandingofvalidmolecularcongurationsthatsatisfytheconstraintsisnecessarytounderstandviralself-assembly,proteincrystallization,prionaggregation,ligandanddrugdockingetc.Sotoolsthatcanhelpanalyze,visualizeallcongurationsofinteractingmolecularunitsarenecessary.Inanassembly,atreeoftetheringconstraintsholdtogetherthemolecularunits. 16

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Bi-Tetherbetweenmiandmj 1 ],asquaredconvexcongurationspacecanbeconstructed,ifG[Fis3-realizablein3D.Apairoftetherconstraintsbetweentwomoleculesremoves2degreeoffreedom.Foranassemblywithtwomolecularunitsacteduponbyapairoftetherconstraintsatleast4moreparametersarenecessarytobecomeonerigidbody. 17

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3-2 ,wecanseemanywaystochooseparametersbetween6atomstoform3-tree.LetFconstitutealltheparameterschosenbetweenalltethertiedmolecularunits.TheFigure 3-2 isanactiveconstraintgraphwithnon-edgeparametersforanassemblywithtwomolecularunitswith3atomseach. 3x3moleculeactiveconstraintgraphby'JamesPence' 18

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SamplingByStepSize((G;),F,step size) size) size 1 oftheabovealgorithmwouldendupperformingsuchcomputations.Thisleadstothequestion,Isthereawaytoavoidsuchexpensivecomputationswhilesampling?Chapter4addressesthisquestion. 3 ofsamplingalgorithmdiscussedabove,non-edgeismadeanedgeandsamplingiscalledagainonthenewgraph.WhensamplingthegraphinFigure 1-1 ,ife1isthelastnon-edgetobecomeanedgethenforanyorderchosen,everyexecutionofStep 1 solvesonlynon-linearinequalities.Earlierargumentshowed,thereexistsanorderwhereatleastforoneofthenon-edge,Step 1 solveslinearinequalities.Aparticularorderisbenecialtosomenon-edgesasthiswouldletthemfallintolinearclass.Characteristicsthatcanidentifysuchnon-edgeswouldcontributetoimprovedsampling.Thisleadstothequestion,whatarethecharacteristicsofnon-edges,inapartial3-tree,thatswitchintolinearclassonadditionofedges,suchthatthegraphalongwithnewedgesisalsoapartial3-tree?InChapter4,weproveanexactcharacterizationforthesenon-edges. 19

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1-2 hasaboundingboxexpressiblebylinearinequalitiesforthecongurationspaceconsistingofthedottededgesasparameters.Butwedon'tknowawaytoidentifyagraphwithsuchaboundingbox.Thisleadstothequestion,whatarethecharacteristicsofagraphthathasaboundingboxexpressiblebylinearinequalities?InChapter4,wepartiallyanswerthenegativeofthisquestionbygivingacharacteristicforpartial2-treesthatcannothaveaboundingboxexpressiblebylinearinequalities. 20

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Asdiscussedinthepreviouschapter,ndingnon-edgeboundsisanimportantstepforsamplingcongurationspaces.Inthenextsectionweprovideanalgorithmthatcomputesecientboundsfornon-edges.Todemonstratecomplexityofthealgorithm,inthissection,weillustrateagradationincomplexityofcomputingstraightforwardboundsfornon-edgesinpartial3-trees. 1 ],theCayleycongurationspaceofany3-realizablegraphin3Dissquaredconvex.3-treesare3-realizableandsotherangeofanon-edgeby[ 1 ]shouldbeacontinuousintervalinsquaredconvexspace. 1 ]observesforanon-edgef,ifalltheminimal2-sumcomponentsofG[fcontainingfarepartial2-treesthenrangeoffcanbeexpressedasasetoflinearinequalities. 4.1.2.1Nonsharingnon-edge 4-1 (a)eachtetrahedronhas5edgesandonenon-edgef.Boundsonfcanbecomputedbysolvingaquadraticinonevariable.Themaximumorminimumvalueanon-edgeinFigure 4-1 (a)cantake,iswhenthetetrahedronisofzerovolume.Insquareddistancespaceeitherequatingthevolumedeterminantofatetrahedrontozeroorusingtheequationofacircletondtheco-ordinatesofthetwopointsbetweenwhicharetheboundsweareinterestedinresultsinaquadraticinonevariable. 4-1 (b)boundsforasinglenon-edgesharedbetweenmultipletetrahedronsiscomputablebysolvingsystemofquadraticinequalitiesinasinglevariable.These 21

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Fact1. ThegraphinFigure 4-2 includingthedottededgeisapartial2-tree.Hencetheresultinggraphafter2-sumwithtetrahedron,T,alongthedottededgeremainsapartial3-tree.Nowremovethedottededge.RefertothisgraphasG.Fromthefactabove,a3-treecanbeextendedfromGonlyifnon-edgeexistbetweentheremoveddottededgevertexpair.RepeattheabovestepsforeachremainingedgeofT.Thiswaywecanobtainatetrahedronwith0to6dottededges.Inalltheguresbelow,adottedlinerepresents2-sumwithFigure 4-2 4-3 (a),takeatetrahedronsimilartotheonein(a)andruna3-sumonthecorrespondingsharedtriangleof(a).Repeatthisuntiltherearentetrahedrons.Repeatthesamefor(b).Themaximumorminimumvalueforanynon-edgehappenswhenthetetrahedronhas0volume.Equatingthevolumedeterminanttozerofortwoparameters(xandy)resultsinacubicinequalityintwovariables. 22

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4-3 has (a) non-adjacentnon-edges,thenthedeterminantcontainstheterms(x2;y2;xy;x;y). (b) adjacentnon-edgesthenthedeterminantcontainstheterms(x2y;y2x;xy;x;y). Astherearentetrahedrons,wecanobtainboundsforanynon-edgebysolvingthesystemofcubicinequalities.Thenumberofnon-edgesthatexistinthesharedtriangleisexactlythenumberofvariablessharedbetweenthesystemofinequalities. 4-4 (a),takeatetrahedronsimilartotheonein(a)andrun3-sumonthecorrespondingsharedtriangleof(a).Repeatthisuntiltherearentetrahedrons.Repeatthesamefor(b),(c).Amaximumorminimumvalueforanynon-edgehappenswhenthetetrahedronhas0volume.Assuringnon-negativityofthethevolumedeterminantforthreeparameters(x;y;z)resultsinacubicinequalitiesinthreevariables. IfeachtetrahedroninFigure 4-4 has (a) allthenon-edgesadjacenttooneanother,thenthedeterminantcontainstheterms(x2;y2;z2;xy;yz;zx;zy;x;y;z) (b) atleastonenon-edgeisnotadjacenttoanother,thenthedeterminantcontainstheterms(x2z;xz2;xyz;y2;xy;xz;yz;x;y;z). Astherearentetrahedrons,wecanobtainboundsforanynon-edgebysolvingatmostncubicinequalities.Thenumberofnon-edgesthatexistinthesharedtriangleisexactlythenumberofvariablessharedbetweenthesystemofinequalities. 4-5 (a,b,c), 4-6 (a,b), 4-7 havesimilarcubicsystemwith4;5;6variablesrespectively. ForFigure 4-5 ,ifeachtetrahedronhastwo (a) non-adjacentedges,thenthedeterminantcontainstheterms(vy2;wx2;xw2;yv2;wxv;vyw;vyx;wyx;vw;xy;yw;yv;wx;x;y;v;w;). (b) adjacentedges,thenthedeterminantcontainstheterms(xy;wyx;xw2;xv;yv;wv;wx2;wxv;xyv;v2;y2;yw;wx;v;x;y). 23

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4-6 ,thedeterminantcontainstheterms(vyx;wxv;vy2;wx2;wyx;vyw;yv2;xw2;vw;vy;wx;xv;xz;yz;wxz;xyz;z2;wz;z;vyz;vwz;vz) ForFigure 4-7 ,thedeterminantcontainstheterms(vyw;wxv;vy2;wx2;wyx;vyw;yv2;xw2;wxz;xyz;uxz;uyz;vuz;wuz;uz2;u2z;uyw;uyv;wux;uxv;vyz;vwz) 24

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Onenon-edgesharedbetweentetrahedrons Fourpathsbetweentwovertices Cubicsystemwithtwovariables(a)sharingtwovariables(b)sharingonevariable Cubicsystemwiththreevariables(a)sharingthreevariable(b)sharingtwovariables(c)sharingonevariable 25

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Cubicsystemwithfourvariables(a)sharingthreevariable(b)sharingtwovariables(c)sharingonevariable Cubicsystemwithvevariables(a)sharingthreevariables(b)sharingtwovariable Cubicsystemwithsixvariables,sharingatmostthreevariables 26

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ThischapteraddressesthequestionsraisedinChapter2. Denition8. Lemma5.1.1. 5-1 )is2-sumwithatrianglet,thenthereexistsanon-edge,f,whereG[t[fisa3-treeand, (a) alltheedgesornon-edgeswithnon-sharingpropertyinGexcept(va,vb),(va,v2),(vb,v2)arepreservedinG[t[f. (c) thetetrahedronfcorrespondstohasonlyonenon-edge,i.eitself Proof. 5-1 .Gisa3-tree.(v1,va,v2,vb)isatetrahedron.Bothofthesesharethe3-sumtriangle(va,v2,vb),whichisa3-sum.HenceG[t[fisa3-tree.Intheaboveconstruction,theedgessharedinG[t[fareonly(va,vb),(va,v2),(vb,v2).Hencethenon-sharingpropertyfortherestoftheedgesornon-edgesincludingfisstillapplicable. (a) everynon-edgeinFhasnon-sharingpropertyin3-tree,G[F

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Non-sharingnon-edgef Thetetrahedroncorrespondingtoeachnon-sharingnon-edgeinFhasonlyonenon-edgei.eitself Proof. Pickaleaftriangle,callitVk+1.Refertoonetrianglethatis2-sumwithVk+1asVk.WithoutVk+1,GVk+1isa2-treewithkvertices,denoteitG0.Byinductionhypothesis,thereexistsasetFwhereG0[Fisa3-treeandeverynon-edgeinFhasnon-sharingpropertyandthetetrahedroncorrespondingtoeverynon-edgeinFhasasinglenon-edge,namelyitself.Vk+1isin2-sumwithtriangle,Vk,nowpartofG0[F.FromLemma 5.1.1 ,thereexistsanon-edgefwhereG0[F[fisa3-tree,preservingnon-sharingpropertyforalledgesornon-edgesinG0[F,exceptedgesinVk.Nowfisthenewnon-sharingnon-edgeinG0[F[f.Callthisnew3-tree,G.FromLemma 5.1.1 thetetrahedronfcorrespondstohasonlyfasnon-edge 28

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Proof. 1 toavoidsolvingforasystemofinequalitieswithhigherdegree,terms,variables. 5.1.2 ,therealwaysexistsF2suchthatG[F1[F2isa3-treeandeverynon-edgeinF2hasnon-sharingproperty. 1 ,pickaF2onlywhennoneofF1'sareleft.FixinganysubsetofF1,sayF,beforeF2ensuresG[Fasapartial2-tree.From[ 1 ],theprojectionofcongurationspaceonanynon-edgeofF1Fcanbeexpressedasacollectionoftriangularinequalities.SoeachF1iscomputablebysetoflinearinequalitiesforanyarbitraryorderchosenforF1. Anon-sharingedgeina3-treecorrespondstoonlyonetetrahedron.Aswemadesureanynon-edgeinF2isxedonlyafterallofF1isxed,G[F1willbea2-treebeforeF2isxed.AsG[F1is2-tree,byTheorem 5.1.2 ,eachnon-edgeinF2,correspondstoanuniquetetrahedronwithitbeingtheonlynon-edge.Socomputingrangeforthisnon-edgeinF2isascomplexassolvingCayleydeterminantforonevariable,whichisaquadraticinsinglevariable. Denition9.

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Theorem5.2.1. Proof. 1 ],whenappliedtoG[xresultsinoneofthetwobasecasesdiscussedinTheorem5:2.AssignmentprovidedtothebasecasesasshowninLemma5:5of[ 1 ]provesexistenceofawherenon-edgefcannotbecomputedbytriangularinequalities. (a) (b) Proof. 30

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1 ] 5.2.3 ,G1[xisapartial3-tree. EdgesinxarebetweenverticesofG1.HencecomponentsofS1[S2areinvarianttoadditionofedgesetx.Fromtheaboveconstruction,aminimal2-sumcomponentcontainingfinG1[xisapartial2-tree.Thisprovesalltheminimal2-sumcomponentsofG[f[xcontainingfarepartial2-trees. S1containsk+1components.PickGk+1[ffromS1.v2;v1inGk+1haveonlyonevertex-disjointpath.Sothereexistsatleastonearticulationpoint,ak+1.Likebasecase,n=1,addedgestoGk+1.Referthemasxk+1.Edgesinxk+1arebetweenverticesofGk+1.HenceS1[S2areinvarianttoadditionofedgesetxk+1.Fromtheaboveconstruction,weknowaminimal2-sumcomponentcontainingfinGk+1[xk+1[fisapartial2-tree.ThisleavesonlyknumberofcomponentsinS1.ByinductionhypothesisthereexistsasetxkofedgessuchthateachcomponentinS1resultsinapartial2-tree.Referxk+1[xkasx.ByTheorem5:2in[ 1 ],non-edgefinG[xiscomputablebycollectionoftriangularinequalities.

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5-2 .Duetov3beinganarticulationpoint,novertexofGahasapathofedgestoGbwithoutcontainingv3.PickapathinGbconnectingv2tov3.Includev2;v3intothepathandcallitP1. ComponentofS1 32

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1 inthesamplingalgorithmvariesbasedonthechoiceofnon-edgesandtheorderinwhichtheyarexed.Forinstance,forthenon-edgefinFigure 5-4 ,alltheminimal2-sumcomponentscontainingfhaveonepath,whichmeansthereexistsanxby 5.2.2 .Byaddingedgee1,allminimal2-sumcomponentscontainingfinG[x[farepartial2-trees.Thise1isthexforG[fmentionedin 5.2.2 .Byincludinge1aspartoftheCayleycongurationspaceandxingfafterxinge1,weensurethereexistsanordersuchthatboundsoffinStep 1 canbecomputedbysolvinglinearinequalities.Thisisimportantbecause,foreveryorderthesamegraphwithachoiceofnon-edgesshowninFigure 5-3 orxinge1lastintheFigure 5-4 resultsinStep 1 solvingnon-linearinequalitiesineveryiteration.ThecharacteristicdescribedinTheorem 5.2.2 ,whentrueforagraph,avoidssuchachoiceandorder,therebyimprovingsampling. ThereexistsanorderwhereStep1solveslinearinequalitieswhilesamplingtheCayleycongurationspaceconsistingofthedottededges Denition10.

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Foreveryorder,everyiterationofStep1solvesnon-linearinequalitieswhilesamplingtheCayleycongurationspaceconsistingofthedottededges Lemma5.3.1. (a) anypathconnectingPitoPjshouldcontainv1orv2 anynon-edge,f,inthepathconnectingPitoPj,notinvolvingv1orv2isnon-linear Proof. 1 ]onGtoreducetobasecases,alsodiscussedinTheorem5:2of[ 1 ].Inboththebasecasesoneoftheminimal2-sumcomponentcontainingv1;v2isatetrahedron.HencecontradictingtheassumptionthatGisapartial2-tree. 34

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(b) 5-5 .fconsistsofonevertexfromPiandotherfromPj.Alsobetweentheverticesofnon-edgefwecanseetwovertex-disjointpaths,(vfatov1tovfb),(vfatov2tovfb).v1;v2actsasapathbetweenthesetwopaths.UsingtheproofofTheorem5:2,thisgraphcanbereducedtothebasecases.Lemma5:5provesexistenceofaforwhichfisnon-linear. Proof. 35

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Non-edgefbetweenPi;Pj Removalofv1andv2fromGresultsinasetofconnectedcomponentsH1:::Hk.ReferGiasthegraphinducedbyHitogetherwithv1andv2.FromeachHi,thereexistsanedgetobothv1andv2,becausemissinganedgeresultsinexistenceof1-sumcomponentssharingeitherv1orv2.vertices(G)=vertices(H1:::Hk)[v1[v2.AsallverticesinGiareconnected, 5.3.1 saystherecannotexistmorethanonevertex-disjointpath.SoP1::PkandH1:::Hkwillhaveaonetoonerelation.LetusassumethePsandHswiththesameindexhavearelation.ThisimpliesPiHi. Apropertyof3-tree,anytwonon-adjacentverticeshaveexactly3vertex-disjointpathsbetweenthem.Weknowfrom 5.3.1 thatalinearnon-edgeoredgecannotexistbetweenPi,Pj(i6=jand1i;jk).Sothereshouldexistatleast3vertex-disjointpathsbetweenverticesvi,vj(vifromPiandvjfromPj).From 5.3.1 ,anedgeoralinearnon-edgebetweenHiandHx(i6=xand1xk)cannotexist.SootherthanbetweenverticesofHi,anypathbetweenviandverticesofGcontaineitherv1orv2,therebyrestrictingthenumberofvertex-disjointpathstotwo. 36

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3-2 doesnothave1-sumcomponents,buthavevertexpairswith3vertex-disjointpaths.Accordingto 5.3.2 ,itisthecharacteristicofgraphsthatcannothaveaboundingboxexpressiblebylinearinequalities.Ontheotherhand,incaseofthegraphinFigure 1-2 ,foreveryvertexpair(e)with3vertex-disjointpathsthereexista1-sumcomponentsharingoneofitsvertexwithe.Sothisgraphdoesnothavethecharacteristicsmentionedin 5.3.2 ,whichmeansaboundingboxexpressiblebylinearinequalitiesmayormaynotexist.ButwecanseethatallthedottededgesmatchupwiththecharacteristicsmentionedinTheorem5:2of[ 1 ].Hencethereexistsaboundingboxexpressiblebylinearinequalities. 37

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AcongurationofagraphGinddimensionrepresentsGwithuniquesetofdistancevaluesbetweenallofitsverticesmaintainingtheedgevaluesandrealizabilityinddimension.Problemslikeproteinfolding,macromolecularassemblyhaveEDCSrepresentationsin3Danddesiredsolutionfortheseproblemsareusuallyasetoraseriesofcongurations.OnemethodemployedtoobtainthesecongurationsisscanningorsearchingthroughtheCayleycongurationspace.EcientstructurefortheCayleycongurationspaceisnecessaryforsampling.IfG[Fisd-realizableind-dimension,aresultof[ 1 ]guaranteesasquaredconvexstructureforCayleycongurationspace.Obtainingboundsforparametersiscrucialforsampling.Usingthedescriptionofpartial3-trees,weshowedgradationinthecomplexityforcomputingboundsforvarioustypesofgraphs.Thecomplexityrangesfromsolvingasystemoflinearinequalitiestoasystemofcubicinequalities. Toavoidsolvingforsuchhigherdegreepolynomials,rstweprovedthereexistsaconstructionof3-treeforevery2-treesuchthatallthenon-edgesintheresulting3-treehavenon-sharingproperty.Usingthisnon-sharingproperty,wethenprovedthatthereexistsanorderandchoiceofnon-edgesforanypartial2-treesuchthatndingboundsforthecompleteCayleycongurationspaceiseithersolvingasystemoflinearinequalitiesoraquadraticinsinglevariable.Weprovedthisbyensuringallthenon-edgesthathavelinearcharacteristics,F1,arexedbeforeanynon-edgeswithnon-sharingproperty,F2,isxed. Samplingalgorithmaddsedgestothegraphforwhichthecongurationspaceisbeingsampled.Edgesaddedinaparticularorderisbenecialforcomputingboundsforsomenon-edges.Wegaveanexactcharacteristicsfornon-edges,whoseboundsareexpressiblebylinearinequalitiesonadditionofedgesthatpreservethepartial3-tree 38

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Boundingboxisagoodapproximationofcongurationspace.Boundingboxexpressiblebylinearinequalitieshasfasterdescriptionalgorithm.Inchapter4,weprovedanecessarybutnotsucientconditionforpartial2-treesthathavelinearlyexpressibleboundingbox.Amoreinterestingquestionandanopenproblemis,Characterizepartial2-treesGthatcanbecompletedintoa3-treeusingasetofnonedgesS,suchthatforeachuinS,G[uisstilla(partial)2-tree.

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[1] M.SitharamandH.Gao,\Characterizinggraphswithconvexandconnectedcongurationspaces,"DiscreteandComputationalGeometry,2008. [2] L.M.Blumenthal,TheoryandApplicationsofDistanceGeometry.ChelseaPublishingCompany,1970. [3] M.BelkandR.Connelly,\Realizabilityofgraphs,"DiscreteandComputationalGeometry,vol.37,pp.125{137,2007. [4] J.E.Graver,CountingonFrameworks:MathematicstoAidtheDesignofRigidStructures.TheMathematicalAssociationofAmerica,2001. 40

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UgandharChittamuruwasbornintoanagriculturalfamilyinVenkannapalem,Nellore(District),India.HegrewupinandaroundGudur,asmalltowninNellore(District),untilhegraduatedfromhighschool.Hewentontopursueabachelor'sincomputersciencefromVelloreInstituteofTechnology,Indiaandgraduatedin2005.LaterheworkedinCognizantTechnologySolutionsbetween2005and2007beforepursuingmaster'satUniversityofFlorida,Gainesville.Duringmaster'sheheldapart-timewebdeveloperpositioninGeomaticsdepartmentandFloridaMuseumofNaturalHistory.Hewillbegraduatingwithmaster'sdegreeincomputerengineeringinDecember2010. 41