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# Molecular Modeling BINF 739

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This 33 page Class Notes was uploaded by Nathanael Schowalter on Monday September 28, 2015. The Class Notes belongs to BINF 739 at George Mason University taught by Jeffrey Solka in Fall. Since its upload, it has received 61 views. For similar materials see /class/215253/binf-739-george-mason-university in BioInformatics at George Mason University.

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Date Created: 09/28/15

BINF 739 SPRING 2007 SOLKAWELLER TREES AND SPANNING TREES BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 31 Characterization and Properties of Trees Def Recall a m is a connected graph with no cycles A Not Tree BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 31 Characterization and Propelties of Trees Thmrrm LLB LN I be ngrnpll Willi n wrlu39nt Then hr following statements in rqlllflilrnl l 39l39 l a lrn39 2 I mnmms no rycllx39 nml luau n A l wlgm J l39 is L39o nu lml ml has n A I Agra l I I r39umm lml inn Wen Mg N 1 lllPiil 39 BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 31 Characterization and Propelties N of Trees Table 311 Summary of Properties of a Tree Ton n vertices 739is connected Tcontains no cycles Given any two vertices uand vof 77 there is a unique u vpath Every edge in Tis a cutedge and its deletion results in a subgraph having exactly two components Tcontains n 1 edges Tcontains at least two vertices of degree 1 if n gt 2 Adding an edge between two vertices of Tyields a graph with exactly one cycle BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 31 Characterization and Properties of Trees Recall from Section 14 Def The eccentricity of a vertex v in a graph G denoted eccv is the distance from v to a vertex farthest from V That is eccv maxXEVG d v x BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 31 Characterization and Properties of Trees Recall from Section 14 Def The radius of a graph G denoted radG is the minimum of the vertex eccentricities That is radG minXEVG 60006 Def A central vertex v of a graph Gis a vertex with minimum eccentricity Thus ecc V radG The center of a graph G denoted 26 is the subgraph induced on the set of central vertices of G BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 31 Characterization and Properties of Trees Lemma 319 Let Tbe a tree with at least thress vertices va is a leaf of Tand w is its neighbor then eccv eccw 1 If v is a central vertex of T then degv gt 2 BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 31 Characterization and Properties of Trees Corollary Jordan 1869 Let T be an nvertex tree Then the center ZG is either a sindgle vertex or a single edge BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 31 Characterization and Properties of Trees Tree Isomorphism and Automorphism Remember that graph invariants must be preserved under isomorphism automorphism Center of a graph must map to the center Leaf and the image of a leaf must be the same distance from the center Unequal diameters BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 31 Characterization and Properties of Trees Tree Isomorphism and Automorphism Leaves are all different distances from the center and must be preserved under isomorphism BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 31 Characterization and Properties of Trees Thm 3114 A sequence ltd 112 dngt of n gt 2 positive integers is tree graphic iff id 2n 2 i1 BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 31 Characterization and Properties of Trees Trees as Subgraphs Thm 3115 Let Tbe any tree on n vertices and let Gbe a simple graph such that dm 7 gt nJ Then Tis a subgraph of G BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 32 Rooted Trees Def A directed tree is a digraph whose underlying r E2133 BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 32 Rooted Trees Def A rooted tree is a directed tree having a distinguished vertex r called the root such that for every other vertex 1 there is a directed path from rto v Phylogenetic Tree of Life Bacteria Archaea Eucarya Green Filamentous ba teria Spiro hates Plants Cilia es Planctomyce F Iagellates Pyrodicu39c Bacteroide d Cytuphaga Tnchomona s Thermomga MIC rospond la Aquifex Dlplomonads Spanning Trees 32 Rooted Trees bruMth39si nx BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees I 32 Rooted Trees 97 Virginie 37 g39 A 1 Seiosa E Virginica V Vilgimca PW216 7 PWlt16 7 e 5 Versicniur Virginica 5 Versicolor 0 2 5 5 Peisi mm BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 32 Rooted Trees Root Computer Screen Figure Figure I Uicontrol Children STYLES checkbox i 77777 71 7 7 7 7 iwiiT T T ii mm lumen l i l i i Rectangl Line i Light i Irane 1 i 1 7 1 7 1 llsfbax 7 71 ponunmanu lmasei Surface Patch Text Justbutton Uimenu Uicontextmenu Axes radlnhulton slider text ULI II IJJ JPIIIIB LUUI SolkaWeller Trees and Spanning Trees 32 Rooted Trees Def An mary tree m gt 2 is a rooted tree in which every vertex has m or fewer children Def A complete mary tree is an mary tree in which every internal vertex has m or fewer children BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 32 Rooted Trees 141515 l7lSl9 0212211 2425 2t 7 8 9 3031 13 34 3s 16 17 A complete 3ary tree 7 Weisstein Eric W quotComplete Ternary Treequot From MathWaraLA Wolfram Web Resource L quotwnlfram rnmF y reehtml BINF 739 Spring 2007 SolkaWeller Trees an Spanning Tres 32 Rooted Trees Recursive Property of a Binary Tree If Tis a binary tree of height h then its left and right subtrees both have heights less than or equal to h1 and equality holds for at least one of them LHII39L III r15 BINF 739 Spring 2007 SolkaWeller Trees and Spanning Tres 32 Rooted Trees quotBinary tree structured vector quantization approach to clustering and visualizing microarray dataquot by Sulton et all Bioinformatics Vol 18 no 90001 2002 Pages 5111 8119 l iu m by mum l lusli img reprcscnnng he mumw m m um mum ml mm c Two milerem replesenlulmns ul lliL um igglnmclillll39c clwcnug wit umig dwdrugmm BINF 739 Spring 2007 SolkaIWeller Trees and Spanning Trees 33 BinaryTree Traversals Def The level order of an o ordered tree is a listing of the vertices in the top to a 3 bottom left to right order of a standard plan drawing of 0 that tree Level order traversal yields F B G A D I C E H http en wikipedia orgwikiTreetraversal BINF 739 Spring 2007 SolkaIWeller Trees and Spanning Trees 33 BinaryTree Traversals left preorder traversal o visitnode print nodevalue 0 9 If nodelel c null then visitnodelel c o o o if noderight null then visitnoderight F B A D C E G I H httpenwikipediaorgwikiTreetraversa BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 33 BinaryTree Traversals postorder postfix or reversePolish traversal Similar to the preortler is the post or er where eac root node is visited after all ofits children In both cases values in the left subtree are printed before values in the right subtree visitnode if nodeleft null then visitnodeJeft if noderight null then visitnoderight print nodevalue 0 e 00 519 Postorder LRV traversal yields A C E D B H I F r httpenwikipediaorgwikiTreetra BINF 739 pws 7 SolkaWeller Trees and Spanning Trees 33 BinaryTree Traversals 0 6 GD 0 O Q 0 0 0 Pre abcdf Postab cde BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 33 BinaryTree Traversals Pre ab cde left walk first abc39de Post ab cd e left walk last BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees SPAN NI NG TREES BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees Introduction Why does one wish to compute a spanning tree One needs a strategy to visit every node in a graph Approaches Depth first Breadth first Primms minimum spanning tree algorithm Dijkstra s shortest path method MSTs provide one with a way to analyze the cycle structure of a graph BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees I 41 Tree Growing Terminology For a given tree Tin a graph G the edges and vertices of Tare called tree edges and tree vertices and the edge and vertices of Gthat are not in Tare called non tree edges and non tree vertices BINF 739 Spring 2007 SolkaWeller 7 Trees and panning Trees 41 Tree Growing Def A frontier edge for a given tree Tin a graph is a nontree edge with one edge point in 7 called the tree endpoint and one endpoint not in 7 its nontree endpoint grapllmrlgulelll u word are him nml m m rill i Nu m Munrme BINF 739 Spring 2007 SolkaWeller 7 Trees and Spanning Trees 7 rr mm rmmr m n 41 TreeGrowing Example 411 cm ed m umm m s nun rrvdgzs shuwn m m up hnlnil39xsurrJ 12 v 4 r a Sumuw39 mm m wlnr N39mrlF r 4 n lmlmu YmHnfl39mun a V anim le mun or Admprvxr l n r mlapplyquotupdalrb runhz 39 mnmrm Arlnh hrvurmu39lh addullmnum rmulwr he m 9 uml rmmml mm m wm 1 N m mm mm u Ham mu her main an m m rm BINF 739 Ebrmg 2007 soTkaWeuer 7 Trees and Sparrrung Trees 41 TreeGrowing Mum 011 WehGlnwiug d yap 39 n mum mm r L vIuIJlthun nvllltyl vml rymmlhlvu I n 1 mm um r n r er a m mm m dd dc Wm r rrmlu rm T II mm Im r xmnrk IMMEth wumumrmrnmm m rw l mm ngnllmw My nmn Hr mm my mum u mn nmuun u mun mntr hllhlnuuh rumWm va m m rmquot My Mum MTm Jivmuul BINF 739 Ebrmg 2007 soTkaWeuer 7 Trees and Sparrrung Trees m rm MM m a raw Flgwu 3 1mm Dmllllwd by quotm instance er Tueamwing UM rs s Hr rg We Spanning Trees 42 DepthFirst and BreadthFirst Search Def Let S be the current set of frontier edges The function dfs nextEdge is defined as follows dfs nextEdgeGS selects and returns as its value the frontier edge whose tree ednpoint has the largerst discovery number If there is more than one such edge then dfs nextEdgeGS selects the one determined by the default priority BINF 739 Spring 2007 SolkaWeller 7 Trees and Spanning Trees 42 DepthFirst and BreadthFirst Search Algorithm 421 Depth First Search Input a connected graph G a starting vertex Vin VG Output an ordered spanning tree Tof Gwith root v Initialize tree Tas vertex V Initialize 5as the set of proper edges incident on M While 5 0 Let e d s nexz EdgeG5 Let Wbe the nontree endpoint of edge e Add edge eand vertex Wto tree I updateFronterm 5 Return tree T BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 42 DepthFirst and BreadthFirst Search U1 X3 26 y5 Wm t4 V0 Depthfirst search BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 42 DepthFirst and BreadthFirst Search Recursive DepthFirst Search DF5V While vhas an undiscovered neighbor Let wbe an undiscovered neighbor of V DF5W Return BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 42 Depth First and Breadth First Search A depth first search DFS explores a path all the way to a leaf before backtracking and exploring another path For example after searching A G then B then D the search backtracks and tries another path from B 0 0 0 Node are explored in the order A J L BDEHLMNIOPCFGJKQ N will be found before J of libel o 9 31m 739 Spring 2007 SolkaWeller Trees and Spanning Trees 42 Depth First and Breadth First Search Algorithm 422 BreadthFirst Search Input a connected graph G a staring vertex Vin lg Output an ordered spanning tree 739of Gwith root v Initialize tree Tas vertex V Initialize Sas the set of proper edges incident on V While 5 0 Let e bEnextfdgeKZS Let Wbe th nontree undiscovered endpoints of edge e Add edge sand vertex Wto tree T updateFrontierGS Return tree 7 BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 42 Depth First and Breadth First Search U1 X3 26 y5 Wm t4 V0 Breadthfirst search BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 20 42 Depth First and Breadth First Search A breadth rst search BFS explores nodes nearest the root before exploring nodes further away For example after searching A then B then C the search proceeds with D E F G Node are explored in the order A BCDEFGHIJKLMNOPQ J will be found before N BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 42 Depth First and Breadth First Search How to do breadthfirst searching Put the root node on a queue while queue is not empty remove a node from the queue if node is a goal node return success put all children of node onto the queue return failure Just before starting to explore level n the queue holds althe nodes at level In a typical tree the number of nodes at each level increases exponentallywith the depth Memory requirements may be infeasible When this method succeeds it doesn t give the path 31m 739 Spring 2007 SolkaWeller Trees and Spanning Trees 21 42 Depth First and Breadth First Search How to do depthfirst searching Put the root node on a stack while stack is not empty remove a node from the stack if node is a goal node return success put all children of node onto the stack return failure At each step the stack contains a path of nodes from the root The stack must be large enough to hold the longest possible path that is the maximum depth of search BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 42 Depth First and Breadth First Search A fast algorithm for determining the best combination of local alignments to a query sequencequot Gavin C Conant and Andreas Wagner BMC Bioinformatics 2004 562 Presents an application of DFS to sequence alignment BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 43 Minimum Spanning Trees and Shortest Paths Recall Section 16 a weighted graph is a graph in which each edge is assigned an edgeweight The Minimum Spanning Tree Problem Let G be a connected weighted graph Find a spanning tree of G whose total edgeweight is minimum A Weighted Graph BINF15 9DQrrlhr39637Spannmg Tree SolkaWeller Trees and Spanning Trees 43 Minimum Spanning Trees and Shortest Paths Def Let 5 be the current set of frontier edges The function PrmnextEa ge is defined as follows PrmnextEa geG5 selects and returns as its value the frontier edge with smallest edgeweight If there is more than one such edge then PrmnextEa geG5 selects the one determined by the default priority BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 23 43 Minimum Spanning Trees and Shortest Paths Algorithm 431 Prim s Minimum Spanning Tree Input a weighted connected graph Gand a stalting veltex v Output a minimum spanning tree 739 Initialize tree Tas a vertex v Initialize Sas the set of proper edges incident on v While 5 0 Let e PrmnexbfdgeC75 Let w be the nontree endpoint of edge e Add edge eand vertex wto tree 77 updateFronterC75 Return tree 739 BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 43 Minimum Spanning Trees and Shortest Paths Salulion 521 restln D A15 5 away Bis 9Ei515ard Fl 5 Ollhese 5 is We e an DA BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 24 43 Minimum Spanning Trees and Shortest Paths Fringe 0 Solution seen 591 Descri ptiun F is 6 G 15 the smailEsL so we highlight the vertex F and the arc DF 7awayframAEi515 and C EVE G AVDF highlighied in red The aigarilhm carries an as abuve Vemax B which Is 7 awa Mickie highiigmea Herein arzDB is mm C E G A D F E BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 43 Minimum Spanning Trees and Shortest Paths Mm seen i i i Snlulinn set nesciipizaii Fringe The sigehih hihhiiiihi 1 7 names eh as abnve Venex B which is 7 away hem A is highiightad Here the arc DB is we niishhihiim him QEG AD EB imhiscssewecsricho r Fi h si i 39 meirieihihg vemces have been used use beiweeh c E and G c is a away from B E is 7 away from B and G is ii away hem 39 s h ih AD EB him 26 E BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 25 43 Minimum Spanning Trees and Shortest Paths l Descripiiun quotm Fringe 5quotquotquot seen Hera llia uiilyvaiiices available are C and G C is 5 away from E and G is 9 away from E C is cliuseii so it is H G A D F B Tlin arr Rr W E 0 iii Mama lian ancillan immF Fi mm nweliigliliglililaiid ADFB llieaic EG mm m a null iiiill E C G illiasweiglil39 39 39 H quotSpanning Trees 7 V 43 Minimum Spanning Trees and Shortest Paths Agglomerative Algorithm Bottom Up or Clumping Start Clusters C1 C2 Cn each with 1 data point 1 Find nearest pair Ci Cj merge Ci and Cj delete C and decrement cluster count by 1 If number of clusters is greater than 1 then go back to step 1 BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 26 43 Minimum Spanning Trees and Shortest giaths Intercluster Dissimilarity Choices Furthest Neighbor Complete Linkage Nearest Neighbor Single Linkage V Group Average BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 43 Minimum Spanning Trees and Shortest Paths Dendrograms glomerative 0 1 2 3 1 12 12345 2 345 3 4 45 4 3 2 1 divisive BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 27 43 Minimum Spanning Trees and Shortest Paths Dendrogram ii Ii 2N3 Ms M s HGleE 1011 A Llondlogram an rrpmsnni iho rolturs of Illuarrhlml lllslcring algae rilllms 39 H nlmioll 39 i in iii iiliiidi 0 iiseli oi mane Palms in and x happen in b0 ine most similar and am morgod at ii n lanrmrn R arthDual ekrFl l 1 Ir r up Classi ulion opyriglil mm by loliii Will y amp Sons liir BINF 739 Spring 2007 lkaWeller 7 Trees an Spanning Tree d 43 Minimum Spanning Trees and IL Shortest Paths B All the singlelin kage clusters could be obtained by deleting the edgs of the MST starting from the largst one I I 910731261114523 Adapted from Course Cluster Analysis and Other Unsupervised Learning Meinods Siai 593 E Speakers Rebeccu Nugentli Larissa Sianberryz Department of 1 Siaiisiies 2 Radiology 39 r 39 i l 39 BINF 739 Spring 2007 SolkaWeller 7 Trees an Spanning Tree d 43 Minimum Spanning Trees and Shortest Paths Applications of MSTbased Clustering to 4 Gene Expression Data Ying Xu Victor Olman and Dong Xu Clustering gene expression data using a graphtheoretic approach an application of minimum spanning trees Bionforma cs Vol 18 no 4 pp 536545 2002 Ying Xu Victor Olman and Dong Xu Minimum spanning trees for gene expression clustering Genome Informatics 12 24 33 2001 BINF 739 Spring 2007 SolkaWeller Trees and panning Trees 43 Minimum Spanning Trees and Shortest Paths Def Let Ube a subset of vertices in a connected edgeweighted graph G The Steinertree problem is to find a minimumweight tree subgraph of Gthat contains all the vertices of U Steiner Tree Problems in the Analysis of Biological Networks Nadja Betzler Arbeitsbereich fiir theoretische InformatikFormale Sprachen Feb 2006 BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 29 43 Minimum Spanning Trees and Shortest Paths Finding the Shortest Path Dijkstra s Algorithm Shortest Path Problem Let s and t be two vertices of a connected weighted graph Find a path from s to t whose total edge weight is minimum Le a shortest s t path BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 43 Minimum Spanning Trees and Shortest Paths Dijkstra s Algorithm Input a weighted connected graph G and a starting vertex 5 Output a shortestpath tree T with roots Initialize tree T as vertex 5 Initialize S as the set of proper edges incident on s l While S 0 Let e DijkstranextEdgeGS Let w be the nontree endpoint of edge e Add edge e and vertex w to tree T updateFrontierGS Return tree T BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 43 Minimum Spanning Trees and Shortest Paths Dijkstra s Algorithm ISOMAP Revisited Shortest Path R3 Geodesic and ISOMAP Geodesic and Manifold Geodesic Associated Nearest Neighbor 31Spring 2007 SolkaWeller Trees and Spanning Trees 43 Minimum Spanning Trees and Shortest Paths Dijkstra s Algorithm Step 391 Ennetrntt neighhnrhnntl graph Define the graph G ever all data points he tnnnetting painta i anti j ii at measured by dxliljl they are deter than e elenrnapl nr if i it are at the it neareat neighhnrs all j Klnarnalalw Set edge lengths equal tn atth 2 En rnnnte ehnrteet paths initialize daft dying ii tj are lintell he an edge digtail ethenrrise Then tar each 1ralne nil ll a N in turn replace all entriea d lig lay minld lt dugTM d ltl l The matrix at final helmet DE dGltjjll will tnntain the ehnrtest path detainees hetneen all airs ai nninta G in i9 Censtrnet dtlirnensinnal embedding tet hp he the pth eigenvalue in decreasing Driller all the rnatrin ring if and e he the i th tnrnpnnent ell the pth eigeneeetnr Then Eel the pth campenent all tl ddihtehtitrhal enertlinate eerter y enltlal tn pr BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 47 Matroids and the Greedy Algorithm Kruskal s Algorithm An alternative strategy for the BINF 739 Spring 2007 4 5 SolkaWeller Trees and 6 7 Spanning Trees Using the MST to Help Identify nteresting Associations Minimum Spanning Tree Inter Class Edges for TWO Ellvarlate Normal Samples 3 The blue points come roma bivariate Gaussian distribution with mean 11 and a 2x2 identity covariance matrix The blac points 39 bivariate Gaussian distribution with mean 11 and a 2x2 identity covariance matrixThese cross The red edges are the ones of P 1539 55 W quota imam edges Will be used In 47 39 our sc eme to 39 facilitate the celerated discoverl process 2 SolkaWeller Trees and Spanning Trees 32 A Visualization Framework to Aid in the Identification of Interesting Associations Blue is anthropology and archaeology A Bit of an Indepth Look at The Use of Minimal Spanning Trees in Subspace Clustering J Diggans and J L Solka quotCluster Subspace Identification Via Conditional Entropy Calculationsquot Interface 2004 Computational Biology and Bioinformaticss 36th SYMPOSIUM ON THE INTERFACE May 26 29 2004 BINF 739 Spring 2007 SolkaWeller Trees and Spanning Trees 33

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