Graph Drawing Graph Drawing Tutorial Isabel F - PDF document

Graph DrawingGraph Drawing TutorialIsabel F. CruzWorcester Polytechnic InstituteRoberto Tamassia Graph Drawing Graph Drawing models, algorithms, and systems for theapplicationstosoftwareengineering(classhierarchies),knowledgerepresentationhierarchies), Graph Drawing drawing n on the geometricrepresentation of vertices and edges drawing drawing Graph Drawing strong visibilityrepresentationdrawing abc abdec Graph Drawing directed acyclic graphs are usually drawnin such a way that all edges ÒßowÓ in thesame direction, e.g., from left to right, orsuchupwarddrawingsvisualize hierarchical relationships, suchnot every planar acyclic digraph admits aplanar upward drawing Graph Drawingdisplay devices and the human eye have Graph Drawing ¥Theangularresolutionofastraight-line drawing is the smallest angleapplicationsandinthenetworks. ¥A ------ Graph Drawing some drawings are better than others incharacterize readability by means of Graph Drawing Trade-Offstesting upward planarity is NP-harddrawing: min # crossingsmax symmetries Graph Drawing Graph Drawing speciÞc graph (e.g., place Òmostareprovidedasadditionalinput to a graph drawing algorithmthe drawingboundary of the drawingdraw a subgraph with a prescribedkeep a group of vertices ÒcloseÓ together Graph DrawingLayoutofthegraphgeneratedaccording. E.g.User-deÞnedalgorithmic approachE.g., [Tamassia Di Battista Batini 87] Graph DrawingLayout of the graph speciÞed by aLayout generated by theSomenaturalaesthetics(e.g.,planarity)Lack of a powerful language for thefacts, or with a set notation) Graph Drawing BookonGraphDrawingbyG.DiBattista,P.Eades,R. Tamassia, and I. G. Tollis, ISBN 0-13-301615-3,, (available in August 1998).Roberto TamassiaÕs WWW pageTutorial on Graph Drawingand Roberto Tamassia (about 100 pages)Annotated Bibliographyon Graph Drawing(more than 300 entries, up to 1993)Battista, Eades, Tamassia, and Tollis.Geometry: Theory and Applications,(about 8,000 BibTeX entries, including mostpapers on graph drawing, updated quarterly)Proceedings of the Graph Drawing(Springer-Verlag, LNCS)Graph Drawing Chapters Graph Drawing Trees Graph Drawing Drawings of Rooted Treesthe usual drawings of rooted trees are(parents above children)itisdesirabletominimizetheto displaylevel drawings may require Graph DrawingAlgorithm for Binary Treeslevel grid drawingdisplay ofsymmetries and of isomorphicparent in between left and right childparents not always centered on children 123456789 1011 Graph Drawing Algorithm for Binary Treesreesndraw the left subtreedraw the right subtreeplace the drawings of the subtrees atway between the childrenifthereisonlyonechild,placetherootathorizontal distance 1 from the child Graph Drawing for Binary Trees level drawingdisplay of symmetries and of isomorphiccan be extended to draw general rootedtrees (e.g., root is placed at the averagex-coordinate of its children) Graph Drawing Tree Drawing Algorithm minimum width drawingdrawing constructed by the algorithm minimizing the width is NP-hard ifinteger coordinates are required Graph DrawingArea-EfÞcient Drawings of Treesplanar straight-line orthogonal upwardgrid drawing of a binary tree withwing of a binary tree with[Shiloach 76]draw the Graph DrawingArea-EfÞcient Drawings of Treesplanar straight-line upward grid drawingsAVL treesVL trees[Crescenzi Penna Piperno 95] Graph Drawing Area-EfÞcient Drawings of Treesplanar polyline upward grid drawings[Garg Goodrich Tamassia 93] Graph Drawing Drawings of Treesrequirement of planar upward straight-line drawings of treesupwardlevel[RT 83]upwardardupwardardupwardardnon-upwardardnon-upwardleaves-on-hull Graph Drawing of Binary Treesofadrawingisthemaximumofitsdrawings of binary trees:achieved for (nonupward) planar straight-line drawings of binary trees?upward, straight-linelevel[RT 83]upward, polylineard, polylineupward, straight-lineAVL treesVL treesupward, straight-line Graph DrawingDrawings of Binary Trees withrecursivewindingtechnique[CGKT96]:let N be number of nodes in the tree,andN()bethenumberofnodesintheforeachnode,swapchildrentohavepath such that:draw the left subtrees on the path draw recursively the subtrees Graph DrawingRecursive Winding DrawingNlogNO(NlogN T" Graph Drawing Tip-OverDrawingsofRootedTreesTip-over drawings are upward planarorthogonal drawings such that the childrenWidely used in organization charts.Allow to better Þt the drawing in a personnel purchasing Graph Drawing Trees display the parent-child relationship by the inclusion betweenisothetic rectangles.Closely related to tip-over drawings.Used for displaying compound graphs (e.g.,Allow to better Þt the drawing in ainternationaldomesticEurope Australia Western air reservations Graph Drawing Area of Tip-Over and InclusionEades,LinandLin(1992)studyofthearearequirement of tip-over and inclusiondrawings of rooted trees.Thedimensionsofthenodelabelsaregiven of the drawing is:drawing. Graph Drawing How to Draw Free Treescycles and do not represent hierarchicalrelationships (e.g., spanning trees)Leveldrawingsofrootedtreesyieldwith minmax distance from the leaves),which gives a rooted tree T'construct a level drawing)toobtainfromradial drawing Graph Drawing Graph Drawing PlanarDrawingsandEmbeddings is a class oftopologically equivalent planar drawingsof edges around eachvertexbounding eachfacetriconnected planar graphs have a unique Graph Drawing TestingPlanaritytestingandconstructingaplanar[Hopcroft Tarjan 74][de Fraysseix Rosenstiehl 82]ysseix Rosenstiehl 82][Even Tarjan 76]arjan 76][Chiba Nishizeki Ozawa 85] Graph Drawing [Hopcroft Tarjan 74]: planarity testing and[Fary 48, Stein 51, Steinitz 34, Wagner 36]:straight-line drawingPlanar straight-line drawings may need[deFraysseixPachPollack88,Schnyder89,grid drawings can be constructed in O(n) Graph Drawing)-area drawings may have[Garg Tamassia 94]:Trade-off (area vs. angular resolution): 1n Odlogd3ç÷ Wcrn Graph Drawing[Malitz Papakostas 92]: the angularGood angular resolution can be achievedfor special classes of planar graphs:[Malitz Papakostas 92][Garg Tamassia 94][Garg Tamassia 94]can we achieve 7dæö Graph Drawing givenplanargraphofdegree4,wewanttoÞnd a planar orthogonal drawing of G with Graph Drawing [Tamassia 87][Di Battista Liotta Vargiu 93][Tamassia Tollis 89]O(n)-timeapproximationwithO(n)bends[Garg Tamassia 93] Graph Drawing na unit of ßow is a 90° anglena vertex (source) produces 4 unitsna face f (sink) consumes 2 deg(f)- 4units (deg(f) + 4 for the external face)nEdges transport ßow across faces 12 211 Graph Drawing nvertex-face arcs: ßow³ 1, cost = 0222322 111111111 Graph Drawing nface-face arcs: ßow³ 0, cost = 11111112 Graph Drawing 444214 1 2 3 Graph Drawing EulerÕs formuladrawing of Goptimal drawing [Tamassia 87] Computing thenetwork N, and takes Graph Drawing upperboundonthenumberofbendsondrawing can be expressed by settingE.g., we can prescribe a maximum of 2bendsonagivenedgebysettingequalto Graph Drawing A drawing has the minimum number ofclosed curve C such that(#edgescrossedbyCfrom90or180If such a curve exists, ÒrotatingÓ theportion of the drawing inside C reduces Graph Drawing on each facevertices cannot be traversed by CC traverses edge from 270°Þ DFC traverses edge from 90°Þ DFbends removed going ÔÔinwardÕÕ andinserted going ÔÔoutwardÕÕC is a closed curveHence, 234432105 Graph Drawing Visibility Representationpreliminarystepfordrawingalgorithms Graph Drawing FromVisibilityRepresentationsto Graph Drawing 1.Construct visibility representation2.Transform visibility representationinto a preliminary drawing3.Apply bend-stretching transformations4.Compact orthogonal representation Graph Drawing Graph Drawing Upward Planarity Testingupward planarity testing for ordered sets[Kelly 87, Di Battista Tamassia 87]:upward planarity is equivalent tosubgraph inclusion in a planar st-digraph(planaracyclicdigraphwithonesourceand[Kelly 87, Di Battista Tamassia 87]:upward planarity is equivalent to upward Graph DrawingPlanarity TestingestingMannino 91]nO(n2)-time forÞxed embeddingn[Hutton Lubiw 91]nO(n2)-timeforsingle-sourcedigraphsn[Bertolazzi Di Battista ManninoTamassia 93][Garg Tamassia 93] Graph DrawingSince an upward planar digraph is aneedtoknowhowtodrawplanarst-digraphsIfGisaplanarst-digraphwithouttransitiveedges, we can use the right (y)1234567891005792613841001210 012345678910 Graph DrawingUpward,planar,straight-line,O(n)area are displayed Graph DrawingAvariationoftheleft/rightnumberingyieldsdominance drawings withedges and are displayed as bends (upwardplanar polyline drawings) Graph DrawingWe can use the techniques for dominancedrawings also for undirected planar graphs:construct a dominance drawing of G' Graph Drawing Graph DrawingAlgorithmicStrategiesforDrawingif the graph is nonplanar,use one of the drawing algorithms fore.g.,GIOTTO[TamassiaBatiniDiBattista87]use one the drawing algorithms fore.g., Graph Drawing1.try adding the edges one at a time, and2.construct a planar embedding of the3.add the nonplanar edges, one at a time, tothe embedding, minimizing each time thenumberof(shortestpathin Graph DrawingTopological Constraints in thedraw the graph such that the havedraw the graph such the vertices Graph DrawingGIOTTO planarization bend minimization Graph Drawing Graph Drawingin GIOTTOtopological contraints have Graph DrawingPlanarization Techniquestostraight-line,orthogonalandpolyline drawingssupportedbytheoreticalresultsonplanardrawings in practice,alsofor large may alterthe userÕs mental map Graph Drawing nreplace the edges bysprings with unitnatural lengthnconnect nonadjacent vertices withadditional springs with inÞnite naturallengthnrecall that the springs attract theendpoints when stretched, and repel thestate is eventually reached) Graph Drawing Graph DrawingOther Force-Directed Techniques[Kamada Kawai 89]drawing is equal to theirfor each pair of vertices (u,v) use a[Fruchterman Reingold 90]given drawing region acts as wall[Davidson Harel 89]vertex distribution, edge-lengths, andgiven drawing region acts as wall Graph Drawingdrawings of the same graph constructedwith the technique of [Davidson Harel 89] Graph DrawingForce-Directed Techniques of the drawing into theuserÕs mental mapoften able to detect and displayfewtheoreticalresultsonthequalityofthedrawings producedpolyline drawings Graph DrawingConstraints in Force-DirectedTechniqueswe can constrain each vertex to remaincanbesatisÞedprovidedtheycanbeexpressedbymeansofforcese.g, Graph Drawing pindowntheverticesoftheexternalfaceface length(e)]2nequivalent to thebarycentric mapping[Tutte 60]:p(v) = 1/deg(v) S(v,w) Graph Drawing Graph DrawingLayerassignment:assignverticestolayers arrange vertices on each layer improve the drawing withwing with[Sugiyama Tagawa Toda 81] Graph Drawing[Sugiyama Tagawa Toda 81] Graph Drawing Graph Drawing ¥These approaches cover a broad range ofTightly-coupledcannot be separated from each other.separate module from the algorithms module. ¥Most of the approaches are somewhere in between ...Tightly-coupled approaches ¥The algorithms can be optimized for the particular ¥The problem is well-deÞned. ¥Takes an expert to modify the code (difÞcult ¥User has less ßexibility. Graph Drawing ¥Flexible: the user speciÞes the drawing usingconstraints, and the graph drawing module executes ¥Extensible: progressive changes can be made to thespeciÞcation module and to the algorithms module. ¥Potential Òimpedance mismatchÓ between the twomodules. ¥EfÞciency: more difÞcult to guarantee. Graph Drawing ¥Languages for display speciÞcationy speciÞcation ¥IDEAL [Van Wyk 82] ¥TripTrip ¥GVL [Graham & Cordy 90] ¥GrammarsVisual Grammarsisual Grammars ¥Picture Grammars [Golin and Reiss 90] ¥Attribute GrammarsAttribute Grammars§ ¥Layout Graph Grammars [Brandenburg94] [Hickl94] ¥Relational Grammars [Weitzman &Wittenburg 94]Visual Constraints ¥U-term languageU-term language ¥Sketching[Gleicher93][Gross94] Visual Used in GD and Visual Graph Drawing nGraphicalobjectsaredeÞnedbyexample,andhave aConstraints are associated with the classesObject-oriented (message passing,Visual programming language.[Van Wyk 82]Textual speciÞcation of constraints.abstract data types, and adding constraints.Uses complex numbers to specify coordinates..nVisual language to specify the display ofprogram data structures.display of a linked list is the display of thedisplay of the rest of the list. Graph Drawing [Brandenburg 94] [Hickl 94]drawing graphslayout speciÞcation for its productionsby repeated applications of its productions,agraphgrammargenerateslabeledgraphs,which deÞne its graph languageclass of layout graph grammars for whichoptimal graphdrawingsH-tree layouts of complete binary treeshv-drawings of binary treesNFA state transition diagrams from Graph Drawing ¥Production rules use constraints. ¥Terminals are: (e.g., rectangle, circle, text)(e.g., arrow) ¥spatial relationships between objects arein the grammar (e.g., over, left_of) ¥More expressive relationships : ¥Complexity of parsing has been studied. over (rectangle rectangle:(rx,ty)(lx,by) rectangle1rectangle2 Graph Drawing [Weitzman & Wittenburg 93, 94] ¥Generalization of attribute string grammarspositionsin2Dand3D,topologicalconnectivity,Text Text Text Number Image ¥Constraints are solved with DeltaBlue (U. ofWashington) for non-cyclic constraints. (2Text(Author-Of21)) Graph Drawing Visual Grammarsisual Grammars ¥Contex-free grammar. ¥Symbols are visual, and are visually annotated. ¥The interpretation of the visual symbols is left ® *bar-list* textline Graph Drawing ¥BriarBriar Constraint-based drawing program: ¥Direct manipulation drawing techniques. ¥Makes relationships between graphical objects ¥Performance concerns in solving constraints. ¥Spatial Relation PredicatesSpatial Relation Predicates ¥Applicationsincluderetrievalofbuildingsfromanarchitecture database. (CONTAINS BOX CIRCLE)(CONTAINS BOX TRIANGLE)(IMMEDIATELY-RIGHT-OF CIRCLE TRIANGLE) Graph Drawingwingnframework for visualizing abstract objectsand relations.constraint-based object layout system original textual representation Visual Mapping layout library Graph Drawing nlayout-aestheticconcernssubordinatedtoalignment, zoning, symmetry, T-shape,layout task as aoptimalitycriteriaderivedfromlayout- Graph Drawing Visual Graph Drawingisual Graph DrawingCruz, Tamassia Van Hentenryck 93] approach to graph drawing canVisual speciÞcation of layout:theusershouldnothavetoVisualspeciÞcationofaestheticcriteriarepresentations.: the user should not have togive precise geometric speciÞcations. Graph Drawing ¥Visual constraints. ¥Simplicityandgenericityofthebasicconstructs. ¥Ability to specify a variety of displays: graphs, ¥Compatibility with the framework of an object- ¥Recursive visual speciÞcation. GRIDON DEFAULT Vis Lan Graph Drawing EfÞcient Visual Graph Drawing[Cruz Garg 94] [Cruz Garg Tamassia 95]drawing deÞned Òwing deÞned Òna set ofconstraints is generated by theapplication of the visual rules to the inputgraphnvarious types of drawings can be visuallyexpressed in such a way that the resulting, e.g.,drawingsoftrees(upwarddrawings,boxinclusion drawings)drawings of series-parallel digraphs(delta drawings)drawings of planar acyclic digraphs(visibility drawings, upward planarpolyline drawings) Graph Drawing Tree Layout L [h] L , HR) [v]LR R [h] WL + 1 [h]1 [v] VON Vis Lan TREEDEFAUL [root®N:node;left®L:binTree;right®R:binTree] R [v]HL [v] Graph Drawing Characteristics of the Previous TreeLevel DrawingsUpwardDisplay of symmetries.Display of isomorphic subtrees. Graph Drawing [root®N:node;left®L:binTree;right®R:binTree] VON Vis Lan L [h]max(HL , HR) + 1R R [h] 1 [v] R [v]HL [v] Graph Drawing EfÞcientVisualGraphDrawingDrawing ¥RecognizeefÞciently. ¥Devise algorithms and data structures for ¥Trees (upward drawing, box inclusion drawing). ¥Series-parallel digraphs(delta drawing). ¥Planar acyclic digraphs(visibility drawing,upward planar polyline drawing). ¥Next: ¥Extend above results to other classes of graphsand drawings. ¥Constraint viewpoint: framework for evaluatingconstraints efÞciently. ¥Incorporate these algorithms into a declarativegraph drawing system that uses Graph Drawing Series-parallel graphs / delta-drawings[Bertolazzi,Cohen,DiBattista,Tamassia&Tollis, 92] GG1G1G2G2 u acbd ExampleSeriesParallelcompositioncompositionBase case Graph Drawing MWSOURCESINK , U1 [v]1 [v]ME SOURCESINKUUUMWUU G1XY D [v] SOURCE D [h]D [v]D [v]MWMESINKSOURCE SINK X parallel (x,y)sp-digraph (G1) Graph Drawing planar upward drawingvisibility drawingtessellation drawing Graph Drawing Tessellation Drawing TessellationDrawing v: sourceVertex [ leftFace f : face ;rightFace v: sourceVertex f LE RE ORIGIN TessellationDrawing v: vertex [ leftFace f : face ;rightFace v: vertex f LE F-Language g f: face TessellationDrawing RE f: face [: vertex ;bottomVertex F-Language Graph Drawing Tessellation Drawing TessellationDrawing v2 f MWv1 g ME max ( 1,D) [h,v] ® v1: vertex;: vertex; leftFace f: face; rightFaceg: face ] RE Graph Drawing Visibility Drawing v: sourceVertex [ leftFace f : face ;rightFace v: sourceVertex F-LanguageORIGIN fF 0.5 [h] 0.5 [h] v: vertex [ leftFace f : face ;rightFace v: vertex F-Language fF 0.5 [h] 0.5 [h] Graph Drawing Visibility Drawing f: face f: face VisibilityDrawing e:edge VisibilityDrawing v2 f MWv1 g ME max ( 1,D) [h,v] ® v1: vertex;: vertex; leftFace f: face; rightFaceg: face ] RE MN Graph Drawing Upward Polyline Drawing v: sourceVertex [ leftFace f : face ;rightFace v: sourceVertex f LE RE ORIGIN v: vertex [ leftFace f : face ;rightFace v: vertex F-Language LE RE Graph Drawing Upward Polyline Drawing f: face f: face PolylineDrawing F-Language 1 [v] 1 [v]LBUB PolylineDrawing MN v2 f MWv1 g ME max ( 1,D) [h,v] ® v1: vertex;: vertex; leftFace f: face; rightFaceg: face ] RE Graph Drawing ¥New approach, therefore much left toexplore, in particular: ¥New speciÞcation languages. ¥Reducing the Òimpedance mismatch.Ó ¥Design of user interfaces, andapplications. ¥IdentiÞcationoflevelsofcomplexityindrawing graphs (e.g., with graphgrammars, constraint languages). ¥Expressiveness of the speciÞcationlanguages,inparticularofdeclarativeand visual languages. ¥ReÞnement of thehierarchy, so that we can have a trueÒtool boxÓ for the declarative, loosely-coupled approach. Graph Drawing Graph Drawing(Brown University, USA)Roberto TamassiaGDToolkit(University of Passau, Germany)Michael HimsoltGraphViz(AT&T Research)