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## CSE 6405 Graph Drawing

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**8**7 5 6 4 3 2 1 CSE 6405 Graph Drawing**Text Books**• T. Nishizeki and M. S. Rahman, Planar Graph Drawing, World Scientific, Singapore, 2004. • G. Di Battista, P. Eades, R. Tamassia, I. G. Tollies, Graph Drawing: Algorithms for the visualization of Graphs, Prentice-Hall Inc., 1999.**Marks Distribution**• Attendance 10 • Participation in Class Discussions 5 • Presentation 20 • Review Report/Survey Report/ Slide Prepration 10 • Examination 55**Presentation**A paper (or a chapter of a book) from the area of Graph Drawing will be assigned to you. You have to read, understand and present the paper. Use PowerPoint slides for presentation.**Presentation Format**• Problem definition • Results of the paper • Contribution of the paper in respect to previous results • Algorithm and methodology including outline of the proofs • Future works, open problems and your idea**Presentation Schedule**• Presentation time: 25 minutes • Presentation will start from 5th week.**Graphs and Graph Drawings**STATION STATION STATION STATION STATION ATM-HUB ATM-RT ATM-RT ATM-SW STATION STATION ATM-HUB ATM-SW ATM-RT ATM-SW STATION STATION ATM-HUB STATION ATM-SW ATM-SW STATION STATION ATM-HUB ATM-HUB ATM-RT STATION STATION STATION STATION STATION STATION A diagram of a computer network**Symmetric**Eades, Hong Objectives of Graph Drawings Nice drawing structure of the graph is easy to understand structure of the graph is difficult to understand To obtain a nice representation of a graph so that the structure of the graph is easily understandable.**8**7 5 6 4 3 2 1 Objectives of Graph Drawings Diagram of an electronic circuit 5 Wire crossings 7 4 8 3 1 2 not suitable for single layered PCB suitable for single layered PCB The drawing should satisfy some criterion arising from the application point of view.**Drawing Styles**Planar Drawing A drawing of a graph is planar if no two edges intersect in the drawing. It is preferable to find a planar drawing of a graph if the graph has such a drawing. Unfortunately not all graphs admit planar drawings. A graph which admits a planar drawing is called a planar graph.**Polyline Drawing**A polyline drawing is a drawing of a graph in which each edge of the graph is represented by a polygonal chain.**Straight Line Drawing**Plane graph**Straight Line Drawing**Straight line drawing Plane graph**Straight Line Drawing**Straight line drawing Plane graph Each vertex is drawn as a point.**Straight Line Drawing**Straight line drawing Plane graph Each vertex is drawn as a point. Each edge is drawn as a single straight line segment.**Every plane graph has a straight line drawing.**Wagner ’36 Fary ’48 Straight Line Drawing Polynomial-time algorithm Straight line drawing Plane graph Each vertex is drawn as a point. Each edge is drawn as a single straight line segment.**Box-orthogonal Drawing**Orthogonal drawing Rectangular Drawing Box-rectangular Drawing**A 46**F 8 G 19 B 65 E 23 I 12 J 14 H 37 K 27 D 56 C 11 Octagonal drawing**Grid Drawing**• When the embedding has to be drawn on a raster device, real vertex coordinates have to be mapped to integer grid points, and there is no guarantee that a correct embedding will be obtained after rounding. • Many vertices may be concentrated in a small region of the drawing. Thus the embedding may be messy, and line intersections may not be detected. • One cannot compare area requirement for two or more different drawings using real number arithmetic, since any drawing can be fitted in any small area using magnification.**Visibility drawing**A visibility drawing of a plane graph G is a drawing of G where each vertex is drawn as a horizontal line segment and each edge is drawn as a vertical line segment. The vertical line segment representing an edge must connect points on the horizontal line segments representing the end vertices.**A 2-visibility drawing**A 2-visibility drawing is a generalization of a visibility drawing where vertices are drawn as boxes and edges are drawn as either a horizontal line segment or a vertical line segment**Properties of graph drawing**Area. A drawing is useless if it is unreadable. If the used area of the drawing is large, then we have to use many pages, or we must decrease resolution, so either way the drawing becomes unreadable. Therefore one major objective is to ensure a small area. Small drawing area is also preferable in application domains like VLSI floorplanning. Aspect Ratio. Aspect ratiois defined as the ratio of the length of the longest side to the length of the shortest side of the smallest rectangle which encloses the drawing.**Bends. At a bend, the polyline drawing of an edge changes**direction, and hence a bend on an edge increases the difficulties of following the course of the edge. For this reason, both the total number of bends and the number of bends per edge should be kept small. Crossings. Every crossing of edges bears the potential of confusion, and therefore the number of crossings should be kept small. Shape of Faces. If every face has a regular shape in a drawing, the drawing looks nice. For VLSI floorplanning, it is desirable that each face is drawn as a rectangle.**Symmetry. Symmetry is an important aesthetic criteria in**graph drawing. A symmetryof a two-dimensional figure is an isometry of the plane that fixes the figure. Angular Resolution. Angular resolution is measured by the smallest angle between adjacent edges in a drawing. Higher angular resolution is desirable for displaying a drawing on a raster device.**Applications of Graph Drawing**Floorplanning VLSI Layout Circuit Schematics Simulating molecular structures Data Mining Etc…..**VLSI Floorplanning**B A F E C G D Interconnection graph**VLSI Floorplanning**B B A A F F E E C G C G D D VLSI floorplan Interconnection graph**VLSI Floorplanning**B B A A F F E E C G C G D D VLSI floorplan Interconnection graph**VLSI Floorplanning**B B A A F F E E C G C G D D VLSI floorplan Interconnection graph**B**A F E G C D VLSI Floorplanning B B A A F F E E C G C G D D VLSI floorplan Interconnection graph Dual-like graph**B**B A A F F E E G G C C D D VLSI Floorplanning B B A A F F E E C G C G D D VLSI floorplan Interconnection graph Dual-like graph Add four corners**B**B A A F F E E G G C C D D VLSI Floorplanning B B A A F F E Rectangular drawing E C G C G D D VLSI floorplan Interconnection graph Dual-like graph Add four corners**Rectangular Drawings**Plane graph G of Input**Rectangular Drawings**corner Rectangular drawing of G Plane graph G of Output Input**Rectangular Drawings**corner Rectangular drawing of G Plane graph G of Output Input Each vertex is drawn as a point.**Rectangular Drawings**corner Rectangular drawing of G Plane graph G of Output Input Each vertex is drawn as a point. Each edge is drawn as a horizontal or a vertical line segment.**Rectangular Drawings**corner Rectangular drawing of G Plane graph G of Output Input Each vertex is drawn as a point. Each edge is drawn as a horizontal or a vertical line segment. Each face is drawn as a rectangle.**VLSI Floorplanning**B B A A F F E Rectangular drawing E C G C G D D VLSI floorplan Interconnection graph**VLSI Floorplanning**B B A A F F E Rectangular drawing E C G C G D D VLSI floorplan Interconnection graph Unwanted adjacency Not desirable for MCM floorplanning and for some architectural floorplanning.**B**B A F A F E G C E C G D D MCM Floorplanning Sherwani Architectural Floorplanning Munemoto, Katoh, Imamura Interconnection graph**B**B A F A F E G C E C G D D MCM Floorplanning Architectural Floorplanning Interconnection graph**B**A F G E C D B B A F A F E G C E C G D D MCM Floorplanning Architectural Floorplanning Interconnection graph Dual-like graph**B**A F G E C D B B A F A F E G C E C G D D MCM Floorplanning Architectural Floorplanning Interconnection graph Dual-like graph**B**B A A F F G G E E C C D D B B A F A F E G C E C G D D MCM Floorplanning Architectural Floorplanning Interconnection graph Dual-like graph**B**B A A F F G G E E C C D D Box-Rectangular drawing B B A F A F E dead space G C E C G D D MCM Floorplanning Architectural Floorplanning Interconnection graph Dual-like graph**Applications**Entity-relationship diagrams Flow diagrams