CSE 6405 Graph Drawing 1 2 3 4 6 8 5 7. Text Books T. Nishizeki and M. S. Rahman, Planar Graph Drawing, World Scientific, Singapore, 2004. G. Di Battista,

CSE 6405Graph Drawing

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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 definitionResults of the paperContribution of the paper in respect to

previous resultsAlgorithm and methodology including

outline of the proofsFuture works, open problems and your

idea

Presentation Schedule

• Presentation time: 25 minutes

• Presentation will start from 5th week.

Graphs and Graph Drawings

A diagram of a computer network

ATM-SW

ATM-SW

ATM-SW

ATM-SW

ATM-SW

ATM-HUBATM-RT

ATM-RT

ATM-HUB

ATM-HUB

ATM-RT

ATM-RT

ATM-HUB

STATIONSTATION

STATIONSTATION

STATION

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STATIONATM-HUB

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STATION

Objectives of Graph Drawings

To obtain a nice representation of a graph so that the structure of the graph is easily understandable.

structure of the graph is difficult to understand

structure of the graph is easy to understand

Nice drawing

Symmetric Eades, Hong

The drawing should satisfy some criterion arising from the application point of view.

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not suitable for single layered PCB

suitable for single layered PCB

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Objectives of Graph Drawings

Diagram of an electronic circuit

Wire crossings

Drawing Styles

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.

Planar Drawing

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


Plane graphStraight line drawing


Each vertex is drawn as a point.




Each edge is drawn as a single straight line segment.




Each edge is drawn as a single straight line segment.


Every plane graph has a straight line drawing.

Wagner ’36 Fary ’48Polynomial-time algorithm

Convex drawing

Orthogonal drawing Box-orthogonal Drawing

Rectangular Drawing Box-rectangular Drawing

Octagonal drawing

A 46

B 65

C 11D 56

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H 37

G 19

I 12 J 14

K 27

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.

Grid Drawing

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 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

A 2-visibility drawing

Properties of graph drawing

Area. A drawing is useless if it is unreadable. If the used areaof 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 Layout

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VLSI FloorplanningVLSI Floorplanning

Interconnection graph

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Interconnection graph VLSI floorplan

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Dual-like graph

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Dual-like graph Add four corners

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Dual-like graph Add four corners

Rectangular drawing

Rectangular Drawings

Plane graph G of 3

Input


Rectangular drawing of GPlane graph G of 3

Input Output

corner



Each vertex is drawn as a point.Input Output

corner



Each edge is drawn as a horizontal or a vertical line segment.


corner



Each edge is drawn as a horizontal or a vertical line segment.

Each face is drawn as a rectangle.


corner

Not every plane graph has a rectangular drawing.

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Rectangular drawing

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Rectangular drawing

Unwanted adjacency

Not desirable for MCM floorplanning andfor some architectural floorplanning.

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MCM FloorplanningMCM Floorplanning SherwaniSherwani

Architectural FloorplanningArchitectural Floorplanning Munemoto, Katoh, ImamuraMunemoto, Katoh, Imamura

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MCM FloorplanningMCM FloorplanningArchitectural FloorplanningArchitectural Floorplanning

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Dual-like graph

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Dual-like graph

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Dual-like graph

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Box-Rectangular drawing


Dual-like graph

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dead space

Applications

Entity-relationship diagramsFlow diagrams

Applications

Circuit schematics

Minimization of bends reduces the number of “vias” or “throughholes,” and hence reduces VLSI fabrication costs.

A planar graph

planar graph non-planar graph

Planar graphs and plane graphs

An embedding is not fixed.A planar graph may have an exponential number of embeddings.

A plane graph is a planar graph with a fixed embedding.

　 different plane graphs

same planar graph

・・・・

Graph Drawing Data Mining

Internet Computing

Social Sciences

Software Engineering

Information Systems

Homeland Security

Web Searching

Documents

CSE 6405 Graph Drawing 1 2 3 4 6 8 5 7. Text Books T. Nishizeki and M. S. Rahman, Planar Graph Drawing, World Scientific, Singapore, 2004. G. Di Battista,