1. CSE 6405 Graph Drawing 1 2 3 4 6 8 5 7

2. 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.

3. Marks Distribution Attendance 10 Participation in Class Discussions 5 Presentation 20 Review Report/Survey Report/ Slide Prepration 10 Examination 55

4. 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.

5. 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

6. Presentation Schedule Presentation time: 25 minutes Presentation will start from 5 th week.

7. Graphs and Graph Drawings A diagram of a computer network ATM-SW ATM-HUB ATM-RT ATM-HUB ATM-RT ATM-HUB STATION ATM-HUB STATION

8. 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

9. The drawing should satisfy some criterion arising from the application point of view. 1 2 3 4 5 7 8 not suitable for single layered PCB suitable for single layered PCB 1 2 3 4 6 8 5 7 Objectives of Graph Drawings Diagram of an electronic circuit Wire crossings

10. 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

11. Polyline Drawing A polyline drawing is a drawing of a graph in which each edge of the graph is represented by a polygonal chain.

12. Straight Line Drawing Plane graph

13. Straight Line Drawing Plane graph Straight line drawing

14. Straight Line Drawing Each vertex is drawn as a point. Plane graph Straight line drawing

15. Straight Line Drawing Each vertex is drawn as a point. Each edge is drawn as a single straight line segment. Plane graph Straight line drawing

16. Straight Line Drawing Each vertex is drawn as a point. Each edge is drawn as a single straight line segment. Plane graph Straight line drawing Every plane graph has a straight line drawing. Wagner ’36 Fary ’48 Polynomial-time algorithm

17. Convex drawing

18. Orthogonal drawing Box-orthogonal Drawing Rectangular Drawing Box-rectangular Drawing

19. Octagonal drawing A 46 B 65 C 11 D 56 E 23 F 8 H 37 G 19 I 12J 14 K 27

20. Grid Drawing

21. 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

22. 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.

23. 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

24. 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.

25. 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.

26. 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.

27. Applications of Graph Drawing Floorplanning VLSI Layout Circuit Schematics Simulating molecular structures Data Mining Etc…..

28. VLSI Layout

29. E A B C F G D VLSI Floorplanning Interconnection graph

30. E A B C F G D A B E C F G D VLSI Floorplanning Interconnection graph VLSI floorplan

31. E A B C F G D A B E C F G D VLSI Floorplanning Interconnection graph VLSI floorplan

32. E A B C F G D A B E C F G D VLSI Floorplanning Interconnection graph VLSI floorplan

33. E A B C F G D A B E C F G D VLSI Floorplanning A B E C F G D Interconnection graph VLSI floorplan Dual-like graph

34. E A B C F G D A B E C F G D VLSI Floorplanning A B E C F G D A B E C F G D Interconnection graph VLSI floorplan Dual-like graphAdd four corners

35. E A B C F G D A B E C F G D VLSI Floorplanning A B E C F G D A B E C F G D Interconnection graph VLSI floorplan Dual-like graphAdd four corners Rectangular drawing

36. Rectangular Drawings Plane graph G of Input

37. Rectangular Drawings Rectangular drawing of G Plane graph G of Input Output corner

38. Rectangular Drawings Rectangular drawing of G Plane graph G of Each vertex is drawn as a point. Input Output corner

39. Rectangular Drawings Rectangular drawing of G Plane graph G of Each edge is drawn as a horizontal or a vertical line segment. Each vertex is drawn as a point. Input Output corner

40. Rectangular Drawings Rectangular drawing of G Plane graph G of Each edge is drawn as a horizontal or a vertical line segment. Each face is drawn as a rectangle. Each vertex is drawn as a point. Input Output corner

41. Not every plane graph has a rectangular drawing.

42. E A B C F G D A B E C F G D VLSI Floorplanning Interconnection graph VLSI floorplan Rectangular drawing

43. E A B C F G D A B E C F G D VLSI Floorplanning Interconnection graph VLSI floorplan Rectangular drawing Unwanted adjacency Not desirable for MCM floorplanning and for some architectural floorplanning.

44. A B E C F G D MCM Floorplanning Sherwani Sherwani Architectural Floorplanning Munemoto, Katoh, Imamura Munemoto, Katoh, Imamura E A B C F G D Interconnection graph

45. A B E C F G D MCM Floorplanning Architectural Floorplanning E A B C F G D Interconnection graph

46. MCM Floorplanning Architectural Floorplanning A E B F G C D E A B C F G D Interconnection graph Dual-like graph A B E C F G D

47. MCM Floorplanning Architectural Floorplanning A E B F G C D E A B C F G D Interconnection graph Dual-like graph A B E C F G D

48. MCM Floorplanning Architectural Floorplanning A E B F G C D E A B C F G D A E B F G C D Interconnection graph Dual-like graph A B E C F G D

49. MCM Floorplanning Architectural Floorplanning A E B F G C D E A B C F G D A E B F G C D Box-Rectangular drawing Interconnection graph Dual-like graph A B E C F G D dead space

50. Applications Entity-relationship diagrams Flow diagrams

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

52. A planar graph planar graph non-planar graph

53. 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 ・・・・

54. Graph Drawing Data Mining Internet Computing Social Sciences Software Engineering Information Systems Homeland Security Web Searching