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Drawing Plane Graphs. Takao Nishizeki. Tohoku University. US President. California Governor. US President. California Governor. What is the common feature?. STATION. STATION. STATION. STATION. STATION. ATM-HUB. ATM-RT. ATM-RT. ATM-SW. STATION. STATION. TPDDI. ATM-HUB. ATM-SW.

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Drawing Plane Graphs


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    1. Drawing Plane Graphs Takao Nishizeki Tohoku University

    2. US President California Governor

    3. US President California Governor What is the common feature?

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

    5. 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.

    6. Modern beauty Objectives of Graph Drawings Nice drawing Ancient beauty

    7. 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.

    8. Drawings of Plane Graphs Straight line drawing Convex drawing

    9. Drawings of Plane Graphs Rectangular drawing Box-rectangular drawing Orthogonal drawing

    10. Book Planar Graph Drawing by Takao Nishizeki Md. Saidur Rahman http://www.nishizeki.ecei.tohoku.ac.jp/nszk/saidur/gdbook.html

    11. Straight Line Drawing Plane graph

    12. Straight Line Drawing Straight line drawing Plane graph

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

    14. 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.

    15. 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.

    16. W H Straight Line Drawing W H Straight line drawing Plane graph

    17. Straight Line Grid Drawing Straight line grid drawing. Plane graph In a straight line grid drawing each vertex is drawn on a grid point.

    18. Wagner ’36 Fary ’48 Grid-size is not polynomial of the number of vertices n Straight line grid drawing. Plane graph

    19. Straight Line Grid Drawing Straight line grid drawing. Plane graph de Fraysseix et al.’90

    20. Chrobak and Payne ’95 Linear algorithm Straight line grid drawing. Plane graph de Fraysseix et al.’90

    21. Schnyder ’90 H n-2 n-2 W Upper bound

    22. What is the minimum size of a grid required for a straight line drawing?

    23. Lower Bound

    24. A restricted class of plane graphs may have more compact grid drawing. Triangulated plane graph 3-connected graph

    25. 4-connected ? disconnected not 4-connected

    26. How much area is required for 4-connected plane graphs?

    27. Straight line grid drawing Miura et al. ’01 Input: 4-connected plane graph G Output: a straight line grid drawing Grid Size : H <n/2 = Area: W < n/2 =

    28. Area < 1/4 = Miura et al. ’01 Schnyder ’90 4-connected plane graph G plane graph G n-2 H <n/2 = W < n/2 n-2 = . Area=n Area<n /4 . 2 2 =

    29. The algorithm of Miura et al. is best possible

    30. Triangulate all inner faces Step1: find a 4-canonical ordering G” n=18 n-1=17 n=18 n-1=17 16 16 Step2: Divide G into two halvesG’ and G” 15 15 12 12 14 14 10 10 13 13 11 11 9 n/2=9 8 8 5 5 6 6 7 7 4 4 Step3 and 4 : Draw G’ and G” in isosceles right-angled triangles 3 3 1 2 1 2 G’ G” W/2 G 1 |slope|>1 |slope|>1 |slope|>1 Step5: Combine the drawings of G’ and G” n/2 W/2 |slope|<1 = G’ W < n/2 -1 n/2 -1 = Main idea G

    31. Draw a graph G on the plane “nicely” Straight line drawing A convex drawing is a straight line drawing where each face is drawn as a convex polygon. Convex drawing

    32. Convex Drawing Convex drawing Tutte 1963 Every 3-connected planar graph has a convex drawing. A necessary and sufficient condition for a plane graph to have a convex drawing. Thomassen ’80

    33. Convex Drawing Chiba et al. ’84 O(n) time algorithm Convex drawing Tutte 1963 Every 3-connected planar graph has a convex drawing A necessary and sufficient condition for a plane graph to have a convex drawing. Thomassen ’80

    34. Convex Grid Drawing Chrobak and Kant ’97 Input: 3-connected graph Output: convex grid drawing n-2 n-2 Grid Size Area

    35. Convex Grid Drawing Miura et al. 2000 Input : 4-connected plane graph Output: Convex grid drawing H Grid Size W Half-perimeter Area

    36. Area < 1/4 = Miura et al. 2000 Chrobak and Kant ’97 3-connected graph 4-connected graph n-2 H n-2 W Area Area

    37. The algorithm of Miura et al. is best possible H W

    38. 21 20 18 19 17 15 12 14 16 11 13 10 9 8 6 7 3 5 4 2 1 21 20 18 19 16 17 11 12 14 15 10 9 7 8 4: Decide y-coordinates 6 13 3 4 5 1 2 Main idea 20 21 19 18 17 14 15 13 10 16 12 11 9 6 8 7 3 4 5 1 2 1: 4-canonical decomposition 2: Find paths O(n)[NRN97] 3: Decide x-coordinates Time complexity: O(n)

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

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

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

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

    43. 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

    44. 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

    45. 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

    46. Rectangular Drawings Plane graph G of Input

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

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

    49. 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.

    50. 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.