1. Orthogonal Drawings of Series-Parallel Graphs Joint work with Xiao Zhou by Tohoku University Takao Nishizeki

2. Orthogonal Drawings of Series-Parallel Graphs with Minimum Bends Xiao Zhou and Takao Nishizeki by Tohoku University 1.each vertex is mapped to a point 2.each edge is drawn as a sequence of alternate horizontal and vertical line segments 3.any two edges don’t cross except at their common ends planar graph orthogonal drawings

3. Orthogonal Drawings of Series-Parallel Graphs with Minimum Bends Xiao Zhou and Takao Nishizeki by Tohoku University 1.each vertex is mapped to a point 2.each edge is drawn as a sequence of alternate horizontal and vertical line segments 3.any two edges don’t cross except at their common ends planar graphorthogonal drawings bend one bend

4. Orthogonal Drawings of Series-Parallel Graphs with Minimum Bends Xiao Zhou and Takao Nishizeki by Tohoku University 1.each vertex is mapped to a point 2.each edge is drawn as a sequence of alternate horizontal and vertical line segments 3.any two edges don’t cross except at their common ends planar graphorthogonal drawings crossing

5. Orthogonal Drawings of Series-Parallel Graphs with Minimum Bends Xiao Zhou and Takao Nishizeki by Tohoku University 1.each vertex is mapped to a point 2.each edge is drawn as a sequence of alternate horizontal and vertical line segments 3.any two edges don’t cross except at their common ends planar graphorthogonal drawings one bendno bend another embedding bend

6. An orthogonal drawing of a planar graph G is optimal if it has the minimum # of bends among all possible orthogonal drawings of G. orthogonal drawings one bendno bend Optimal orthogonal drawing optimal planar graph bend via-hole, through-hole VLSI design

7. Example 4 bends Orthogonal Drawings drawing bend

8. Example 4 bends Orthogonal Drawings drawing bend Optimal ? No

9. Example embedding Orthogonal Drawings drawing bend flip 4 bends

10. Example embedding flip 4 bends Orthogonal Drawings drawing bend

11. Example embedding flip 4 bends Orthogonal Drawings drawing bend

12. Example embedding flip 4 bends Orthogonal Drawings drawing bend

13. Example embedding 4 bends Orthogonal Drawings drawing bend

14. Example drawing embedding 0 bend optimal 4 bends Orthogonal Drawings drawing bend

15. Orthogonal Drawings drawing bend 4 bends Example drawing embedding 0 bend optimal Given a planar graph Wish to find an optimal orthogonal drawing Find an optimal orthogonal drawing of a given planar graph. Problem

16. An orthogonal drawing of a planar graph G is optimal if it has the minimum # of bends among all possible orthogonal drawings of G. Optimal orthogonal drawing Find an optimal orthogonal drawing of a given planar graph. Problem

17. where degree of vertex v: # of edges incident to v Δ ： max degree Δ =3, n=7 Known results The problem: NP-complete for planar graphs of Δ ≦ 4 A. Garg, R. Tamassia, 2001 If Δ ≧ 5, then no orthogonal drawing. d(v)=2 d(v)=3 2 2 2 2 3 2 3

18. Known results The problem: NP-complete for planar graphs of Δ ≦ 4 A. Garg, R. Tamassia, 2001 D. Battista, et al. 1998 If Δ ≦ 3, O( n 5 log n ) time where n: # of vertices Δ =3, n=7

19. Known results The problem: NP-complete for planar graphs of Δ ≦ 4 A. Garg, R. Tamassia, 2001 If Δ ≦ 3, O( n 5 log n ) time D. Battista, et al. 1998 If Δ ≦ 4, O( n 4 ) time If Δ ≦ 3, O( n 3 ) time D. Battista, et al. 1998 For biconnected series-parallel graphs

20. If Δ ≦ 4, O( n 4 ) time If Δ ≦ 3, O( n 3 ) time D. Battista, et al. 1998 For biconnected series-parallel graphs Connected graphs ?

21. If Δ ≦ 4, O( n 4 ) time If Δ ≦ 3, O( n 3 ) time D. Battista, et al. 1998 For biconnected series-parallel graphs Connected graphsNon-connected graphs

22. Connected graphsNon-connected graphs Biconnected graphs

23. Known results The problem: NP-complete for planar graphs of Δ ≦ 4 A. Garg, R. Tamassia, 2001 If Δ ≦ 3, O( n 5 log n ) time D. Battista, et al. 1998 If Δ ≦ 4, O( n 4 ) time If Δ ≦ 3, O( n 3 ) time D. Battista, et al. 1998 For biconnected series-parallel graphs

24. Our results If Δ ≦ 3, O( n ) time our result For series-parallel (multiple) graphs Non-connectedConnected

25. Our results If Δ ≦ 3, O( n ) time our result much simpler and faster than the known algorithms For series-parallel graphs

26. Series-Parallel Graphs (a) if are SP graphs, is a SP graph. then are SP graphs A SP graph is recursively defined as follows: (b) a single edge

27. Series-Parallel Graphs (a) if are SP graphs, is a SP graph. then are SP graphs A SP graph is recursively defined as follows: (b) a single edge

28. Series-Parallel Graphs (a) if are SP graphs, is a SP graph. then are SP graphs A SP graph is recursively defined as follows: (b) a single edge

29. parallel-connection Series-Parallel Graphs (a) if are SP graphs, is a SP graph. then are SP graphs A SP graph is recursively defined as follows: (b) a single edge

30. series-connectionparallel-connection Series-Parallel Graphs (a) if are SP graphs, is a SP graph. then are SP graphs A SP graph is recursively defined as follows: (b) a single edge

31. parallel-connectionseries-connection Series-Parallel Graphs (a) if are SP graphs, is a SP graph. then are SP graphs A SP graph is recursively defined as follows: (b) a single edge

32. Series-Parallel Graphs (a) if are SP graphs, is a SP graph. then are SP graphs A SP graph is recursively defined as follows: (b) a single edge

33. Series-Parallel Graphs Example SP graph

34. Series-Parallel Graphs Example SP graph series-connection

35. Series-Parallel Graphs Example SP graph parallel-connection

36. Series-Parallel Graphs Example SP graph Series-connection

37. Series-Parallel Graphs Example SP graph Parallel-connection

38. Series-Parallel Graphs Example Biconnected SP graphs SP graph biconnected Biconnected graphs G : G – v is connected for each vertex v. vv

39. Series-Parallel Graphs Example SP graph biconnected Biconnected graphs G : G – v is connected for each vertex v. not biconnected v Biconnected SP graphs cut vertex

40. Series-Parallel Graphs Example SP graphs biconnected

41. Series-Parallel Graphs (a) if are SP graphs, is a SP graph. then are SP graphs A SP graph is recursively defined as follows: (b) a single edge

42. (Our Main Idea) Lemma 1 Every biconnected SP graph G of Δ ≦ 3 has one of the following three substructures: (a) a diamond C (b) two adjacent vertices u and v s.t. d ( u )= d ( v )=2 (c) a triangle K 3. (a) u v (b) K3K3 (c) 2 3 2 3 22 2 3 3

43. (Our Main Idea) Lemma 1 Every biconnected SP graph G of Δ ≦ 3 has one of the following three substructures: (a) a diamond C (b) two adjacent vertices u and v s.t. d ( u )= d ( v )=2 (c) a triangle K 3. (a) u v K3K3 (b)(c) 2 3 2 3 22 2 3 3

44. Algorithm( G ) Let G be a biconnected SP graph of Δ ≦ 3. Case (a): ∃ a diamond, Recursively find an optimal drawing. Case (b): ∃ Decompose to smaller subgraphs in series or parallel, and iteratively find an optimal drawing Case (c): ∃ Similar as Case (b).

45. Let G be a biconnected SP graph of Δ ≦ 3. Case (a): ∃ a diamond, Algorithm( G ) Algorithm( ), Recur G to a smaller one find optimal contract Return expand

46. Let G be a biconnected SP graph of Δ ≦ 3. Case (a): ∃ a diamond, Algorithm( G ) Algorithm( ), find optimal contract Return expand contract Example contract drawing

47. Algorithm( G ) Let G be a biconnected SP graph of Δ ≦ 3. If n ( G )< 6, Then find an optimal drawing of G Else If ∃ a diamond, Then Algorithm( ), Recur G to a smaller one find optimal contract Return

48. Algorithm( G ) Let G be a biconnected SP graph of Δ ≦ 3. If n ( G )< 6, Then find an optimal drawing of G Else If ∃ a diamond, Then Algorithm( ), Example Find an optimal drawing of SP graphs without diamonds

49. Algorithm( G ) Let G be a biconnected SP graph of Δ ≦ 3. If n ( G )< 6, Then find an optimal drawing of G Else If ∃ a diamond, Then Algorithm( ), Example Find an optimal drawing of SP graphs without diamonds

50. delete edge Let G be a biconnected SP graph of Δ ≦ 3. Case (a): ∃ a diamond, Algorithm( G ) Algorithm( ), find contract Return expand Case (b): ∃

51. deg=1 2-legged SP Find an opt & U-shape delete edge Case (b): ∃

52. terminals are drawn on the outer face; the drawing except terminals doesn’t intersect the north side Our Main Idea Definition of I-, L- and U-shaped drawings I-shapeL-shapeU-shape 2-legged SP graph s t U-shape

53. Our Main Idea 2-legged SP graph s t Definition of I-, L- and U-shaped drawings L-shapeU-shape terminals are drawn on the outer face; the drawing except terminals intersects neither the north side nor the south side I-shape

54. terminals are drawn on the outer face; the drawing except terminals intersects neither the north side nor the east side Our Main Idea Definition of I-, L- and U-shaped drawings I-shapeL-shapeU-shape 2-legged SP graph s t L-shape

55. Lemma 2 Every 2-legged SP graph without diamond has optimal I-, L- and U-shaped drawings optimal I-shapeL-shape optimal U-shape optimal

56. Lemma 2 Every 2-legged SP graph without diamond has optimal I-, L- and U-shaped drawings I-shape optimal bend L-shape optimal bend U-shape optimal bend

57. Δ ≦ 3 deg=1 2-legged SP Find an opt & U-shape parallel connection series connection delete edge Case (b): ∃ decompose

58. opt & L-shape opt & U-shape opt & L-shape Find an opt & U-shape delete edge Case (b): ∃

59. opt & U-shape opt & I-shape opt & U-shape opt & I-shape opt & U-shape Find an opt & U-shape delete edge Case (b): ∃

60. opt & U-shape opt & I-shape opt & U-shape opt & I-shape Find an opt & U-shape delete edge Return an optimal drawing extend Case (b): ∃

61. Example How to find an optimal U-shaped drawing ? Else If ∃ Then

62. Example Recursively find L-shaped drawings series connection U-shape

63. Example parallel connection recursively L-shapeI-shapeU-shape I-shapeU-shape

64. Example

65. Algorithm( ), Algorithm( G ) u v Case (c): ∃ a complete graph K 3. opt & U-shape Return optimal drawing bend expand Let G be a biconnected SP graph of Δ ≦ 3. Case (a): ∃ a diamond, contraction Case (b): ∃

66. Algorithm( G ) Let G be a biconnected SP graph of Δ ≦ 3. If n ( G )< 6, Then find an optimal drawing of G Else If ∃ a diamond, Then Algorithm( ), Else If ∃ two adjacent vertices u, v s.t. d ( u )= d ( v )=2 Then u v Else ∃ a complete graph K 3. Find an optimal drawing of SP graphs without diamonds

67. Algorithm( G ) Let G be a biconnected SP graph of Δ ≦ 3. If n ( G )< 6, Then find an optimal drawing of G Else If ∃ a diamond, Then Algorithm( ), Else If ∃ two adjacent vertices u, v s.t. d ( u )= d ( v )=2 Then u v Else ∃ a complete graph K 3.

68. An optimal orthogonal drawing of a biconnected SP graph G of Δ ≦ 3 can be found in linear time. multiple edges Our algorithm works well even if G has multiple edges Theorem 1 Conclusions bend

69. An optimal orthogonal drawing of a biconnected SP graph G of Δ ≦ 3 can be found in linear time. Theorem 1

70. An optimal orthogonal drawing of a biconnected SP graph G of Δ ≦ 3 can be found in linear time. Theorem 1 Our algorithm works well even if G is not biconnected. find three types of drawings best one of Optimal drawing = bend 4 bends 2 bends 3 bends

71. An optimal orthogonal drawing of a SP graph G of Δ ≦ 3 can be found in linear time. Theorem 1 Conclusions

72. bend( G ) ≦ n /3 for biconnected SP graphs G of Δ ≦ 3 Grid size ≦ 8 n /9 =width + height width height

74. Optimal orthogonal drawings ? Optimal orthogonal drawing planar graph

75. a 2-bend orthogonal drawing Optimal orthogonal drawing 1-connected SP graph not optimal one bend bend not optimal one bend bend Is this optimal ? ∃ a one-bend orthogonal drawing ?

76. Optimal orthogonal drawing one bend no bend optimalnot optimal bend ∃ a one-bend orthogonal drawing ? 1-connected SP graph

77. Yes one bend no bend optimalnot optimal bend ∃ a one-bend orthogonal drawing ? Is this optimal ? ∃ a 0-bend orthogonal drawing ?

78. orthogonal drawings Optimal orthogonal drawing 0 bend optimal ∃ a 0-bend orthogonal drawing ? 1-connected SP graph

79. 0 bend Optimal orthogonal drawing optimal 0 bend optimal ∃ a 0-bend orthogonal drawing ? orthogonal drawings 1-connected SP graph

80. Optimal orthogonal drawing No optimal no bend optimal no bend crossing ∃ a 0-bend orthogonal drawing ? 1-connected SP graph

81. one bend optimal bend two bends bend 1-connected SP graph

82. An optimal orthogonal drawing of a biconnected SP graph G of Δ ≦ 3 can be found in linear time. Theorem 1 Conclusions Our algorithm works well even if G is not biconnected. min # of bends min # of bends = 1-connected SP graph

83. Conclusions bend( G ) ≦ n /3 for biconnected SP graphs G of Δ ≦ 3 Grid size ≦ 8 n /9 =width + height width height

84. Conclusions bend( G ) ≦ ( n +4)/3for SP graphs G of Δ ≦ 3 Our algorithm works well even if G is not biconnected. Best possible bend( G )= ( n – 8)/3 + 4 = ( n +4)/3 bend

85. Is there an O( n )-time algorithm to find an optimal orthogonal drawing of G ? For series-parallel graphs G with Δ =4 open s t s t s t s t s t s t optimal I-shape optimal L-shape no optimal U-shape

86. Optimal drawing

87. Our Main Idea 2-legged SP graph A SP graph G is 2-legged if n ( G ) ≧ 3 and d ( s )= d ( t )=1 for the terminals s and t. s t

88. Our Main Idea 2-legged SP graph A SP graph G is 2-legged if n ( G ) ≧ 3 and d ( s )= d ( t )=1 for the terminals s and t. s t Definition of I-, L- and U-shaped drawings L-shapeU-shape terminals are drawn on the outer face; the drawing except terminals intersects neither the north side nor the south side I-shape

89. terminals are drawn on the outer face; the drawing except terminals intersects neither the north side nor the east side Our Main Idea 2-legged SP graph A SP graph G is 2-legged if n ( G ) ≧ 3 and d ( s )= d ( t )=1 for the terminals s and t. Definition of I-, L- and U-shaped drawings I-shapeL-shapeU-shape s L-shape t

90. terminals are drawn on the outer face; the drawing except terminals doesn’t intersect the north side Our Main Idea 2-legged SP graph A SP graph G is 2-legged if n ( G ) ≧ 3 and d ( s )= d ( t )=1 for the terminals s and t. s t Definition of I-, L- and U-shaped drawings I-shapeL-shapeU-shape t not U-shapeU-shape

91. Lemma 2 The following (a) and (b) hold for a 2-legged SP graph G of Δ ≦ 3 unless G has a diamond: (a) G has three optimal I-, L- and U-shaped drawings (b) such drawings can be found in linear time. optimal U-shaped drawings optimal I-shaped drawings

92. Lemma 2 The following (a) and (b) hold for a 2-legged SP graph G of Δ ≦ 3 unless G has a diamond: (a) G has three optimal I-, L- and U-shaped drawings (b) such drawings can be found in linear time. optimal I-shapeL-shape optimal U-shape optimal

93. Definition of Diamond Graph (a) if are diamond graphs, is a diamond graph. then is a diamond graph A Diamond graph is recursively defined as follows: (b) a path with three vertices

94. Definition of Diamond Graph (a) if are diamond graphs, is a diamond graph. then is a diamond graph A Diamond graph is recursively defined as follows: (b) a path with three edges

95. Diamond Graph (a) is a diamond graph. If and are diamond graphs, (b) then is a diamond graph

96. Diamond Graph (a) is a diamond graph. If and are diamond graphs, (b) then is a diamond graph

97. If G is a diamond graph, then (a) G has both a no-bend I-shaped drawing and a no-bend L-shaped drawing (b) every no-bend drawing is either I-shaped or L-shaped. Lemma 1 I-shape L-shape ∃ no-bend U-shaped drawing ? NO 1-bend U-shaped drawing

98. If G is a diamond graph, then (a) G has both a no-bend I-shaped drawing and a no-bend L-shaped drawing (b) every no-bend drawing is either I-shaped or L-shaped. Lemma 1 no-bend I-shaped drawings

99. If G is a diamond graph, then (a) G has both a no-bend I-shaped drawing and a no-bend L-shaped drawing (b) every no-bend drawing is either I-shaped or L-shaped. I-shaped L-shaped Lemma 1 no-bend I-shaped drawings Such drawings can be found in linear time.

100. Lemma 2 The following (a) and (b) hold for a 2-legged SP graph G of Δ ≦ 3 unless G is a diamond graph (a) G has three optimal I-, L- and U-shaped drawings (b) such drawings can be found in linear time. I-shape L-shape diamond graph no-bend drawing I-shape L-shape not a diamond graph U-shape

101. Lemma 2 The following (a) and (b) hold for a 2-legged SP graph G of Δ ≦ 3 unless G is a diamond graph (a) G has three optimal I-, L- and U-shaped drawings (b) such drawings can be found in linear time. Proof: by an induction on # of vertices. Parallel-connection Series-connection Suppose that the lemma holds true for and

102. Lemma 2 The following (a) and (b) hold for a 2-legged SP graph G of Δ ≦ 3 unless G is a diamond graph (a) G has three optimal I-, L- and U-shaped drawings (b) such drawings can be found in linear time. optimal U-shaped drawings optimal I-shaped drawings

103. Lemma 2 The following (a) and (b) hold for a 2-legged SP graph G of Δ ≦ 3 unless G is a diamond graph (a) G has three optimal I-, L- and U-shaped drawings (b) such drawings can be found in linear time. Proof: by an induction on # of vertices. Parallel-connection Series-connection Suppose that the lemma holds true for and

104. Known results In the fixed embedding setting: Min-cost flow problem n : # of vertices For plane graph: O( n 2 log n ) time R. Tamassia, 1987

105. Known results In the fixed embedding setting: improved n : # of vertices For plane graph: O( n 2 log n ) time R. Tamassia, 1987 O( n 7/4 log n ) time A. Garg, R. Tamassia, 1997

106. Known results In the fixed embedding setting: For plane graph: O( n 2 log n ) time R. Tamassia, 1987 O( n 7/4 log n ) time A. Garg, R. Tamassia, 1997 In the variable embedding setting: NP-complete for planar graphs of Δ ≦ 4 A. Garg, R. Tamassia, 2001

107. Known results In the fixed embedding setting: For plane graph: O( n 2 log n ) time R. Tamassia, 1987 O( n 7/4 log n ) time A. Garg, R. Tamassia, 1997 In the variable embedding setting: NP-complete for planar graphs of Δ ≦ 4 A. Garg, R. Tamassia, 2001 If Δ ≦ 3, O( n 5 log n ) time D. Battista, et al. 1998

108. Lemma 3 Every biconnected SP graph G of Δ ≦ 3 has one of the following three substructures: (a) a diamond C (b) two adjacent vertices u and v s.t. d ( u )= d ( v )=2 (c) a complete graph K 3. (a) G

109. G Lemma 1 Every biconnected SP graph G of Δ ≦ 3 has one of the following three substructures: (a) a diamond (b) two adjacent vertices u and v s.t. d ( u )= d ( v )=2 (c) a complete graph K 3. G G G bend( G )=bend( G )bend( G )=bend( G )=0 (Our Main Idea)

110. Proof bend( G )=bend( G ) G G

111. Proof G G bend( G ) ≦ bend( G ) Given an optimal drawing of G

112. Proof G G bend( G ) ≧ bend( G ) Given an optimal drawing of G Case 1: bend( ) = 0 Case 2: bend( ) = 1 Case 3: bend( ) ≧ 2 omitted

113. Lemma 1 Every biconnected SP graph G of Δ ≦ 3 has one of the following three substructures: (a) a diamond (b) two adjacent vertices u and v s.t. d ( u )= d ( v )=2 (c) a complete graph K 3. u v G G G bend( G )=bend( G ) (Our Main Idea) No diamonds ∃ an optimal U- shape drawing

114. G Lemma 2 For each SP graph G If n ≧ 4, Δ ≦ 3 and d(s)=d(t)=1, then ∃ an optimal U-shape drawing u v G G bend( G )=bend( G ) No diamonds ∃ an optimal U- shape drawing s t optimal U-shaped drawing s t G

115. terminals are drawn on the outer face; the drawing except terminals doesn’t intersect the north side Our Main Idea 2-legged SP graph A SP graph G is 2-legged if n ( G ) ≧ 3 and d ( s )= d ( t )=1 for the terminals s and t. s t Definition of I-, L- and U-shaped drawings I-shapeL-shapeU-shape

116. Lemma 1 Every biconnected SP graph G of Δ ≦ 3 has one of the following three substructures: (a) a diamond (b) two adjacent vertices u and v s.t. d ( u )= d ( v )=2 (c) a complete graph K 3. K3K3 G G bend( G )=bend( G )+1 (Our Main Idea) No diamonds G ∃ an optimal U- shape drawing

117. Lemma 2 K3K3 G G G bend( G )=bend( G )+1 No diamonds G ∃ an optimal U- shape drawing For each SP graph G If n ≧ 4, Δ ≦ 3 and d(s)=d(t)=1, then ∃ an optimal U-shape drawing s t optimal U-shaped drawing s t

118. Optimal orthogonal drawings ? Optimal orthogonal drawing 1-connected planar graphnon

119. deg=1 2-leg SP Find an opt & U-shape U-shape not U-shapeU-shape parallel connection series connection delete edge Case (b): ∃