1. Department of Computer Science and Engineering Bangladesh University of Engineering and Technology M. Sc. Engg. Thesis Md. Emran Chowdhury (040805068P) Supervisor: Prof. Dr. Md. Saidur Rahman

2. 2 ▒ Problem Definition Contents ▒ Motivation ▒ Previous Results and Our Results ▒ Upward Point-Set Embedding ▒ Orthogonal Point-Set Embedding ▒ Conclusion and Future Works

3. 3 Point-Set Embedding a c b d f e S a c b d f e G Each vertex is placed at a distinct point Input

4. 4 f c b d a e a c b d f e S G Point-Set Embedding Each vertex is placed at a distinct point Each edge is drawn by straight or poly line Output Bend

5. Upward Point-Set Embedding f c b d a e a c b d f e S G Each edge is drawn upward Input Each vertex is placed at a distinct point Output

6. 6 Each edge is drawn upward Upward Point-Set Embedding a c b d f e G’ S f c b d a e a c b d f e G a c b d f e G S f c b d a e f c b d a e S G’ has no upward point-set embedding on S Not every graph has upward point-set embedding on a fixed point-set

7. 7 1 3 4 2 a c d b Upward Point-Set Embedding with mapping a c d b S G φ

8. 8 a c d b S G a c d b φ a c b d S φ’φ’ No upward point-set embedding with this mapping Upward Point-Set Embedding with mapping Finding upward point-set embedding with mapping is a real challenge

9. 9 ab c d e jk g h f planar graph G point-set in the plane Orthogonal Point-Set Embedding Input

10. 10 ab c d e jk g h f Output j k f h g e c d a b Orthogonal Point-Set Embedding Each edge is drawn as a sequence of vertical and horizontal line segments

11. 11 ▒ Problem Definition Contents ▒ Motivation

12. 12 a b c d e f a b c d f e Interconnection Graph Point-Set Embedding In VLSI design, often the places for the modules are fixed, we have to connect the modules w. r. t. the inter connection graph Motivation

13. 13 a b c d e f Motivation Interconnection Graph VLSI Layout a b c d f e It is always desirable to reduce the number of bends Point-Set Embedding

14. 14 ▒ Problem Definition Contents ▒ Motivation ▒ Previous Results and Our Results

15. 15 Previous Results and Our Results Problem Graph class Authors Results Giordano et. al. ’07 upward point-set embedding Upward planar digraphs at most two bends per edge Giordano, Liotta, and Whiteside ’09 upward point-set embedding with mapping Upward planar digraphs at most 2n-3 bends per edge This Thesis upward point-set embedding with mapping Upward planar digraphs at most n-3 bends per edge upper bound on total number of bends Upward Point-Set Embedding

16. 16 Rahman et. al. ’99 Orthogonal drawing Cubic 3-connected plane graphs bend optimal drawing Rahman and Nishizeki ’02 Orthogonal drawing plane graphs with  ≤ 3 bend optimal drawing Rahman, Nishizeki and Naznin ’03 Orthogonal drawing plane graphs with  ≤ 3 no bend drawing Previous Results and Our Results But, they did not consider the point-set embedding Problem Graph class Authors Results Orthogonal Drawing Time complexity = O(n)

17. 17 Previous Results and Our Results Kaufman and Wiese ’02 Point-set embedding General plane graphs 2 bends per edge One can draw the edge orthogonally But, the size of the vertices may increase Problem Graph class Authors Results Poly-line Point-Set Embedding Time complexity = O(n 2 )

18. 18 This Thesis Orthogonal point-set embedding 3-connected cubic planar graphs at most (5n+4)/2 bends in total Previous Results and Our Results Kaufman and Wiese ’02 Point-set embedding General plane graphs 2 bends per edge One can draw the edge orthogonally This Thesis Orthogonal point-set embedding with mapping 4-connected planar graphs at most 6n bends in total Tight upper bound Problem Graph class Authors Results Orthogonal Point-Set Embedding But, the size of the vertices may increase Time complexity = O(n 2 ) Time complexity = O(n)

19. 19 ▒ Problem Definition Contents ▒ Motivation ▒ Previous Results and Our Results ▒ Upward Point-Set Embedding

20. 20 Upward Point-Set Embedding SG Input Upward Topological Book Embedding v1v1 v3v3 v4v4 v2v2 v5v5 1 2 3 4 5 1 2 3 4 5 v1v1 v3v3 v4v4 v2v2 v5v5 1 2 3 4 5 Upward Point-set Embedding

21. 21 a c b d Upward Topological Book Embedding a c d b S G Spine Left Page Right Page The vertices on the spine The edges on the pages Digraph Upward Topological Book Embedding

22. 22 G contains directed hamiltonian path 1 3 2 4 7 6 5 A directed path containing all the vertices 1 2 3 4 5 6 7 Upward Topological Book Embedding

23. 23 G contains directed hamiltonian path 1 3 2 4 7 6 5 1 2 3 4 5 6 7 Upward Topological Book Embedding

24. 24 1 2 3 4 5 6 7 Upward Topological Book Embedding

25. 25 1 2 3 4 5 6 7 Upward Topological Book Embedding 1 2 3 4 5 6 7

26. 26 1 2 3 4 5 6 7 Upward Topological Book Embedding 1 2 3 4 5 6 7 The drawing ….. has no edge crossings since it has the same embeddingas the original graph has no spine crossing has 1 bend per edge

27. 27 G does not contain directed Hamiltonian path 1 3 2 4 7 6 5 a b c d e Upward Topological Book Embedding

28. 28 a b c d e 1 3 2 4 7 6 5 Upward Topological Book Embedding G does not contain directed Hamiltonian path

29. 29 a b c d e 1 3 2 4 7 6 5 Upward Topological Book Embedding G does not contain directed Hamiltonian path

30. 30 a b c d e 1 3 2 4 7 6 5 a b c d e 1 2 3 4 5 6 7 Upward Topological Book Embedding

31. 31 1 3 2 4 7 6 5 a b c d e a b c d e 1 2 3 4 5 6 7 Upward Topological Book Embedding Input digraph 1 2 3 4 5 6 7 Each spine crossing corresponds to a dummy vertex

32. 32 Calculation of number of Bends i i+1 i+2 j-2 j-1 j Spine crossing from i to j is at most j-i-2 The edge (1, n) has no crossings Spine Crossings per edge is at most n-4

33. 33 Calculation of number of Bends Spine crossing from i to j is at most j-i-2 The edge (1, n) has no crossings Spine Crossings per edge is at most n-4 Bends per edge is at most n-3

34. 34 Calculation of number of Bends Spine crossing from i to j is at most j-i-2 The edge (1, n) has no crossings Spine Crossings per edge is at most n-4 Bends per edge is at most n-3 Total number of spine crossings =2(n-4)+3(n-5)+... +k(n-2-k)+p(n-3-k) where p, k are integers Number of edges which crosses the spine={k(k+1)/2}-1+p

35. 35 ▒ Problem Definition Contents ▒ Motivation ▒ Previous Results and Our Results ▒ Upward Point-Set Embedding ▒ Orthogonal Point-Set Embedding

36. 36 Orthogonal Point-Set Embedding 3-connected cubic planar graphs 4-connected planar graphs (  ≤ 4) 4-connected 4-regular planar graphs

37. 37 3-connected cubic planar graph 3-connected cubic planar graph G with HC v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 v 10

38. 38 point-set in the plane v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 v 10 3-connected cubic planar graph G with HC Plane embedding G’ of graph G 3-connected cubic planar graph

39. 39 v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 v 10 3-connected cubic planar graph Plane embedding G’ of graph G Inner edges Outer edges

40. 40 v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 v 10 Inner vertices 3-connected cubic planar graph Plane embedding G’ of graph G

41. 41 p 10 p9p9 p6p6 p8p8 p7p7 p5p5 p3p3 p4p4 p2p2 p1p1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 3-connected cubic planar graph v1v1 v 10

42. 42 p 10 p9p9 p6p6 p8p8 p7p7 p5p5 p3p3 p4p4 p2p2 p1p1 v2v2 v3v3 v4v4 v6v6 v7v7 v8v8 v9v9 3-connected cubic planar graph v1v1 v 10 Case 1: Inner edges in left page v5v5 We have to consider two cases Case 1: Inner edges in left page Case 2 : Inner edges in right page

43. 43 p 10 p9p9 p6p6 p8p8 p7p7 p5p5 p3p3 p4p4 p2p2 p1p1 v2v2 v3v3 v4v4 v6v6 v7v7 v8v8 v9v9 3-connected cubic planar graph v1v1 v 10 Case 1: Inner edges in left page Count L = 6 v5v5 Nice points (L)

44. 44 Count L = 6 p 10 p9p9 p6p6 p8p8 p7p7 p5p5 p3p3 p4p4 p2p2 p1p1 v2v2 v3v3 v4v4 v6v6 v7v7 v8v8 v9v9 3-connected cubic planar graph v1v1 v 10 Count R = 2 Case 2: Inner edges in right page v5v5 Nice points (R )

45. 45 3-connected cubic planar graph v2v2 v3v3 v4v4 v6v6 v7v7 v8v8 v9v9 v1v1 v 10 v5v5 Count L = 6 Count R = 2 p 10 p9p9 p6p6 p8p8 p7p7 p5p5 p3p3 p4p4 p2p2 p1p1

46. 46 p 10 p9p9 p6p6 p8p8 p7p7 p5p5 p3p3 p4p4 p2p2 p1p1 v2v2 v3v3 v4v4 v6v6 v7v7 v8v8 v9v9 3-connected cubic planar graph v1v1 v 10 v5v5 From pigeonhole principle….. Either count L or count R is at least = (n-2)/2which edges canbe drawn with 1 bend Left nice points Right nice points Total bends = 1.(n-2)/2+2.(3n/2-(n-2)/2-1)+3 = n/2-1+3n-n+2-2+3 = n/2+2n+2 = (5n+4)/2 Computation of number of bends

47. 47 4-connected planar graph G v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 1 2 3 4 5 6 Point-set S 4-connected planar graph

48. 48 Plane embedding G’ of graph G v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 1 2 3 4 5 6 Point-set S 4-connected planar graph

49. 49 v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 1 2 3 4 5 6 4-connected planar graph Plane embedding G’ of graph G Inner edges Outer edges v1v1 v2v2 v3v3 v4v4 v5v5 v6v6

50. 50 v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 4-connected planar graph Inner edges Outer edges v1v1 v2v2 v3v3 v4v4 v5v5 v6v6

51. 51 v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 1 2 3 4 5 6

52. 52 1 2 3 4 5 6 Middle vertex Left vertex Right vertex G is 4-regular, each vertex is incident to exactly four edges v1v1 v2v2 v3v3 v4v4 v5v5 v6v6

53. 53 1 2 3 4 5 6 Orthogonal Point-set Embedding Total bends = 3.(2n-1) + 3 = 6n Computation of number of bends v1v1 v2v2 v3v3 v4v4 v5v5 v6v6

54. 54 Tight Example 1 2 3 4 5 6 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6

55. 55 Tight Example 1 2 3 4 5 6 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 Each vertex of G is mapped to a point i in S

56. 56 ▒ Problem Definition Contents ▒ Motivation ▒ Previous Results and Our Results ▒ Upward Point-Set Embedding ▒ Orthogonal Point-Set Embedding ▒ Conclusion and Future Works

57. 57 Conclusion upward planar digraph 3-connected cubic planar graphs 4-connected 4-regular planar graphs n-3 bends per edge (5n+4)/2 bends in total 6n bends in total Quadratic Linear

58. 58 Design a fast algorithm for checking upward point-set embedding Minimize the number of bends in upward point-set embedding Find necessary and sufficient condition for orthogonal point-set embedding Reduce the number of bends for 3-connected cubic planar graphs Find Universal Point-Sets for sub-classes of planar graphs Future Works

59. 59 Reference Hal91 J. H. Halton, “On the thickness of graphs of given degree”, Information Sciences, Vol. 54, pp. 219-238, 1991. CAR09M. E. Chowdhury, M. J. Alam, and M. S. Rahman, “On Upward Point- Set Embedding of Upward Planar Digraphs”, Proc. of the 16 th Mathematics Conference of Bangladesh Mathematical Society, 2009. RNN99 M. S. Rahman, S. Nakano and T. Nishizeki, “A linear algorithm for bend- optimal orthogonal drawings of triconnected cubic plane Graphs”, Journal of Graph Alg. and Appl., http://jgaa.info, 3(4), pp. 31-62, 1999. RN02 M. S. Rahman and T. Nishizeki, “Bend-minimum orthogonal drawings of plane 3-graphs”, In Proc. International Workshop on Graph Theoretic Concepts in Computer Science (WG '02), Lect. Notes in Computer Science, Springer, Vol. 2573, pp. 367-378, 2002.

60. 60 KW02 M. Kaufmann, and R. Wiese, “Embedding vertices at points: Few bends suffice for planar graphs”. Journal of Graph Algorithms and Applications, 6(1), pp. 115–129 (2002) GLMS07 F. Giordano, G. Liotta, T. Mchedlidze,and A, Symvonis, “Computing Upward Topological Book Embeddings of Upward Planar Digraphs”, In proceedings of International Symposium on Algorithms and Computation (ISAAC 2007), Springer, Lecture Notes in Computer Science, Vol. 4835, pp. 172–183, 2007. GLW09 F. Giordano, G. Liotta, and S. H. Whitesides, “Embeddability Problems for Upward Planar Digraphs”, In the proceedings of The 16th International Symposium on Graph Drawing (GD 2008), Springer, Lecture Notes in Computer Science, Vol. 5417, pp. 242–253, 2009. RNN03 M. S. Rahman, T. Nishizeki, and M. Naznin, “Orthogonal drawings of plane graphs without bends”, Journal of Graph Alg. and Appl., http://jgaa.info, 7(4), pp. 335-362, 2003. Reference

61. 61 ThankYou

62. 62 visual analysis of self-modifiable code, based on computing a sequence of drawings whose edges are defined at run-time [Hal91] Motivation Upward Point-set Embedding with mapping That alters its own instructions while it is executing-usually to reduce the instruction path length and improve performance.

63. 63 Motivation The graphs are specified one at a time The vertex locations for the output graphs are determined by the first graph

64. 64 1 2 3 4 5 6 Now we draw the edges of G’-C except long edge Case vertex type Drawing x i+1 > x i x i+1 < x i 1middle URUULU 2left RURULU 3right LURULU

65. 65 Case vertex type Drawing x i+1 > x i x i+1 < x i 1middle URUULU 2left RURULU 3right LURULU 1 2 3 4 5 6 Now we draw the edges of G’-C except long edge

66. 66 1 2 3 4 5 6 Now we draw the edges of G’-C except long edge v i is a middle vertex vertex type ( v j ) middle left right (v i, v j ) is inner otherwise other edge of v j is (v j, v k ) y i > y k otherwise y i > y k otherwise Condition LUR RUL LUR LURD RUL RULD Drawing

67. 67 vertex type ( v j ) middle left right (v i, v j ) is inner otherwise other edge of v j is (v j, v k ) y i > y k otherwise y i > y k otherwise Condition LUR RUL LUR LURD RUL RULD Drawing 1 2 3 4 5 6 v i is a middle vertex

68. 68 1 2 3 4 5 6 v i is a left vertex vertex Type of v j middle left y j > y l > y i otherwise other edge of v j is (v j, v k ) y j > y k > y i > y l Condition LUR UR URD LURD UR LUR Drawing (v i, v l ) is other inner edge y j > y k > y l > y i y k > y j > y i > y l > y k y k > y j > y l > y i > y k

69. 69 y j > y k > y l > y i vertex Type of v j middle left y j > y l > y i otherwise other edge of v j is (v j, v k ) Condition LUR UR URD LURD UR LUR Drawing (v i, v l ) is other inner edge y k > y j > y i > y l > y k y k > y j > y l > y i > y k 1 2 3 4 5 6 v i is a left vertex y j > y k > y i > y l

70. 70 1 2 3 4 5 6 v i is a right vertex vertex Type of v j middle right y j > y l > y i otherwise other edge of v j is (v j, v k ) y j > y k > y i > y l Condition RUL UL ULD RULD UL RUL Drawing (v i, v l ) is other inner edge y j > y k > y l > y i y k > y j > y i > y l > y k y k > y j > y l > y i > y k

71. 71 vertex Type of v j middle right y j > y l > y i otherwise other edge of v j is (v j, v k ) y j > y k > y i > y l Condition RUL UL ULD RULD UL RUL Drawing (v i, v l ) is other inner edge y j > y k > y l > y i y k > y j > y i > y l > y k y k > y j > y l > y i > y k 1 2 3 4 5 6 v i is a right vertex

72. 72 1 2 3 4 5 6 We now draw the long edge (v 1, v n ) vertex type of v n middle DRULD Drawing vertex type of v 1 left NA right otherwise DRULD DRUL right otherwise DLUL DLUR

73. 73 vertex type of v n middle DRULD Drawing vertex type of v 1 left NA right otherwise DRULD DRUL right otherwise DLUL DLUR 1 2 3 4 5 6 Orthogonal Point-set Embedding