Presentation is loading. Please wait.

Presentation is loading. Please wait.

WALCOM 2012February 16, 2012 Stephane Durocher Debajyoti Mondal Department of Computer Science University of Manitoba.

Published by Modified over 9 years ago

Similar presentations

Presentation on theme: "WALCOM 2012February 16, 2012 Stephane Durocher Debajyoti Mondal Department of Computer Science University of Manitoba."— Presentation transcript:

1 WALCOM 2012February 16, 2012 Stephane Durocher Debajyoti Mondal Department of Computer Science University of Manitoba

2 a b c d e f g h i A plane graph G A point set P 1 WALCOM 2012February 16, 2012

3 a b c d e f g h i A plane graph G An embedding of G on P a b c d e f g h i 2 WALCOM 2012February 16, 2012

4 Reference Graph Class Time complexity 3 WALCOM 2012February 16, 2012

5 WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) A chain 4 y c1c1 x y x c1c1 c2c2 c|S|c|S| 9 10 11 S

6 WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) 5 y x c1c1 c2c2 c|S|c|S| 9 10 11 S y x

7 WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) 6 y x c1c1 c2c2 c|S|c|S| 9 10 11 S y x G

8 WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) 7 y x G B B B B P

9 WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) 8 y x G P x y

10 WALCOM 2012February 16, 2012 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} 3-Partition  Point-Set Embeddability (2-connected graphs) 9 y x G P x y

11 WALCOM 2012February 16, 2012 10 Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 G Does G contain a Hamiltonian Cycle? Does G contain a Hamiltonian Cycle? Does G admits a 1-bend PSE on P ? Does G admits a 1-bend PSE on P ? P

12 WALCOM 2012February 16, 2012 11 Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v4v4 v5v5 G P v1v1 v3v3 v6v6 v2v2 v1v1 v5v5 v3v3 v2v2 v4v4 v6v6 v1v1 v5v5 v2v2 v3v3 v4v4 v6v6 If G contains a Hamiltonian Cycle, then G admits a 1-bend PSE on P

13 WALCOM 2012February 16, 2012 12 Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v4v4 v5v5 G P v1v1 v3v3 v6v6 v2v2 v1v1 v5v5 v2v2 v3v3 v4v4 v6v6 If G admits a 1-bend PSE on P, then G contains a Hamiltonian Cycle e How to get rid off bends? How to get rid off bends?

14 WALCOM 2012February 16, 2012 13 Point-Set Embeddability is NP-hard for 3-Connected Graphs G / G

15 WALCOM 2012February 16, 2012 14 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 G Does G contain a Hamiltonian Cycle? Does G contain a Hamiltonian Cycle? Does G / admits a PSE on P ? Does G / admits a PSE on P ? P G / ? ? ?

16 WALCOM 2012February 16, 2012 15 P G / P ?

17 WALCOM 2012February 16, 2012 16 h a d c b e f g a d c b e f g h G h a d c b e f g G /

18 WALCOM 2012 17 a d c b e f g h h a d c b e f g G / G A B P

19 WALCOM 2012 18 a d c b e f g h h a d c b e f g G / G a dcbh g e f A B P

20 WALCOM 2012 A B A B 19 a dcbh g e f G /

21 WALCOM 2012 A B 20 G / h a d c b e f g h a d c b e f g 12 points 12 14 10

22 WALCOM 2012 21 a d c b e f g h h a d c b e f g G / G Does G contain a Hamiltonian Cycle? Does G contain a Hamiltonian Cycle? Does G / admit a PSE on some point set among P 1, P 2, …, P k ? Does G / admit a PSE on some point set among P 1, P 2, …, P k ? February 16, 2012

23 WALCOM 2012February 16, 2012 A plane 3-tree G f g h k m a b c d e A construction for G 22 a b c d e f g h k m k e c g m Nishat et al. (2010), Durocher et al. (2011), Moosa and Rahman (2011) Point-set embeddability can be tested for plane 3-trees in O(n 4/3 + ɛ ) time.

24 WALCOM 2012February 16, 2012 23 a b c d a b c e f g a b c e f h i j a b c f h i j k l m

25 WALCOM 2012February 16, 2012 24 a b c d a b c e f g a b c e f h i j a b c f h i j k l m a b c d a c b g f e a c b f e h j i a c b f i j l h k m

26 WALCOM 2012February 16, 2012 A klee graph G 25 b a c q o p n m l k j i g h e f d r b c a d q fo p n m r h g e ij k l Does G admits a convex point-set embedding on P ? P

27 WALCOM 2012February 16, 2012 A klee graph G 26 b a c q o p n m l k j i g h e f d r Does G admits a convex point-set embedding on P ? P

28 WALCOM 2012February 16, 2012 27 b a c q o p n m l k j i g h e f d r b a c q o p n m l k j i g h e f d r

29 WALCOM 2012February 16, 2012  Is PSE NP-hard for 4-connected graphs?  Convex PSE algorithms for general klee graphs.  PSE algorithms for klee graphs without convexity constraint.  Is PSE NP-hard for 4-connected graphs?  Convex PSE algorithms for general klee graphs.  PSE algorithms for klee graphs without convexity constraint. 28

30

31 WALCOM 2012February 16, 2012 A plane 3-tree G 4 v1v1 v2v2 v3v3 v4v4 v8v8 v5v5 v6v6 v7v7 v9v9 v 10 v1v1 v2v2 v3v3 v4v4 v8v8 v5v5 v6v6 v7v7 v9v9

32 WALCOM 2012February 16, 2012 A plane 3-tree G 4 f1f1 f2f2 f3f3 f4f4 f1f1 f3f3 f4f4 e f d f1f1 f2f2 f4f4 q o p f2f2 f3f3 n m l k j i g h f4f4 r

33 WALCOM 2012February 16, 2012 A plane 3-tree G 4 b f1f1 f2f2 f3f3 f4f4 a c q o p n m l k j i g h e f d f1f1 f3f3 f4f4 e f d f1f1 f2f2 f4f4 q o p f2f2 f3f3 n m l k j i g h f4f4 r r

34 y = 0 y = -1 y = 1 (c) h a d c b e f g M / (a) A B (d) (b) a dcbh g e f L PiPi

35 WALCOM 2012February 16, 2012 20 Tricks… a d c b e f g h h a d c b e f g Instead of the blue vertices use blue cycles. Replace black vertices with some special graph structure. Use simpler input instances. h a d c b e f g

36 WALCOM 2012February 16, 2012 20

37 WALCOM 2012February 16, 2012 20 Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 G Does G contain a Hamiltonian Cycle? Does G contain a Hamiltonian Cycle? Does G admits a 1-bend PSE on P ? Does G admits a 1-bend PSE on P ? P

38 WALCOM 2012February 16, 2012 20 Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v5v5 v6v6 v8v8 Does G contain a Hamiltonian Cycle? Does G contain a Hamiltonian Cycle? v2v2 v3v3 v1v1 v7v7 v4v4 v1v1 v2v2 v8v8 v3v3 v4v4 v7v7 v5v5 v6v6 G

39 WALCOM 2012February 16, 2012 a b c d e f g h i j k l m n o A plane 3-tree G f g h i j k l m n o a b c d e A construction for G 4

40 WALCOM 2012February 16, 2012 a b c d e f g h i j k l m n o A plane 3-tree G f g h i j k l m n o a b c d e The representative vertex of G k l e A plane 3-tree A construction for G c o c g m n d A plane 3-tree 5

41 h a d c b e f g a d c b e f g h M M / (a) (b) (c) a / b / c / r st

Download ppt "WALCOM 2012February 16, 2012 Stephane Durocher Debajyoti Mondal Department of Computer Science University of Manitoba."

Similar presentations

Rahnuma Islam Nishat Debajyoti Mondal Md. Saidur Rahman Graph Drawing and Information Visualization Laboratory Department of Computer Science and Engineering.

Rahnuma Islam Nishat Debajyoti Mondal Md. Saidur Rahman Graph Drawing and Information Visualization Laboratory Department of Computer Science and Engineering.
Poly-Logarithmic Approximation for EDP with Congestion 2

Poly-Logarithmic Approximation for EDP with Congestion 2
Debajyoti Mondal Department of Computer Science University of Manitoba Department of Computer Science University of Colorado Denver Stephane Durocher Department.

Debajyoti Mondal Department of Computer Science University of Manitoba Department of Computer Science University of Colorado Denver Stephane Durocher Department.
GD 2011September 21, 2011 Stephane Durocher Debajyoti Mondal Md. Saidur Rahman Sue Whitesides Rahnuma Islam Nishat Dept. of Computer Science and Engg.

GD 2011September 21, 2011 Stephane Durocher Debajyoti Mondal Md. Saidur Rahman Sue Whitesides Rahnuma Islam Nishat Dept. of Computer Science and Engg.
Approximating the two-source minimum routing cost spanning trees Bang Ye Wu Shu-Te University.

Approximating the two-source minimum routing cost spanning trees Bang Ye Wu Shu-Te University.
Perfect Matching for Biconnected Cubic Graphs in O(nlog 2 n) Time Krzysztof Diks & Piotr Stańczyk.

Perfect Matching for Biconnected Cubic Graphs in O(nlog 2 n) Time Krzysztof Diks & Piotr Stańczyk.
Directed Graphs. Given vertices u and v of a digraph G , we say that u reaches v (and v is reachable from u ) if G  has a directed path from u to.

Directed Graphs. Given vertices u and v of a digraph G , we say that u reaches v (and v is reachable from u ) if G  has a directed path from u to.
Debajyoti Mondal, Rahnuma Islam Nishat, Md. Saidur Rahman and Md. Jawaherul Alam Graph Drawing and Information Visualization Laboratory Department of Computer.

Debajyoti Mondal, Rahnuma Islam Nishat, Md. Saidur Rahman and Md. Jawaherul Alam Graph Drawing and Information Visualization Laboratory Department of Computer.
Stephane Durocher 1 Debajyoti Mondal 1 Rahnuma Islam Nishat 2 Sue Whitesides 2 1 Department of Computer Science, University of Manitoba 2 Department of.

Stephane Durocher 1 Debajyoti Mondal 1 Rahnuma Islam Nishat 2 Sue Whitesides 2 1 Department of Computer Science, University of Manitoba 2 Department of.
Graph Drawing and Information Visualization Laboratory Department of Computer Science and Engineering Bangladesh University of Engineering and Technology.

Graph Drawing and Information Visualization Laboratory Department of Computer Science and Engineering Bangladesh University of Engineering and Technology.
Minimum-Segment Convex Drawings of 3-Connected Cubic Plane Graphs Sudip Biswas Debajyoti Mondal Rahnuma Islam Nishat Md. Saidur Rahman Graph Drawing and.

Minimum-Segment Convex Drawings of 3-Connected Cubic Plane Graphs Sudip Biswas Debajyoti Mondal Rahnuma Islam Nishat Md. Saidur Rahman Graph Drawing and.
1 Debajyoti Mondal 2 Rahnuma Islam Nishat 2 Sue Whitesides 3 Md. Saidur Rahman 1 University of Manitoba, Canada 2 University of Victoria, Canada 3 Bangladesh.

1 Debajyoti Mondal 2 Rahnuma Islam Nishat 2 Sue Whitesides 3 Md. Saidur Rahman 1 University of Manitoba, Canada 2 University of Victoria, Canada 3 Bangladesh.
Graph Algorithms: Minimum Spanning Tree 1 2 3 4 6 5 10 1 5 4 3 2 6 1 1 8 We are given a weighted, undirected graph G = (V, E), with weight function w:

Graph Algorithms: Minimum Spanning Tree We are given a weighted, undirected graph G = (V, E), with weight function w:
Computability and Complexity 15-1 Computability and Complexity Andrei Bulatov NP-Completeness.

Computability and Complexity 15-1 Computability and Complexity Andrei Bulatov NP-Completeness.
Convex Grid Drawings of 3-Connected Plane Graphs Erik van de Pol.

Convex Grid Drawings of 3-Connected Plane Graphs Erik van de Pol.
Approximation Algorithms: Concepts Approximation algorithm: An algorithm that returns near-optimal solutions (i.e. is provably good“) is called an approximation.

Approximation Algorithms: Concepts Approximation algorithm: An algorithm that returns near-optimal solutions (i.e. is "provably good“) is called an approximation.
Reduction Techniques Restriction Local Replacement Component Design Examples.

Reduction Techniques Restriction Local Replacement Component Design Examples.
Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.

Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.
Joint with Christian KnauerFreie U., Berlin Andreas SpillnerJena Takeshi TokuyamaTohoku University Alexander WolffUniversity of Karlsruhe Algorithms for.

Joint with Christian KnauerFreie U., Berlin Andreas SpillnerJena Takeshi TokuyamaTohoku University Alexander WolffUniversity of Karlsruhe Algorithms for.
1 University of Denver Department of Mathematics Department of Computer Science.

1 University of Denver Department of Mathematics Department of Computer Science.

Similar presentations

Ads by Google