1. WADS 2013 August 12, 2013 Department of Computer Science University of Manitoba Stephane Durocher Debajyoti Mondal

2. A plane graph G 1 WADS 2013 August 12, 2013 A point set S Input a b c d e f g h i b c d e f g h i

3. An embedding of G on S 2 WADS 2013 August 12, 2013 b h i c d e a f g Output A plane graph G A point set S Input a b c d e f g h i b c d e f g h i

4. A plane graph G 3 WADS 2013 August 12, 2013 A point set S Input a b e b c d f

5. A plane graph G 4 WADS 2013 August 12, 2013 A point set S Input a b e b c d f b a c b a c b a c d d d

6. Plane 3-trees Nishat et al.(2010)  O(n 2 ) Moosa & Rahman(2011)  O(n 4/3 + ɛ ) This Presentation  O(n lg 3 n)  2-bend embeddability  1/ √ n Approximation 5 WADS 2013 August 12, 2013 Outerplanar graphs Gritzmann et al. (1991), Castañeda & Urrutia (1996)  O(n 2 ), Bose (2002)  O(n lg 3 n) NP-complete Cabello (2006 )  2-outerplanar, Nishat et al. (2011 )  partial 3-tree, Durocher & M.(2012 )  3-connected, Biedl &Vatshelle (2012 )  3-connected, fixed treewidth Not Embeddable

7. 6 WADS 2013 August 12, 2013 a b c d e f A plane 3-tree

8. 7 WADS 2013 August 12, 2013 a b c d e f A plane 3-tree a b c Insert e a b c d a b c d e Insert d Insert f

9. 8 WADS 2013 August 12, 2013 Plane 3-Tree Representative Vertex a b c d e f b c e d a d b f d c a

10. 9 WADS 2013 August 12, 2013 a b d Convex Hull f h A Plane 3-Tree G A Point-Set S g c e

11. 10 WADS 2013 August 12, 2013 b c d e a f g h c a b A Plane 3-Tree G A Point-Set S a b c

12. 11 WADS 2013 August 12, 2013 d c b n 1 =4 n 2 =4 n 3 =5 3 6 4 a c d e a f g h b

13. 12 WADS 2013 August 12, 2013 b c d e a a c b d 4 4 5 f g h n 1 =4 n 2 =4 n 3 =5 Nishat et al. (2010): The mapping of the representative vertex is unique.

14. 13 WADS 2013 August 12, 2013 b c d e a a c b d 4 4 5 f g h n 1 =4 n 2 =4 n 3 =5 T(n) = T(n 1 ) + T(n 2 ) + T(n 3 ) + O(n 2 ) = O(n 3 )

15. 14 WADS 2013 August 12, 2013 T(n) = T(n 1 ) + T(n 2 ) + T(n 3 ) + O(n 2 ) = O(n 3 ) T(n) = T(n 1 ) + T(n 2 ) + T(n 3 ) + O(n) = O(n 2 ) T(n) = T(n 1 ) + T(n 2 ) + T(n 3 ) + min{n 1, n 2, n 3 }. n 1/3+ ɛ = O(n 4/3+ ɛ ) T(n) = T(n 1 ) + T(n 2 ) + T(n 3 ) + min{n 1 +n 2, n 2 +n 3, n 3 +n 1 }. lg 2 n = O(n lg 3 n) Moosa and Rahman (COCOON 2011) Nishat et al. (GD 2010) This Presentation

16. 15 WADS 2013 August 12, 2013 W. Steiger and I. Streinu (1998) Given a triangular set S of n points in general position, in O(n) time one can construct a new point m such that the sub-triangles contain prescribed number of points of S. (i+j+k) -3 = 10 = n m x z y i = 4 j = 5 k = 4 (i+j+k) -3 = 10 = n m' x z y i = 4 j = 5 k = 4

17. 16 WADS 2013 August 12, 2013 The partition of the interior points into subtriangles is unique ! (i+j+k) -3 = 10 = n m x z y i = 4 j = 5 k = 4 (i+j+k) -3 = 10 = n m' x z y i = 4 j = 5 k = 4

18. 17 WADS 2013 August 12, 2013 b c d e a a c b m n 1 - 1 n 2 + 1 n 3 - 1 f g h n 1 =4 n 2 =4 n 3 =5

19. 18 WADS 2013 August 12, 2013 b c d e a a c b m n 1 - 1 n 2 + 1 n 3 - 1 f g h n 1 =4 n 2 =4 n 3 =5 d

20. 19 WADS 2013 August 12, 2013 b c d e a a c b d n1n1 n2n2 n3n3 f g h n 1 =4 n 2 =4 n 3 =5 T(n) = T(n 1 ) + T(n 2 ) + T(n 3 ) + O(n) = O(n 2 )

21. 20 WADS 2013 August 12, 2013 a c b n1n1 n2n2 n3n3 a c b n1-1n1-1 n3-1n3-1 n2-1n2-1 u v n 3 ≤ n 2 ≤ n 1 The representative vertex must lie inside the green region.

22. 21 WADS 2013 August 12, 2013 a c b n1n1 n2n2 n3n3 a c b n3-1n3-1 n3-1n3-1 n3-1n3-1 u v n 3 ≤ n 2 ≤ n 1 The green region contains O(n 3 ) points. The representative vertex and its incident edges must lie inside the green region. r s

23. 22 WADS 2013 August 12, 2013 a c b n1n1 n2n2 n3n3 a c b u v n 3 ≤ n 2 ≤ n 1 The green region contains O(n 3 ) points. The representative vertex and its incident edges must lie inside the green region.

24. 23 WADS 2013 August 12, 2013 a c b n1n1 n2n2 n3n3 a c b u v Finding a valid mapping in S with partition (n 1, n 2, n 3 )  Finding a valid mapping in S / with partition (n 1 -x 1, n 2 -x 2, n 3 ) or, (n 3 +1, n 3 +1, n 3 ) x1x1 x2x2

25. 24 WADS 2013 August 12, 2013 Select O(n 2 +n 3 ) candidate points in O((n 2 +n 3 ) lg 2 n) time Find the required mapping in the reduced point set in O(n 3 ) time T(n) = T(n 1 ) + T(n 2 ) + T(n 3 ) + O(min{n 1 +n 2, n 2 +n 3, n 3 +n 1 }. lg 2 n ) = O(n lg 3 n)

26. 20 WADS 2013 August 12, 2013 a b c d f p e A 2-bend point-set embedding of G on S Output A plane 3-tree G A point set S Input M. Kaufmann and R. Wiese (2002) Every plane graph admits a 2-bend point set embedding with O(W 3 ) area on any set of n points in general position.

27. 21 WADS 2013 August 12, 2013 a b c a b c d e f a b c d e f a b c d a b c e d f

28. 22 WADS 2013 August 12, 2013 a b c e d f A plane 3-tree G S

29. 23 WADS 2013 August 12, 2013 a b c e d f S S(Γ ) = 3 A plane 3-tree G p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p1p1 p3p3 p5p5

30. 24 WADS 2013 August 12, 2013 a b c e d f S S(Γ * ) = 4 S(Γ ) Approximation factor = __________ S(Γ * ) S(Γ ) = 3 A plane 3-tree G p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p1p1 p3p3 p5p5 p1p1 p3p3 p5p5 p6p6

31. 25 WADS 2013 August 12, 2013 Variable embedding: Is there a subquadratic-time algorithm for testing point-set embeddability of plane 3-trees in variable embedding setting? 1-Bend Point-Set Embeddability: Is it always possible to find 1-bend point set embeddings for plane 3-trees? Approximation: Is it possible to approximate point-set embeddability of plane 3-trees within a constant factor ?