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On Plane Constrained Bounded-Degree Spanners Prosenjit Bose, Rolf Fagerberg, Andr e van Renssen and Sander Verdonschot Carleton University, University of Southern Denmark April 15, 2012 Sander Verdonschot (Carleton University) Constrained

  1. On Plane Constrained Bounded-Degree Spanners Prosenjit Bose, Rolf Fagerberg, Andr´ e van Renssen and Sander Verdonschot Carleton University, University of Southern Denmark April 15, 2012 Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 1 / 15

  2. Geometric Spanners Given: Goal: Set of points in the plane Approximate the complete Euclidean graph Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 2 / 15

  3. Geometric Spanners Given: Goal: Set of points in the plane Approximate the complete Euclidean graph Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 2 / 15

  4. Geometric Spanners Given: Goal: Set of points in the plane Approximate the complete Euclidean graph shortest path ≤ k · Euclidean distance Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 2 / 15

  5. Geometric Spanners Small spanning ratio Planarity Bounded degree Small number of hops Low total edge length Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 3 / 15

  6. Geometric Spanners Small spanning ratio Planarity Bounded degree Small number of hops Low total edge length Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 4 / 15

  7. Plane Spanners Empty square ( L 1 ) Delaunay triangulation ≤ 3 . 16 (Chew - 1986) = 2 . 61 (Bonichon et al. - 2012) Empty circle ( L 2 ) Delaunay triangulation ≤ 5 . 08 (Dobkin et al. - 1987) ≤ 2 . 42 (Keil, Gutwin - 1992) Empty equilateral triangle Delaunay triangulation = 2 (Chew - 1989) Equivalent to half- θ 6 -graph (Bonichon et al. - 2010) Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 5 / 15

  8. Plane Bounded-Degree Spanners Degree k Authors 27 10.02 Bose et al. - 2005 23 7.79 Li, Wang - 2004 17 28.54 Bose et al. - 2009 14 3.53 Kanj, Perkovi´ c - 2008 6 98.91 Bose et al. - 2012 6 6 Bonichon et al. - 2010 Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 6 / 15

  9. Constrained Geometric Spanners Given: Goal: Set of points in the plane V Approximate visibility graph Set of constraints ⊆ V × V Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 7 / 15

  10. Constrained Geometric Spanners Given: Goal: Set of points in the plane V Approximate visibility graph Set of constraints ⊆ V × V Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 7 / 15

  11. Constrained Geometric Spanners Given: Goal: Set of points in the plane V Approximate visibility graph Set of constraints ⊆ V × V Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 7 / 15

  12. Constrained Geometric Spanners Given: Goal: Set of points in the plane V Approximate visibility graph Set of constraints ⊆ V × V Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 7 / 15

  13. Constrained Geometric Spanners k B.D. Plane Authors Graph 1 + ǫ Clarkson - 1987 1 + ǫ Das - 1997 � 5.08 � Karavelas - 2001 Delaunay triangulation 2.42 Bose, Keil - 2006 Delaunay triangulation � 2 � Our result Half- θ 6 -graph 6 Our result Half- θ 6 -graph � � Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 8 / 15

  14. Half- θ 6 -graph 6 Cones around each vertex: 3 positive, 3 negative Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 9 / 15

  15. Half- θ 6 -graph Connect to ‘closest’ vertex in each positive cone Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 9 / 15

  16. Half- θ 6 -graph Connect to ‘closest’ vertex in each positive cone Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 9 / 15

  17. Half- θ 6 -graph Connect to ‘closest’ vertex in each positive cone Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 9 / 15

  18. Constrained Half- θ 6 -graph Connect to ‘closest’ visible vertex in each positive cone Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 10 / 15

  19. Constrained Half- θ 6 -graph Connect to ‘closest’ visible vertex in each positive cone Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 10 / 15

  20. Constrained Half- θ 6 -graph Connect to ‘closest’ visible vertex in each positive sub cone Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 10 / 15

  21. Constrained Half- θ 6 -graph Connect to ‘closest’ visible vertex in each positive sub cone Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 10 / 15

  22. Constrained Half- θ 6 -graph Connect to ‘closest’ visible vertex in each positive sub cone Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 10 / 15

  23. Spanning ratio Theorem The constrained half- θ 6 -graph is a 2-spanner of the visibility graph Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 11 / 15

  24. Spanning ratio Theorem The constrained half- θ 6 -graph is a 2-spanner of the visibility graph Proof by induction on the area of the equilateral triangle Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 11 / 15

  25. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  26. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  27. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  28. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  29. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  30. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  31. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  32. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  33. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  34. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  35. Spanning ratio Induction hypothesis: there is a path of length at most one side plus the longer top segment If the larger side is empty, the length is at most one side plus the shorter top segment Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 12 / 15

  36. Bounded-Degree subgraph Theorem The constrained half- θ 6 -graph has a bounded degree subgraph that is a 6-spanner of the visibility graph Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

  37. Bounded-Degree subgraph Theorem The constrained half- θ 6 -graph has a bounded degree subgraph that is a 6-spanner of the visibility graph Sander Verdonschot (Carleton University) Constrained Bounded-Degree Spanners April 15, 2012 13 / 15

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