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Theta-3 is connected Oswin Aichholzer 1 Sang Won Bae 2 Luis Barba 34 - PowerPoint PPT Presentation

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Theta-3 is connected Oswin Aichholzer 1 Sang Won Bae 2 Luis Barba 34 Prosenjit Bose 3 Matias Korman 5 e van Renssen 3 Andr Perouz Taslakian 6 Sander Verdonschot 3 1 Graz University of Technology 2 Kyonggi University 3 Carleton University 4

  1. Theta-3 is connected Oswin Aichholzer 1 Sang Won Bae 2 Luis Barba 34 Prosenjit Bose 3 Matias Korman 5 e van Renssen 3 Andr´ Perouz Taslakian 6 Sander Verdonschot 3 1 Graz University of Technology 2 Kyonggi University 3 Carleton University 4 Universit´ e Libre de Bruxelles 5 Universitat Polit´ ecnica de Catalunya 6 American University of Armenia 25th Canadian Conference on Computational Geometry Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 1 / 16

  2. θ -graphs Partition plane into cones Add edge to ‘closest’ vertex in each cone Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 2 / 16

  3. θ -graphs Partition plane into cones Add edge to ‘closest’ vertex in each cone Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 2 / 16

  4. Geometric Spanners Graphs with short detours between vertices For every u and w , there is a path with length ≤ t · | uw | Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 3 / 16

  5. Previous Work Clarkson 1987 θ -graphs with > 8 cones are spanners Keil 1988 Ruppert & Seidel 1991 θ -graphs with > 6 cones are spanners Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 4 / 16

  6. Previous Work Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 5 / 16

  7. Previous Work Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 6 / 16

  8. Previous Work Clarkson 1987 θ -graphs with > 8 cones are spanners Keil 1988 Ruppert & Seidel 1991 θ -graphs with > 6 cones are spanners θ 2 and θ 3 are not spanners El Molla 2009 Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 7 / 16

  9. Previous Work Clarkson 1987 θ -graphs with > 8 cones are spanners Keil 1988 Ruppert & Seidel 1991 θ -graphs with > 6 cones are spanners θ 2 and θ 3 are not spanners El Molla 2009 Bonichon et al. 2010 θ 6 is a planar 2-spanner Barba et al. 2013 θ 4 and θ 5 are spanners Bose et al. Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 7 / 16

  10. Connectedness Even θ -graphs Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 8 / 16

  11. Connectedness Odd θ -graphs Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 9 / 16

  12. Connectedness Odd θ -graphs Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 9 / 16

  13. Connectedness Odd θ -graphs Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 9 / 16

  14. Connectedness Theta-routing does not work in θ 3 Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 10 / 16

  15. Properties Edges in the same cone cannot cross Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 11 / 16

  16. Properties Edges in the same cone cannot cross Edges cannot cross empty cones Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 11 / 16

  17. Paths Unique up-path from each vertex Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

  18. Paths Paths can form barriers Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

  19. Paths Paths can form barriers Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

  20. Paths Up-paths cannot cross up-barriers Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

  21. Paths Up-paths cannot cross up-barriers Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

  22. Paths Up-paths cannot cross up-barriers Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

  23. Paths Up-paths cannot cross up-barriers Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

  24. Paths Up-paths cannot cross up-barriers Other paths can be forced to cross up-barriers Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 12 / 16

  25. Lemma Special configuration of up-sinks Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

  26. Lemma Special configuration of up-sinks . . . 0 Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

  27. Lemma Special configuration of up-sinks . . . 0 Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

  28. Lemma Special configuration of up-sinks ⇒ they are connected . . . 0 Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

  29. Lemma Special configuration of up-sinks ⇒ they are connected . . . k Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

  30. Lemma Special configuration of up-sinks ⇒ they are connected . . . k Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

  31. Lemma Special configuration of up-sinks ⇒ they are connected . . . k Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

  32. Lemma Special configuration of up-sinks ⇒ they are connected . . . k − 1 Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

  33. Lemma Special configuration of up-sinks ⇒ they are connected . . . k − 1 Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 13 / 16

  34. Proof Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 14 / 16

  35. Proof Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 14 / 16

  36. Proof Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 14 / 16

  37. Proof Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 14 / 16

  38. Proof Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 14 / 16

  39. Proof Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 14 / 16

  40. Proof Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 14 / 16

  41. Proof Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 14 / 16

  42. Proof Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 14 / 16

  43. Conclusion θ 3 is connected Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 15 / 16

  44. Conclusion θ 3 is connected Properties hold for Yao 3 as well ⇒ Yao 3 is connected Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 15 / 16

  45. Future work Sander Verdonschot (Carleton University) Theta-3 is connected CCCG 2013 16 / 16

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