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Higher Order City Voronoi Diagrams Andreas Gemsa 1 , D.T. Lee 2 , 3 , - PowerPoint PPT Presentation

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Higher Order City Voronoi Diagrams Andreas Gemsa 1 , D.T. Lee 2 , 3 , Chih-Hung Liu 1 , 2 , Dorothea Wagner 1 1 Karlsruhe Institute of Technology (KIT), 2 Academia Sinica, 3 National Chung Hsing University. Institute of Theoretical Informatics

  1. City Metric city metric: L 1 metric and transportation network shortest path: walk L 1 metric quickest path? walk walk city metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  2. City Metric city metric: L 1 metric and transportation network Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  3. City Metric city metric: L 1 metric and transportation network Def.: transportation network graph G = ( V C , E C ) planar, straight-line embedding only horizontal and vertical edges Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  4. City Metric city metric: L 1 metric and transportation network G Def.: transportation network graph G = ( V C , E C ) planar, straight-line embedding only horizontal and vertical edges Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  5. City Metric city metric: L 1 metric and transportation network G Def.: transportation network graph G = ( V C , E C ) planar, straight-line embedding only horizontal and vertical edges speed off the network: 1 speed on the network: v > 1 Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  6. City Metric city metric: L 1 metric and transportation network G Def.: transportation network graph G = ( V C , E C ) planar, straight-line embedding only horizontal and vertical edges speed off the network: 1 speed on the network: v > 1 Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  7. City Metric city metric: L 1 metric and transportation network G Def.: transportation network graph G = ( V C , E C ) planar, straight-line embedding only horizontal and vertical edges speed off the network: 1 speed on the network: v > 1 c := | V C | Complexity of G : O ( c ) Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  8. City Metric city metric: L 1 metric and transportation network G q Def.: transportation network graph G = ( V C , E C ) planar, straight-line embedding only horizontal and vertical edges speed off the network: 1 speed on the network: v > 1 c := | V C | p Complexity of G : O ( c ) distance between two points? Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  9. City Metric city metric: L 1 metric and transportation network G q Def.: transportation network graph G = ( V C , E C ) planar, straight-line embedding only horizontal and vertical edges L 1 speed off the network: 1 speed on the network: v > 1 c := | V C | p Complexity of G : O ( c ) distance between two points? Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  10. City Metric city metric: L 1 metric and transportation network G q Def.: transportation network graph G = ( V C , E C ) planar, straight-line embedding only horizontal and vertical edges L 1 speed off the network: 1 speed on the network: v > 1 c := | V C | p Complexity of G : O ( c ) distance between two points? Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  11. City Metric city metric: L 1 metric and transportation network G q Def.: transportation network graph G = ( V C , E C ) planar, straight-line embedding only horizontal and vertical edges L 1 speed off the network: 1 speed on the network: v > 1 c := | V C | p Complexity of G : O ( c ) distance between two points? d C ( p, q ) quickest path Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  12. Higher Order City Voronoi Diagrams Q: Higher Order Voronoi Diagrams? Q: City Voronoi Diagrams? Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  13. Our Contribution Euclidean-Metric City-Metric Structural Complexity Structural Complexity Θ( n + c ) first-order first-order Θ( n ) k th -order k th -order Θ( k ( n − k )) O ( k ( n − k ) + kc ) Ω( n + kc ) farthest-site Θ( n ) Θ( nc ) farthest-site [Bae et al., 2012] Time Complexity Time Complexity O (( n + c ) log n ) first-order O ( n log n ) first-order O ( k 2 ( n + c ) log( n + c )) O ( k 2 n log n ) k th -order k th -order O ( nc log n log 2 ( n + c )) farthest-site O ( n log n ) farthest-site [Bae et al., 2012] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  14. Our Contribution Euclidean-Metric City-Metric Structural Complexity Structural Complexity Θ( n + c ) first-order first-order Θ( n ) k th -order k th -order Θ( k ( n − k )) O ( k ( n − k ) + kc ) Ω( n + kc ) farthest-site Θ( n ) Θ( nc ) farthest-site [Bae et al., 2012] Time Complexity Time Complexity O (( n + c ) log n ) first-order O ( n log n ) first-order O ( k 2 ( n + c ) log( n + c )) O ( k 2 n log n ) k th -order k th -order O ( nc log n log 2 ( n + c )) farthest-site O ( n log n ) farthest-site [Bae et al., 2012] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  15. Wavefront Propagation – Euclidean Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  16. Wavefront Propagation – Euclidean Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  17. Wavefront Propagation – Euclidean Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  18. Wavefront Propagation – Euclidean Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  19. Wavefront Propagation – Euclidean Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  20. Wavefront Propagation – Euclidean Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  21. Wavefront Propagation – L 1 Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  22. Wavefront Propagation – L 1 Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  23. Wavefront Propagation – L 1 Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  24. Wavefront Propagation – L 1 Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  25. Wavefront Propagation – L 1 Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  26. Wavefront Propagation – L 1 Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  27. Wavefront Propagation – L 1 Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  28. Wavefront Propagation – L 1 Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  29. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  30. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  31. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  32. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  33. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  34. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  35. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  36. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  37. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  38. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  39. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  40. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  41. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  42. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  43. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  44. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  45. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  46. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  47. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  48. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  49. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  50. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  51. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  52. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  53. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  54. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  55. Wavefront Propagation – City Metric O ( c ) activation points [Aichholzer et al., 2004] Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  56. Wavefront Propagation – City Metric Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  57. Wavefront Propagation – City Metric arrow Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  58. Wavefront Propagation – City Metric arrow Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  59. Wavefront Propagation – City Metric arrow r q q ′ p activation point Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  60. Wavefront Propagation – City Metric arrow r q q ′ p activation point Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  61. Wavefront Propagation – City Metric arrow r q q ′ p Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  62. Wavefront Propagation – City Metric arrow r q q ′ p Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  63. Wavefront Propagation – City Metric arrow r q q ′ p Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  64. Wavefront Propagation – City Metric arrow r q q ′ p Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  65. Wavefront Propagation – City Metric arrow r q q ′ p Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

  66. Wavefront Propagation – City Metric arrow r q q ′ p Institute of Theoretical Informatics Andreas Gemsa , D.T. Lee, Chih-Hung Liu, Dorothea Wagner – Higher Order City Voronoi Diagrams. Prof. Dr. Dorothea Wagner

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