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A Simple 3-Approximation of Minimum Manhattan Networks Bernhard Fuchs and Anna Schulze TU Braunschweig and Uni K oln CTW 2008 B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 1 / 14 Manhattan networks Given a set of

  1. Prior work Complexity of MMN: Still unknown! Approximation algorithms so far: in O ( n 3 ) [Gudmunsson et al.] 99 8-approx. 4-approx. in O ( n log n ) in O ( n 3 ) [Kato et al.] 02 2-approx. incomplete [Benkert et al.] 04 3-approx. in O ( n log n ) [Chepoi et al.] 05 2-approx. via LP-rounding in O ( n 3 ) [Seibert, Unger] 05 1 . 5-approx. incomplete [Nouioua] — 2-approx. in O ( n log n ) unpublished B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 6 / 14

  2. Prior work Complexity of MMN: Still unknown! Approximation algorithms so far: in O ( n 3 ) [Gudmunsson et al.] 99 8-approx. 4-approx. in O ( n log n ) in O ( n 3 ) [Kato et al.] 02 2-approx. incomplete [Benkert et al.] 04 3-approx. in O ( n log n ) [Chepoi et al.] 05 2-approx. via LP-rounding in O ( n 3 ) [Seibert, Unger] 05 1 . 5-approx. incomplete [Nouioua] — 2-approx. in O ( n log n ) unpublished Our algorithm: A new 3-approximation in O ( n log n ). B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 6 / 14

  3. Prior work Complexity of MMN: Still unknown! Approximation algorithms so far: in O ( n 3 ) [Gudmunsson et al.] 99 8-approx. 4-approx. in O ( n log n ) in O ( n 3 ) [Kato et al.] 02 2-approx. incomplete [Benkert et al.] 04 3-approx. in O ( n log n ) [Chepoi et al.] 05 2-approx. via LP-rounding in O ( n 3 ) [Seibert, Unger] 05 1 . 5-approx. incomplete [Nouioua] — 2-approx. in O ( n log n ) unpublished Our algorithm: A new 3-approximation in O ( n log n ). Much simpler than [Benkert et al.], both algorithm and proof. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 6 / 14

  4. Algorithm outline Our algorithm has two phases: B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 7 / 14

  5. Algorithm outline Our algorithm has two phases: Phase I A horizontal and a vertical sweep adding line segments ‘on-the-fly’. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 7 / 14

  6. Algorithm outline Our algorithm has two phases: Phase I A horizontal and a vertical sweep adding line segments ‘on-the-fly’. Phase II A standard 2-approximation algorithm inside so-called ‘ staircases ’. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 7 / 14

  7. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  8. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  9. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . Add vertical sides of rectangle. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  10. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . Add vertical sides of rectangle. Iterate through rectangles from left to right. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  11. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . Add vertical sides of rectangle. Iterate through rectangles from left to right. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  12. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . Add vertical sides of rectangle. Iterate through rectangles from left to right. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  13. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . Add vertical sides of rectangle. Iterate through rectangles from left to right. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  14. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . Add vertical sides of rectangle. Iterate through rectangles from left to right. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  15. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . Add vertical sides of rectangle. Iterate through rectangles from left to right. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  16. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . Add vertical sides of rectangle. Iterate through rectangles from left to right. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  17. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . Add vertical sides of rectangle. Iterate through rectangles from left to right. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  18. Phase I - The sweep Consider critical rectangles spanned by horizontal , or x-neighbors . Add vertical sides of rectangle. Iterate through rectangles from left to right. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  19. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  20. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  21. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  22. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  23. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  24. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  25. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  26. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  27. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  28. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  29. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  30. Phase I - The sweep Proceed likewise with y-neighbors . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 8 / 14

  31. After the sweep In general, no Manhattan network after sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  32. After the sweep In general, no Manhattan network after sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  33. After the sweep In general, no Manhattan network after sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  34. After the sweep In general, no Manhattan network after sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  35. After the sweep In general, no Manhattan network after sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  36. After the sweep In general, no Manhattan network after sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  37. After the sweep In general, no Manhattan network after sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  38. After the sweep In general, no Manhattan network after sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  39. After the sweep In general, no Manhattan network after sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  40. After the sweep In general, no Manhattan network after sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  41. After the sweep In general, no Manhattan network after sweep. So-called staircases still empty. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  42. After the sweep A In general, no Manhattan network after sweep. So-called staircases still empty. Call A the staircase area . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 9 / 14

  43. After the sweep Definition (staircase) B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  44. After the sweep v 1 v 2 v 3 v 4 Definition (staircase) k sequence points ( v 1 , . . . , v k ). B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  45. After the sweep v 1 v 2 v 3 v 4 b y b x Definition (staircase) k sequence points ( v 1 , . . . , v k ). Two base points b x , b y . ( b x = b y possible.) B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  46. After the sweep v 1 v 2 v 3 v 4 b y b x Definition (staircase) k sequence points ( v 1 , . . . , v k ). Two base points b x , b y . ( b x = b y possible.) For all v i , b x is the x -neighbor of v i in the third quadrant of v i . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  47. After the sweep v 1 v 2 v 3 v 4 b y b x Definition (staircase) k sequence points ( v 1 , . . . , v k ). Two base points b x , b y . ( b x = b y possible.) For all v i , b x is the x -neighbor of v i in the third quadrant of v i . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  48. After the sweep v 1 v 2 v 3 v 4 b y b x Definition (staircase) k sequence points ( v 1 , . . . , v k ). Two base points b x , b y . ( b x = b y possible.) For all v i , b x is the x -neighbor of v i in the third quadrant of v i . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  49. After the sweep v 1 v 2 v 3 v 4 b y b x Definition (staircase) k sequence points ( v 1 , . . . , v k ). Two base points b x , b y . ( b x = b y possible.) For all v i , b x is the x -neighbor of v i in the third quadrant of v i . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  50. After the sweep v 1 v 2 v 3 v 4 b y b x Definition (staircase) k sequence points ( v 1 , . . . , v k ). Two base points b x , b y . ( b x = b y possible.) For all v i , b x is the x -neighbor of v i in the third quadrant of v i . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  51. After the sweep v 1 v 2 v 3 v 4 b y b x Definition (staircase) k sequence points ( v 1 , . . . , v k ). Two base points b x , b y . ( b x = b y possible.) For all v i , b x is the x -neighbor of v i in the third quadrant of v i . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  52. After the sweep v 1 v 2 v 3 v 4 b y b x Definition (staircase) k sequence points ( v 1 , . . . , v k ). Two base points b x , b y . ( b x = b y possible.) For all v i , b x is the x -neighbor of v i in the third quadrant of v i . For all v i , b y is the y -neighbor of v i in the third quadrant of v i . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  53. After the sweep v 1 v 2 v 3 v 4 b y b x Observation The grey shaded areas contain no points. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  54. After the sweep v 1 v 2 v 3 A v 4 c b y b x Observation The grey shaded areas contain no points. No sweep lines lie inside the staircase area A . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  55. After the sweep v 1 v 2 v 3 A v 4 c b y b x Observation The grey shaded areas contain no points. No sweep lines lie inside the staircase area A . ⇒ Points v 3 , . . . , v k − 2 need to be connected to cross point c . B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 10 / 14

  56. After the sweep Lemma Except for staircase areas, all critical pairs of points are connected via shortest paths after the sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 11 / 14

  57. After the sweep A 3 A 1 A 2 A 4 Lemma Except for staircase areas, all critical pairs of points are connected via shortest paths after the sweep. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 11 / 14

  58. After the sweep A 3 A 1 A 2 A 4 Lemma Except for staircase areas, all critical pairs of points are connected via shortest paths after the sweep. The staircase areas A i are bordered by as many line segments from the sweep step as possible, B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 11 / 14

  59. After the sweep A 3 A 1 A 2 A 4 Lemma Except for staircase areas, all critical pairs of points are connected via shortest paths after the sweep. The staircase areas A i are bordered by as many line segments from the sweep step as possible, and are as small as possible. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 11 / 14

  60. Approximation analysis Consider the approximation seperately: inside staircases ( A = � i A i , Phase II), and outside staircases ( A := R 2 \ A , Phase I). B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 12 / 14

  61. Approximation analysis Consider the approximation seperately: inside staircases ( A = � i A i , Phase II), and outside staircases ( A := R 2 \ A , Phase I). Phase I (Area A ) By construction, one of the two lines inserted by the sweep is justified. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 12 / 14

  62. Approximation analysis Consider the approximation seperately: inside staircases ( A = � i A i , Phase II), and outside staircases ( A := R 2 \ A , Phase I). Phase I (Area A ) By construction, one of the two lines inserted by the sweep is justified. ⇒ 2-Approximation. B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 12 / 14

  63. Approximation analysis Consider the approximation seperately: inside staircases ( A = � i A i , Phase II), and outside staircases ( A := R 2 \ A , Phase I). Phase I (Area A ) By construction, one of the two lines inserted by the sweep is justified. ⇒ 2-Approximation?! Unfortunately not: B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 12 / 14

  64. Approximation analysis Consider the approximation seperately: inside staircases ( A = � i A i , Phase II), and outside staircases ( A := R 2 \ A , Phase I). Phase I (Area A ) By construction, one of the two lines inserted by the sweep is justified. ⇒ 2-Approximation?! Unfortunately not: B. Fuchs (TU Braunschweig) Simple 3-Approximation of MMNs CTW 2008 12 / 14

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