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Advanced Algorithms COMS31900 Orthogonal Range Searching Rapha el Clifford Slides by Benjamin Sach Orthogonal range searching A 2D range searching data structure stores n distinct ( x, y ) -pairs and supports: the lookup ( x 1 , x 2 ,

  1. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67

  2. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67

  3. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67

  4. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67

  5. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67

  6. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67

  7. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67

  8. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67 67 > 64 = x 2

  9. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67

  10. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67 k lookups take O (log n + k ) time ( k is the number of points reported)

  11. Starting simple. . . 1D range searching build a sorted array containing the x -coordinates in O ( n log n ) preprocessing (prep.) time and O ( n ) space to perform lookup ( x 1 , x 2 ) . . . find the successor of x 1 by binary search and then ‘walk’ right x 1 = 15 x 2 = 64 3 3 7 11 19 23 27 35 43 53 61 67 n 3 7 11 19 23 27 35 43 53 61 67 k lookups take O (log n + k ) time ( k is the number of points reported) this is called being ‘output sensitive’

  12. Starting simple. . . 1D range searching

  13. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . .

  14. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle

  15. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . half the points are to the left half the points are to the right find the point in the middle

  16. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle

  17. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle . . . and recurse on each half

  18. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle . . . and recurse on each half

  19. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle . . . and recurse on each half

  20. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle . . . and recurse on each half (in a tie, pick the left)

  21. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle . . . and recurse on each half (in a tie, pick the left)

  22. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle . . . and recurse on each half (in a tie, pick the left)

  23. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle . . . and recurse on each half (in a tie, pick the left)

  24. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle . . . and recurse on each half (in a tie, pick the left) We can store the tree in O ( n ) space (it has one node per point)

  25. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . find the point in the middle . . . and recurse on each half (in a tie, pick the left) We can store the tree in O ( n ) space (it has one node per point) It has O (log n ) depth

  26. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . O (log n ) find the point in the middle . . . and recurse on each half (in a tie, pick the left) We can store the tree in O ( n ) space (it has one node per point) It has O (log n ) depth

  27. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . O (log n ) find the point in the middle . . . and recurse on each half (in a tie, pick the left) We can store the tree in O ( n ) space (it has one node per point) It has O (log n ) depth and can be built in O ( n log n ) time

  28. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . O (log n ) find the point in the middle . . . and recurse on each half (in a tie, pick the left) We can store the tree in O ( n ) space (it has one node per point) It has O (log n ) depth and can be built in O ( n log n ) time

  29. Starting simple. . . 1D range searching alternatively we could build a balanced tree. . . O (log n ) find the point in the middle . . . and recurse on each half (in a tie, pick the left) We can store the tree in O ( n ) space (it has one node per point) It has O (log n ) depth and can be built in O ( n log n ) time ( O ( n ) time if the points are sorted)

  30. Starting simple. . . 1D range searching how do we do a lookup?

  31. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2

  32. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 Step 1: find the successor of x 1

  33. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 Step 1: find the successor of x 1

  34. Starting simple. . . 1D range searching how do we do a lookup? x 1 is to the left x 1 x 2 Step 1: find the successor of x 1

  35. Starting simple. . . 1D range searching how do we do a lookup? x 1 is to the right x 1 x 2 Step 1: find the successor of x 1

  36. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 Step 1: find the successor of x 1

  37. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 Step 1: find the successor of x 1 in O (log n ) time

  38. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 Step 2: find the predecessor of x 2

  39. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 Step 2: find the predecessor of x 2

  40. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 Step 2: find the predecessor of x 2

  41. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 Step 2: find the predecessor of x 2

  42. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2

  43. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  44. Starting simple. . . 1D range searching how do we do a lookup? x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  45. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  46. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  47. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  48. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  49. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  50. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  51. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  52. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  53. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  54. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  55. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  56. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  57. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  58. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  59. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  60. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  61. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  62. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path after the split this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output?

  63. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path after the split this is called an off-path edge “it’s all or nothing” x 1 x 2 in O (log n ) time Step 1: find the successor of x 1 in O (log n ) time Step 2: find the predecessor of x 2 which points in the tree should we output? those in the O (log n ) selected subtrees on the path

  64. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path after the split this is called an off-path edge “it’s all or nothing” x 1 x 2

  65. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path after the split this is called an off-path edge “it’s all or nothing” x 1 x 2 as before lookups take O (log n + k ) time ( k is the number of points reported)

  66. Starting simple. . . 1D range searching how do we do a lookup? look at any node on the path after the split this is called an off-path edge “it’s all or nothing” x 1 x 2 as before lookups take O (log n + k ) time ( k is the number of points reported) so what have we gained?

  67. Subtree decomposition root Warning: the root to split path isn’t to scale too big split off-path edge too small x 1 x 2 off-path subtree

  68. Subtree decomposition root Warning: the root to split path isn’t to scale too big split off-path edge too small x 1 x 2 off-path subtree after the paths to x 1 and x 2 split. . .

  69. Subtree decomposition root Warning: the root to split path isn’t to scale too big split off-path edge too small x 1 x 2 off-path subtree after the paths to x 1 and x 2 split. . . any off-path subtree is either in or out

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