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Approximating the Depth via Sampling and Emptiness

Approximating the Depth via Sampling and Emptiness. Approximating the Depth via Sampling and Emptiness. Approximating the Depth via Sampling and Emptiness. Approximating the Depth via Sampling and Emptiness. Example: Range tree. S = Set of points in the plane r = Query Rectangle.

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Approximating the Depth via Sampling and Emptiness

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  1. Approximating the Depth via Sampling and Emptiness

  2. Approximating the Depth via Sampling and Emptiness

  3. Approximating the Depth via Sampling and Emptiness

  4. Approximating the Depth via Sampling and Emptiness

  5. Example: Range tree • S = Set of points in the plane • r = Query Rectangle

  6. Range tree – primary tree

  7. Range tree query • Given interval [a,b], search for a and b • Find where the paths split, look at subtrees along the search Paths split a b

  8. Range tree complexity

  9. Approximating the Depth via Sampling and Emptiness

  10. Approximating the Depth via Sampling and Emptiness

  11. Approximating the Depth via Sampling and Emptiness

  12. Approximating the Depth via Sampling and Emptiness S S’

  13. Approximating the Depth via Sampling and Emptiness

  14. Motivation

  15. Halfplane emptiness

  16. Idea 1

  17. Idea 1 – problems

  18. Idea 1 – problems

  19. Idea 2

  20. - problems Idea 2

  21. Halfspace complexity status

  22. The decision procedure < ? >

  23. Why δ? <

  24. The data structure

  25. Answering a query

  26. Lemma 1

  27. Reminder - Chernoff bound

  28. Lemma 1 - proof

  29. Lemma 1 - proof

  30. Lemma 2

  31. Lemma 2 - cont

  32. Lemma 3

  33. Range counting data structure

  34. Range counting data structure

  35. Answering a query

  36. “Light” depth

  37. “Light” depth – option 1

  38. “Light” depth – option 2

  39. “Heavy” depth

  40. “Heavy” depth

  41. Improved data structure

  42. Answering a query

  43. Answering a query

  44. Answering a query

  45. Answering a query

  46. Answering a query

  47. Answering a query

  48. Coarse search

  49. Coarse search

  50. Coarse search

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