1 / 51

Segment Tree and Its Usage for geometric Computations

Segment Tree and Its Usage for geometric Computations. Shmuel Wimer Bar Ilan Univ., Eng. Faculty Technion, EE Faculty. Motivation. We may be interested in calculating the underling area, perimeter or contour construction. Segment Tree Definition. Introduced by J. L. Bentley in 1977

verne
Download Presentation

Segment Tree and Its Usage for geometric Computations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Segment Tree and Its Usage for geometric Computations Shmuel Wimer Bar Ilan Univ., Eng. Faculty Technion, EE Faculty Segment Tree

  2. Motivation We may be interested in calculating the underling area, perimeter or contour construction Segment Tree

  3. Segment Tree Definition • Introduced by J. L. Bentley in 1977 • Data structure designed to handle intervals on the real line • Intervals end points belong to a fixed set of abscissas • Abscissas can be normalized to range [1,N] without loss of generality by using a lookup table • Given an interval [l,r], the segment tree T(l,r) is a rooted binary tree defined recursively Segment Tree

  4. Segment Tree

  5. Segment Tree

  6. 4,15 4,9 9,15 4,6 6,9 9,12 12,15 4,5 5,6 6,7 7,9 9,10 10,12 12,13 13,15 7,8 8,9 10,11 11,12 13,14 14,15 Segment Tree

  7. Insertion and Deletion Segment Tree

  8. Insertion and Deletion Segment Tree

  9. Segment Tree

  10. 73,77 1,257 65,97 81,97 73,81 1,129 74,75 75,77 77,81 73,75 65,129 97,129 65,81 97,115 97,105 105,115 105,109 105,107 Segment Tree

  11. Segment Tree

  12. Segment Tree

  13. Allocation and De-Allocation • Depends on application. • If we wish to know the cardinality of cover of [B[v],E[v]] then a counter C[v] is associated with node v: • C[v] = C[V]+1 is allocation for INSERT • C[v] = C[V]-1 is de-allocation for DELETE • In many cases C[v] indicates the presence of material, so we’ll be interested in whether C[v] > 0 (material exists) or C[v]==0 (no material). Segment Tree

  14. 1D Measure of Union of Intervals Segment Tree

  15. 1D Measure of Union of Intervals Segment Tree

  16. Segment Tree

  17. 2D Measure (Area) of Union of Rectangles Segment Tree

  18. 2D Measure (Area) of Union of Rectangles Segment Tree

  19. scan-line Segment Tree

  20. Efficient Calculation of m(xi) Segment Tree

  21. Segment Tree

  22. Segment Tree

  23. Perimeter of Union of Rectangles Perimeter is the length sum of: vertical edges horizontal edges Segment Tree

  24. Segment Tree

  25. Segment Tree

  26. Segment Tree

  27. Segment Tree

  28. Segment Tree

  29. Segment Tree

  30. The Contour of Union of Rectangles Segment Tree

  31. Segment Tree

  32. Segment Tree

  33. Segment Tree

  34. Segment Tree

  35. Segment Tree

  36. Segment Tree

  37. Segment Tree

  38. Segment Tree

  39. Segment Tree

  40. Completing the Cycles of Contour Segment Tree

  41. Segment Tree

  42. e3 e6 e8 e1 e10 e5 e7 e2 e9 e4 Segment Tree

  43. There’s no ambiguity in deciding whether to go to left or right triplet when an horizontal edge is decided. It follows that a pair of successive triplets defines horizontal edges. Consequently, once two successive triplets are traversed and define a new horizontal edge, the number of triplets on both the left and the right parts of the list must be even. Therefore, if the index of a triplet is even, its left adjacent triplet is paired, otherwise, the right triplet is paired. Segment Tree

  44. Run-time Complexity Segment Tree

  45. Segment Tree

  46. U’ U” U’ U” |P(U’)+ P(U”)|=7 |P(U’)+ P(U”)|=5 Segment Tree

  47. Segment Tree

  48. Segment Tree

  49. Segment Tree

  50. Segment Tree

More Related