Segment Tree and Its Usage for geometric Computations

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

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34. Segment Tree

35. Segment Tree

36. Segment Tree

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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

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50. Segment Tree