1. Segment Tree and Its Usage for geometric Computations
Shmuel Wimer
Bar Ilan Univ., Eng. Faculty
Technion, EE Faculty
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2. Motivation
We may be interested in calculating the underling area, perimeter or contour construction
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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
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6. 4,15
9,15
4,9
6,9
4,6
12,15
9,12
5,6
4,5
7,9
6,7
10,12
9,10
13,15
12,13
8,9
7,8
11,12
10,11
14,15
13,14
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7. Insertion and Deletion
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8. Insertion and Deletion
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10. 1,257
1,129
65,129
97,129
65,97
65,81
81,97
97,115
105,109
97,105
105,115
105,107
73,81
73,77
73,75
74,75
75,77
77,81
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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).
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14. 1D Measure of Union of Intervals
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15. 1D Measure of Union of Intervals
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17. 2D Measure (Area) of Union of Rectangles
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18. 2D Measure (Area) of Union of Rectangles
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19. scan-line
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20. Efficient Calculation of m(xi)
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23. Perimeter of Union of Rectangles
Perimeter is the length sum of:
vertical edges
horizontal edges
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30. The Contour of Union of Rectangles
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40. Completing the Cycles of Contour
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42. e1 e3 e2 e6 e5 e8 e7 e4 e9
e10
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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.
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44. Run-time Complexity
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46. U’ U”
|P(U’)+ P(U”)|=7 U’ U”
|P(U’)+ P(U”)|=5
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