By Elmar Langetepe

Targeting algorithms and information buildings that experience confirmed to be flexible, effective, basic, and straightforward to enforce, this ebook familiarizes scholars, in addition to practitioners within the box of special effects, with a variety of information buildings.

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Extra info for Geometric Data Structures for Computer Graphics

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7. 2. I. I. For the inductive construction of a d-dimensional range tree, let xi denote the ith coordinate. 8 for a 2D example. A d-dimensional range tree with respect to (X1, X2, …, Xd) is inductively defined as follows. 8: Some parts of the 2D range tree of eight points in 2D. The range tree T and the relevant trees for the nodes u1, u2, and u3 with , , and for the query rectangle R = I1 × I2 with intervals I1 &"unicode">⊂ Y. 11: The inductive construction algorithm of a d-dimensional range tree.

A range query with an axis-parallel rectangle can be answered in time O( + a), where a denotes the size of the answer. As mentioned earlier, the 2D kd-tree can be easily generalized to arbitrary dimension d, splitting the points successively with respect to the given axes in a balanced way. Fortunately, the depth of the balanced tree still is bounded by O(log n), and the tree can be built up in time O(n log n) and space O(n) for fixed dimension d. Therefore, the kd-tree is optimal in space. A rectangular range query can be answered in time .

We have to report the horizontal segments h stabbed by the x-coordinate x of s. For the left endpoints of such a segment h, we have 36 Chapter 2: Orthogonal Windowing and Stabbing Queries Chapter 2: Orthogonal Windowing and Stabbing Queries 37 to take care that the y-coordinate of h lies also in the interval [yl, yu]. 1. We replace the sorted list ML and MR of the left and right endpoints of Smed by two range trees. A 2D range tree ML for the left endpoints of the segments in Smed and a 2D range tree MR for the right endpoints of the segments in Smed.