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Computational Geometry

Computational Geometry. Piyush Kumar (Lecture 5: Range Searching). Welcome to CIS5930. Range Searching : recap. 1D Range search kD trees Range Trees Fractional Cascading. Range Searching. Preprocess a set P of objects for efficiently answering queries.

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Computational Geometry

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  1. Computational Geometry Piyush Kumar (Lecture 5: Range Searching) Welcome to CIS5930

  2. Range Searching : recap • 1D Range search • kD trees • Range Trees • Fractional Cascading

  3. Range Searching • Preprocess a set P of objects for efficiently answering queries. • Typically, P is a collection of geometric objects (points, rectangles, polygons) in Rd. • Query Range, Q : d-rectangles, balls, halfspaces, simplices, etc.. • Either count all objects in P  Q or report the objects themselves. Courtesy Bhosle 

  4. Example: Points in R2 Q1 Q2

  5. Applications • Databases • Spatial databases (G.I.S.) • Computer Graphics • Robotics • Vision

  6. 7 4 12 2 5 8 15 2 4 5 7 8 12 15 19 Range Trees (1D) 6 17 2 4 5 7 8 12 15 19 Counting ? Reporting?

  7. Range Trees (2d) P3 P8 P5 P2 P6 P4 P7 P1 P3 P2 P4 P1 P1 P2 P3 P4 P5 P6 P7 P8

  8. Query and Space Complexity(counting) • 1D • Query : O(log n) Space: O(n) • 2D • Query : O(log n) Space O(nlogn) • Construction time : O(nlogn) • d-D • Query : O(logd-1n) Space: O(nlogd-1n) • Construction time : O(nlogd-1n)

  9. Kd-Trees • a typical struct • Int cut_dim; // dim orthogonal to cutting plane • Double cut_val; // location of cutting plane • Int size; // number of points in subtree • The best implementation of kd-trees I know of: • http://www.cs.umd.edu/~mount/ANN/ • Look at kd_tree.cc and kd_tree.h

  10. kD-Trees (k-dimensional Trees) • 1-d tree : split along median point and recursively build subtrees for the left and right sets. • Higher dimensions : same approach, but cycle through the dimensions. Or, select the next dimension as the one with the widestspread. • Efficiency of query processing drops as dimensions increase (becomes almost linear). However, the space requirement remains linear : O(n.d)

  11. kD-Trees (Contd.) c o m d f l n a b e g j k h i f k h n d i j l a b o c g e m

  12. kD-Trees (Contd.) • Query complexity : How many cells can a query box intersect ? Let us consider a facet of the query • Any axis parallel line can intersect atmost 2 of these 4 cells. • Each of these 4 cells contain exactly n/4 points. Q(n) = 2.Q(n/4) + 1 Q(n) = O(n1/2) i.e. Query answered in O(n1-1/d + m) time where m is the output size

  13. Kd-Tree • Summarizing • Building (preprocessing time): O(n log n) • Size: O(n) • Range queries: O(sqrt(n)+k) • In higher dimensions • Building: O(dnlogn) • Size: O(dn) • Counting Query: O(n1-1/d)

  14. QuadTrees • 4-way partitions • Linear space • Used in real life more than kd-trees

  15. Octrees • 3D Version of Quadtrees • 8 child nodes • Applications • Range searching • Collision detection • Mesh generation • Visibility Culling • Computer games

  16. General Sets of points • We assumed till now that the x,y coordinates of the points are distinct and do not overlap the coordinates of the query. • How do we relax this?

  17. More on the Chalk board…

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