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. Typically, P is a collection of geometric objects (points, rectangles, polygons) in R d. Query Range, Q : d-rectangles, balls, halfspaces, simplices, etc.. Either count all objects in P  Q or report the objects themselves. Courtesy Bhosle

Q1Q1 Example: Points in R 2 Q2Q2

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

Range Trees (1D) Counting ? Reporting?

Range Trees (2d) P4P4 P3P3 P2P2 P1P1 P3P3 P2P2 P4P4 P1P1 P1P1 P2P2 P3P3 P4P4 P8P8 P7P7 P6P6 P5P5 P5P5 P8P8 P6P6 P7P7

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(log d-1 n) Space: O(nlog d-1 n)  Construction time : O(nlog d-1 n)

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:   Look at kd_tree.cc and kd_tree.h

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 widest spread. Efficiency of query processing drops as dimensions increase (becomes almost linear). However, the space requirement remains linear : O(n.d)

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

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

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(n 1-1/d )

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

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

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?

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