computational geometry piyush kumar (lecture 5: range searching) welcome to cis5930
TRANSCRIPT
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 Rd.
Query Range, Q : d-rectangles, balls, halfspaces, simplices, etc..
Either count all objects in P Q or report the objects themselves.
Courtesy Bhosle
Q1
Example: Points in R2
Q2
Applications
Databases Spatial databases (G.I.S.) Computer Graphics Robotics Vision
Range Trees (1D)
2 4 5 7 8 12 15 19
6 17 7
7 19151282 4 5
2
4
5 8
12
15Counting ?Reporting?
Range Trees (2d)
P4P3P2P1
P3
P2
P4
P1
P1
P2
P3
P4
P8P7P6P5
P5 P8
P6
P7
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)
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
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.)
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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
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(n1-1/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?
More on the Chalk board…