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MIT6_851S10_lec06

MIT6_851S10_lec06 - 6.851 Advanced Data Structures Spring...

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6.851: Advanced Data Structures Spring 2010 Lecture 6 February 23, 2010 Dr. Andr´ e Schulz 1 Overview In the last lecture we introduced ray shooting, where we determine which is the first object in a set intersected by a given ray. We overviewed how to solve this problem if our objects are simple polygons. This lecture explores ray shooting more generally, beginning with data structures designed to perform halfspace and simplex range queries such as partition trees, and continuing with an explanation of how to use these data structures to perform ray shooting. 2 Partition Trees Problem. Given a pointset S = { p 1 , p 2 , . . . , p n } , we would like to perform two sorts of queries: 1. Halfspace Range Queries: find properties relating to the subset of S on one side of a line h q (e.g., how many points are above h q ?). 2. Simplex Range Queries: find properties relating to the subset of S inside a simplex t q (e.g., how many points lie inside t q ?). In two dimensions, a simplex is a triangle, and we will use two-dimensional examples for the remainder of these notes. Idea. Partition S into r disjoint subsets S 1 , S 2 , . . . , S r . Each subset S i is associated with a triangle t i that contains the points in that subset (the triangles need not be disjoint). We call this partition Ψ = { ( S 1 , t 1 ) , ( S 2 , t 2 ) , . . . , ( S r , t r ) } . When finding the points in the halfplane defined by a query line h q , we can easily accept or discard as necessary the points lying in a triangle that do not intersect h q . We then recurse on the remaining subsets in triangles intersecting h q . h q discard accept recurse
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The crossing number of Ψ is the maximum number of triangles that can be intersected by some line. We call Ψ fine if | S i | ≤ 2 r n S i , meaning every subset of S has no more than twice the average number of points per subset (i.e., the subsets are fairly equally distributed).
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