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Robotics Research Technical Report: Ray Shooting, Implicit Point Location, and Related Queries in Arrangements of Segments (Classic Reprint) - Hardcover
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Excerpt from Robotics Research Technical Report: Ray Shooting, Implicit Point Location, and Related Queries in Arrangements of Segments
Typically, the queries we are interested in are trivially answered in 0(log n) time, if we are willing to calculate and store A explicitly. Since the combinatorial complexity of A can be quadratic in n in the worst case, this might be too expensive, and we therefore seek alterna tive methods of more efficient but implicit representation of A, which still facilitates reason ably efficient query handling, although not as efficient as the (optimal) 0(log n) time that can be achieved with explicit representation.
As a typical example, consider the following implicit point location problem. Given such an arrangement A, preprocess it so that, given any query point 1, we can calculate quickly certain properties of the face of A containing x. For example, suppose A is formed as the overlay of n (possibly intersecting) triangles, and the property in question is how many triangles contain the query point x, or more generally, if each triangle is associated a certain weight, what is the sum of the weights of the triangles covering x, etc. Clearly, by calculating the arrangement A induced by these triangles, and by further preprocessing it for efficient point location, it is easy to answer such a query in (optimal) 0(log n) time. Can we obtain faster preprocessing and more compact representation of A that will still enable us to answer these queries reasonably efficiently?
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