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3-D Vertical Ray Shooting and 2-D Point Enclosure, Range Searching, and Arc Shooting Amidst Convex Fat Objects

Matthew Katz 1
1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : We present a new data structure for a set of $n$ convex simply-shaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) {\em vertical ray shooting} --- preprocess a set $\K$ of $n$ non-intersecting convex simply-shaped flat objects in 3-space, whose $xy$-projections are fat, for efficient vertical ray shooting queries, (ii) {\em point enclosure} --- preprocess a set $\Cn$ convex simply-shap- ed fat objects in the plane, so that the $k$ objects containing a query point $p$ can be reported efficiently, (iii) {\em bounded-size range searching} --- preprocess a set $\Cn$ convex fat polygons, so that the $k$ objects intersecting a `not-too-large' query polygon can be reported efficiently, and (iv) {\em bounded-size segment shooting} --- preprocess a set $\Cn (iii), so that the first object (if exists) hit by a `not-too-long' oriented query segment can be found efficiently. For the first three problems we construct data structures of size $O(\lambda_s(n) \log^3 n)$, where $s$ is the maximum number of intersections between the boundaries of the ($xy$-projections) of any pair of objects, and $\lambda_s(n)$ is the maximum length of $(n,s)$ Davenport-Schinzel sequences. The data structure for the fourth problem is of size $O(\lambda_s(n) \log^2 n)$. The query time in the first problem is $O(\log^4 n)$, the query time in the second and third problems is $O(\log^3 n + k \log^2 n)$, and the query time in the fourth problem is $O(\log^3 n)$. We also present a simple algorithm for computing a depth order for a set $\K$ as in (i), that is based on the solution to the vertical ray shooting problem. (A depth order for $\K$, if exists, is a linear order of $\K$, such that, if $K_1,K_2 \in \K$ and $K_1$ lies vertically above $K_2$, then $K_1$ precedes $K_2$.) The algorithm is able to determine whether such an order exists, unlike the algorithm of Agarwal~et~al. \cite{AgKS} that might output a false order when a depth order does not exist, and it is often more efficient in practical situations than the latter algorithm.
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Submitted on : Wednesday, May 24, 2006 - 2:31:47 PM
Last modification on : Friday, February 4, 2022 - 3:16:27 AM
Long-term archiving on: : Thursday, March 24, 2011 - 2:13:02 PM


  • HAL Id : inria-00074101, version 1



Matthew Katz. 3-D Vertical Ray Shooting and 2-D Point Enclosure, Range Searching, and Arc Shooting Amidst Convex Fat Objects. RR-2583, INRIA. 1995. ⟨inria-00074101⟩



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