Algorithms for ray-shooting and intersection searching

doi:10.1016/0196-6774(92)90062-H

Journal of Algorithms

Volume 13, Issue 4, December 1992, Pages 670-692

https://doi.org/10.1016/0196-6774(92)90062-H Get rights and content

Abstract

Space-efficient algorithms are presented for some geometric intersection searching problems such as ray-shooting, segment intersection searching, and triangle stabbing. The algorithms for ray-shooting and segment intersection searching improve upon the algorithms given previously. These algorithms also have significantly lower query times than the algorithms previously presented, but the space is larger by a low-degree polylog factor. Efficient dynamic algorithms are also presented for the above problems and for triangle range searching by taking advantage of their decomposability.

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We give new methods for maintaining a data structure that supports ray-shooting and shortest-path queries in a dynamically changing connected planar subdivision $S$ . Our approach is based on a new dynamic method for maintaining a balanced decomposition of a simple polygon via geodesic triangles. We maintain such triangulations by viewing their dual trees as balanced trees. We show that rotations in these trees can be implemented via simple “diagonal swapping” operations performed on the corresponding geodesic triangles, and that edge insertion and deletion can be implemented on these trees using operations akin to the standardsplitandspliceoperations. We also maintain a dynamic point location structure on the geodesic triangulation, so that we may implement ray-shooting queries by first locating the ray's endpoint and then walking along the ray from geodesic triangle to geodesic triangle until we hit the boundary of some region of $S$ . The shortest path between two points in the same region is obtained by locating the two points and then walking from geodesic triangle to geodesic triangle either following a boundary or taking a shortcut through a common tangent. Our data structure usesO(n) space and supports queries and updates inO(log² n) worst-case time, wherenis the current size of $S$ . It outperforms the previous best data structure for this problem by a log nfactor in all the complexity measures (space, query times, and update times).
Algorithms for generalized halfspace range searching and other intersection searching problems
1996, Computational Geometry: Theory and Applications
In a generalized intersection searching problem, a set S of colored geometric objects is to be preprocessed so that, given a query object q, the distinct colors of the objects of S that are intersected by q can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and have many applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented (i.e., axes-parallel) or where the color classes satisfy additional properties (e.g., connectedness). In this paper, efficient algorithms are given for several generalized problems involving objects that are not necessarily iso-oriented. These problems include: generalized halfspace range searching in R^d, for any fixed d >− 2, and segment intersection searching, triangle stabbing, and triangle range searching in R>² for certain classes of line segments and triangles. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search.
Algorithms for generalized halfspace range searching and other intersection searching problems
1996, Computational Geometry: Theory and Applications
In a generalized intersection searching problem, a set S of colored geometric objects is to be preprocessed so that, given a query object q, the distinct colors of the objects of S that are intersected by q can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and have many applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented (i.e., axes-parallel) or where the color classes satisfy additional properties (e.g., connectedness). In this paper, efficient algorithms are given for several generalized problems involving objects that are not necessarily iso-oriented. These problems include: generalized halfspace range searching in $R^{d}$ , for any fixed d ≥ 2, and segment intersection searching, triangle stabbing, and triangle range searching in $R^{2}$ for certain classes of line segments and triangles. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search.
Further results on generalized intersection searching problems: Counting, reporting, and dynamization
1995, Journal of Algorithms
In a generalized intersection searching problem, a set, S, of colored geometric objects is to be preprocessed so that given some query object, q, the distinct colors of the objects intersected by q can be reported efficiently or the number of such colors can be counted efficiently. In the dynamic setting, colored objects can be inserted into or deleted from S. These problems generalize the well-studied standard intersection searching problems and are rich in applications. Unfortunately, the techniques known for the standard problems do not yield efficient solutions for the generalized problems. Moreover, previous work (R. Janardan and M. Lopez, Generalized intersection searching problems, Internat. J. Comput. Geom. Appl.3 (1993), 39-69) on generalized problems applies only to the static reporting problems. In this paper, a uniform framework is presented to solve efficiently the counting/reporting versions of a variety of generalized intersection searching problems in static/dynamic settings. These problems include 1-, 2-, and 3-dimensional range searching, quadrant searching, interval intersection searching, 1- and 2-dimensional point enclosure searching, and orthogonal segment intersection searching.

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^☆: Research partly supported by a grant-in-aid of research from the Graduate School of the University of Minnesota. The second author was also supported in part by NSF Grant CCR-8808574.

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Algorithms for ray-shooting and intersection searching☆

Abstract

J. Comput. System Sci.

Inform. Process. Lett.

Ray shooting and other applications of spanning trees with low stabbing number

Applications of a new space partitioning technique

An optimal algorithm for intersecting line segments in the plane

Fractional cascading. I, A data structuring technique; II. Applications

Algorithmica

The power of geometric duality

BIT

Filtering search: A new approach to query answering

SIAM J. Comput.

Space-efficient ray-shooting and intersection searching: algorithms, dynamization, and applications

Quasi-optimal range searching in spaces with finite VC-dimension

Discrete Comput. Geom.