Fractional cascading: II. Applications

@article{Chazelle2005FractionalCI,
  title={Fractional cascading: II. Applications},
  author={Bernard Chazelle and Leonidas J. Guibas},
  journal={Algorithmica},
  year={2005},
  volume={1},
  pages={163-191}
}
This paper presents several applications offractional cascading, a new searching technique which has been described in a companion paper. The applications center around a variety of geometric query problems. Examples include intersecting a polygonal path with a line, slanted range search, orthogonal range search, computing locus functions, and others. Some results on the optimality of fractional cascading, and certain extensions of the technique for retrieving additional information are also… 

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References

SHOWING 1-10 OF 22 REFERENCES
Linear data structures for two types of range search
TLDR
Two algorithms for pictures of polyhedral scenes are presented that use a reciprocal diagram in the plane to characterize pictures of strict oriented polyhedra in space and a combinatorial algorithm that uses counts on the incidence structure.
An Optimal Worst Case Algorithm for Reporting Intersections of Rectangles
TLDR
This paper investigates the problem of reporting all intersecting pairs in a set of n rectilinearly oriented rectangles in the plane and describes an algorithm that solves this problem in worst case time proportional to n lg n + k, where k is the number of interesecting pairs found.
Multidimensional divide-and-conquer
TLDR
Multidimensional divide-and-conquer is discussed, an algorithmic paradigm that can be instantiated in many different ways to yield a number of algorithms and data structures for multidimensional problems.
The power of geometric duality
TLDR
A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane.
Visibility and intersection problems in plane geometry
TLDR
New data structures for solving various visibility and intersection problems about a simple polygonP onn vertices are developed and anO(logn)-time algorithm for determining which side ofP is first hit by a bullet fired from a point in a certain direction is developed.
Searching and Storing Similar Lists
  • R. Cole
  • Computer Science
    J. Algorithms
  • 1986
Location of a point in a planar subdivision and its applications
TLDR
A search algorithm, called point-location algorithm, is presented, which operates on a suitably preprocessed data structure, and yields interesting and efficient solutions of other geometric problems, such as spatial convex inclusion and inclusion in an arbitrary polygon.
How to Search in History
...
1
2
3
...