# 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…

## 80 Citations

Fractional cascading: I. A data structuring technique

- Computer ScienceAlgorithmica
- 2005

This paper shows that, if ordered lists can be put in a one-to-one correspondence with the nodes of a graph of degreed so that the iterative search always proceeds along edges of that graph, then this structure can be built, called afractional cascading structure, in which all original searches after the first can be carried out at only logd extra cost per search.

Polygonal Path Approximation: A Query Based Approach

- Computer Science, MathematicsISAAC
- 2003

It is shown that the query based approach can be used to obtain a subquadratic time exact algorithm with infinite beam criterion and Euclidean distance metric if some condition on the input path holds.

Polygonal chain approximation: a query based approach

- Computer Science, MathematicsComput. Geom.
- 2005

Non-orthogonal Range Searching: A Review

- Computer Science
- 2008

The long history of nonorthogonal range searching is reviewed from its beginning to the present and the latest advances in this area are shown, including heuristic algorithms, small improvements for higher dimensions, and new problems on range searching with a variety of applications.

2D Generalization of Fractional Cascading on Axis-aligned Planar Subdivisions

- Computer Science, Mathematics2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

This paper shows that it is actually possible to circumvent the lower bound of Chazelle and Liu for axis-aligned planar subdivisions for two-dimensional fractional cascading, and presents a number of upper and lower bounds which reveal that in two-dimensions, the problem has a much richer structure.

Partial Enclosure Range Searching ∗

- Computer Science, Mathematics
- 2015

A new type of range searching problem, called the partial enclosure range searching problem, is introduced in this paper. Given a set of geometric objects S and a query region Q, our goal is to…

Optimal cooperative search in fractional cascaded data structures

- Computer ScienceAlgorithmica
- 2005

This paper shows how to preprocess a variety of fractional cascaded data structures whose underlying graph is a tree so that searching can be done efficiently in parallel.

Hopcroft's Problem, Log-Star Shaving, 2D Fractional Cascading, and Decision Trees

- Computer Science, MathematicsSODA
- 2022

This work revisits Hopcroft’s problem and related fundamental problems about geometric range searching and describes two interesting and different ways to achieve the result, which is randomized and uses a new 2D version of fractional cascading for arrangements of lines and deterministic and uses decision trees in a manner inspired by the sorting technique of Fredman (1976).

Chapter 4 Fractional Cascading

- Computer Science, Mathematics

In this chapter, this algorithm design principle called fractional cascading is studied, which says that many problems can be solved by rst solving (recursively) a subproblem whose size is a constant fraction of the original problem size and then using this solution to get back to a solution of theOriginal problem.

Hierarchical Decompositions and Circular Ray Shooting
in Simple Polygons

- Computer ScienceDiscret. Comput. Geom.
- 2004

A hierarchical decomposition of a simple polygon is introduced that has logarithmic depth, linear size, and its regions have at most three neighbors and can be answered in O(log2 n) query time and O(n log n) space.

## References

SHOWING 1-10 OF 22 REFERENCES

Linear data structures for two types of range search

- Computer ScienceSCG '86
- 1986

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

- Computer ScienceIEEE Transactions on Computers
- 1980

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

- Computer ScienceCACM
- 1980

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

- Mathematics, Computer Science
- 1985

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

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 1989

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.

A Class of Algorithms which Require Nonlinear Time to Maintain Disjoint Sets

- Computer ScienceJ. Comput. Syst. Sci.
- 1979

Location of a point in a planar subdivision and its applications

- Mathematics, Computer ScienceSTOC '76
- 1976

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.