# Dynamic fractional cascading

@article{Mehlhorn2005DynamicFC, title={Dynamic fractional cascading}, author={Kurt Mehlhorn and Stefan N{\"a}her}, journal={Algorithmica}, year={2005}, volume={5}, pages={215-241} }

The problem of searching for a key in many ordered lists arises frequently in computational geometry. Chazelle and Guibas recently introduced fractional cascading as a general technique for solving this type of problem. In this paper we show that fractional cascading also supports insertions into and deletions from the lists efficiently. More specifically, we show that a search for a key inn lists takes timeO(logN +n log logN) and an insertion or deletion takes timeO(log logN). HereN is the…

## 112 Citations

A Lower Bound for Dynamic Fractional Cascading

- Computer ScienceSODA
- 2021

A lower bound of $\Omega( \log n \sqrt{\log\log n})$ is proved on the worst-case query time of dynamic fractional cascading, when queries are paths of length $O(\log n)$.

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.

Dynamic Orthogonal Range Searching on the RAM, Revisited

- Computer ScienceSoCG
- 2017

This work presents a new data structure achieving O(log n/log log n+k) optimal query time and O (log2/3+o(1)n) update time (amortized) in the word RAM model, where n is the number of data points and k is the output size.

Optimal Dynamic Strings

- Computer ScienceSODA
- 2018

This paper presents an efficient data structure for maintaining a dynamic collection of strings under the following operations, and proves that even if the only possible query is checking equality of two strings, either updates or queries take amortized $\Omega(\log n)$ time; hence the implementation is optimal.

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.

Reference Space Query Time Insertion Time Deletion Time Bentley

- Computer Science
- 2019

A fully-dynamic data structure for the planar point location problem in the external memory model with almost-optimal query cost, which almost matches the best known upper bound in the internal-memory model.

Amortized Bounds for Dynamic Orthogonal Range Reporting

- Computer ScienceESA
- 2014

The fundamental problem of 2-D dynamic orthogonal range reporting for 2- and 3-sided queries in the standard word RAM model is considered, and a step forward is made for an important problem that has not seen any progress in recent years.

2D Fractional Cascading on Axis-aligned Planar Subdivisions

- Computer Science, MathematicsArXiv
- 2020

It is shown that it is possible to circumvent the lower bound of Chazelle and Liu for axis-aligned planar subdivisions for fractional cascading, and a number of upper and lower bounds are presented which reveal that in 2D, the problem has a much richer structure.

Worst-Case Efficient Dynamic Geometric Independent Set

- Computer Science, MathematicsESA
- 2021

A data structure is presented that maintains a constant-factor approximate maximum independent set for broad classes of fat objects in d dimensions in sublinear worst-case update time, giving the first results for dynamic independent set in a wide variety of geometric settings.

Sequential Dependency Computation via Geometric Data Structures

- Mathematics, Computer ScienceCCCG
- 2011

## References

SHOWING 1-10 OF 64 REFERENCES

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.

Maintaining order in a generalized linked list

- Computer ScienceActa Informatica
- 2004

A representation for linked lists which allows to efficiently insert and delete objects in the list and to quickly determine the order of two list elements and an algorithm which determines the ancestor relationship of two given nodes in a dynamic tree structure of bounded degree in time O(1).

Dynamization of geometric data structures

- Computer ScienceSCG '85
- 1985

An amortized analysis of update cost in fractional cascading is given and it is shown that insertions take 0( 1) isortized time and insertions and deletions take0( log log X) amortization time.

New Data Structures for Orthogonal Queries.

- Computer Science
- 1979

This report shows that many of the earlier results can be improved by a factor of log N with a slightly modified data structure that enables k-dimensional searches to be performed in O(log N) to the k-1 power) time.

A linear-time algorithm for a special case of disjoint set union

- Computer Science, MathematicsJ. Comput. Syst. Sci.
- 1985

A linear-time algorithm for the special case of the disjoint set union problem in which the structure of the unions (defined by a “union tree”) is known in advance, which gives similar improvements in the efficiency of algorithms for solving a number of other problems.

Rectilinear Line Segment Intersection, Layered Segment Trees, and Dynamization

- Computer ScienceJ. Algorithms
- 1982

A data structure for orthogonal range queries

- Computer Science19th Annual Symposium on Foundations of Computer Science (sfcs 1978)
- 1978

It is shown that a decision tree of height O(dn log n) can be constructed to process n operations in d dimensions, suggesting that the standard decision tree model will not provide a useful method for investigating the complexity of orthogonal range queries.

A Lower Bound on the Complexity of the Union-Split-Find Problem

- Computer ScienceSIAM J. Comput.
- 1988

A $\Theta (\log n)$ bound on the complexity of the Union-Split-Find problem is proved, which shows that the separation assumption can imply an exponential loss in efficiency.