To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Klee's measure problem

From Wikipedia, the free encyclopedia

A set of rectangular ranges ('trellis') whose area has to be measured.

In computational geometry, Klee's measure problem is the problem of determining how efficiently the measure of a union of (multidimensional) rectangular ranges can be computed. Here, a d-dimensional rectangular range is defined to be a Cartesian product of d intervals of real numbers, which is a subset of Rd.

The problem is named after Victor Klee, who gave an algorithm for computing the length of a union of intervals (the case d = 1) which was later shown to be optimally efficient in the sense of computational complexity theory. The computational complexity of computing the area of a union of 2-dimensional rectangular ranges is now also known, but the case d ≥ 3 remains an open problem.

YouTube Encyclopedic

  • 1/3
    Views:
    5 287
    4 448
    834
  • The Art Gallery Theorem (Math)
  • Univalent Foundations: New Foundations of Mathematics | Vladimir Voevodsky
  • Spectra and Invariants under Barycentric subdivision

Transcription

History and algorithms

In 1977, Victor Klee considered the following problem: given a collection of n intervals in the real line, compute the length of their union. He then presented an algorithm to solve this problem with computational complexity (or "running time") — see Big O notation for the meaning of this statement. This algorithm, based on sorting the intervals, was later shown by Michael Fredman and Bruce Weide (1978) to be optimal.

Later in 1977, Jon Bentley considered a 2-dimensional analogue of this problem: given a collection of n rectangles, find the area of their union. He also obtained a complexity algorithm, now known as Bentley's algorithm, based on reducing the problem to n 1-dimensional problems: this is done by sweeping a vertical line across the area. Using this method, the area of the union can be computed without explicitly constructing the union itself. Bentley's algorithm is now also known to be optimal (in the 2-dimensional case), and is used in computer graphics, among other areas.

These two problems are the 1- and 2-dimensional cases of a more general question: given a collection of n d-dimensional rectangular ranges, compute the measure of their union. This general problem is Klee's measure problem.

When generalized to the d-dimensional case, Bentley's algorithm has a running time of . This turns out not to be optimal, because it only decomposes the d-dimensional problem into n (d-1)-dimensional problems, and does not further decompose those subproblems. In 1981, Jan van Leeuwen and Derek Wood improved the running time of this algorithm to for d ≥ 3 by using dynamic quadtrees.

In 1988, Mark Overmars and Chee Yap proposed an algorithm for d ≥ 3. Their algorithm uses a particular data structure similar to a kd-tree to decompose the problem into 2-dimensional components and aggregate those components efficiently; the 2-dimensional problems themselves are solved efficiently using a trellis structure. Although asymptotically faster than Bentley's algorithm, its data structures use significantly more space, so it is only used in problems where either n or d is large. In 1998, Bogdan Chlebus proposed a simpler algorithm with the same asymptotic running time for the common special cases where d is 3 or 4.

In 2013, Timothy M. Chan developed a simpler algorithm that avoids the need for dynamic data structures and eliminates the logarithmic factor, lowering the best known running time for d ≥ 3 to .

Known bounds

The only known lower bound for any d is , and optimal algorithms with this running time are known for d=1 and d=2. The Chan algorithm provides an upper bound of for d ≥ 3, so for d ≥ 3, it remains an open question whether faster algorithms are possible, or alternatively whether tighter lower bounds can be proven. In particular, it remains open whether the algorithm's running time must depend on d. In addition, the question of whether there are faster algorithms that can deal with special cases (for example, when the input coordinates are integers within a bounded range) remains open.

The 1D Klee's measure problem (union of intervals) can be solved in where p denotes the number of piercing points required to stab all intervals[1] (the union of intervals pierced by a common point can be calculated in linear time by computing the extrema). Parameter p is an adaptive parameter that depends on the input configuration, and the piercing algorithm[2] yields an adaptive algorithm for Klee's measure problem.

See also

References and further reading

Important papers

Secondary literature

References

  1. ^ "Adaptive Computational Geometry", F. Nielsen, pdf
  2. ^ "Fast stabbing of boxes in high dimensions", F. Nielsen, Theoretical Computer Science Volume 246, Issues 1–2, 6 September 2000, Pages 53-72 pdf
This page was last edited on 17 December 2023, at 12:13
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.