Geometric Topology in Dimensions 2 and 3

Moïse, Edwin E.

doi:10.1007/978-1-4612-9906-6

“…This is true for dimensions not equal to 4, since n-dimensional topological manifolds admit unique smooth structures when n ≤ 3 and they admit unique Lipschitz structures when n ≥ 5. See [9,27,34,66,85] concerning these topics.…”

Section: Happy Fractalsmentioning

confidence: 99%

Happy fractals and some aspects of analysis on metric spaces

Semmes¹

2003

Publicacions Matemàtiques

View full text Add to dashboard Cite

In this survey we discuss "happy fractals", which are complete metric spaces which are not too big, in the sense of a doubling condition, and for which there is a path between any two points whose length is bounded by a constant times the distance between the two points. We also review some aspects of basic analysis on metric spaces, related to Lipschitz functions, approximations and regularizations of functions, and the notion of "atoms".

show abstract

“…One last point is that many of the results about the fringe data-structure derive from the Jordan Curve theorem [16], [21], and perhaps this could be generalized, for example, to the Voronoi diagram for line-segments [15].…”

Section: Comments and Open Problemsmentioning

confidence: 99%

A nearly optimal deterministic parallel Voronoi diagram algorithm

Cole

¹

,

Goodrich

²

,

Dúnlaing

³

1996

View full text Add to dashboard Cite

Abstract.We describe an n-processor, O(log(n) log log(n))-time CRCW algorithm to construct the Voronoi diagram for a set of n point-sites in the plane.Key Words. Voronoi diagram, Parallel algorithm.1. Introduction. Outline of the Algorithm. The Voronoi diagram is a geometric structure of great computational interest: see [5] for a useful survey. This paper addresses the problem of constructing the diagram in parallel, given as input a set of n points ("sites") in the plane. The Voronoi diagram for a set of sites is the locus of points equidistant from two closest sites: Figure 1 illustrates a diagram with 32 sites.The model of parallelism we assume is a CRCW PRAM, a system of independent processors accessing a shared random-access memory, where the same memory cell can be read by several processors simultaneously (concurrent read) and written by several processors simultaneously (concurrent write). Write-conflicts are resolved arbitrarily: the model of computation is an ARBITRARY CRCW PRAM.Each processor is assumed capable of exact rational and integer arithmetic in unit time.Earlier algorithms [1], [9] were presented for CREW 5 machines. The algorithm in [1] used n processors and took O(log 2 (n)) parallel time; that in [2] used n log(n) processors and took O(log(n) log log(n)) parallel time. Our algorithm reduces the overall work (parallel time × number of processors) to O(n log(n) log log(n)), while maintaining a runtime of O(log(n) log log(n)). Both of these figures are within the factor log log(n)

show abstract

“…13 A Jordan curve is a curve in the plane topologically equivalent to a circle. The Jordan Curve theorem [16], [21] says that such a curve has a definite "inside" and "outside." 14 "Open boundary" is a nonstandard term, used only in this paper.…”

Section: 16mentioning

confidence: 99%

“…It follows that there are at most 2|P| Voronoi vertices. 21 This gives a simple way to allocate space for Vor(P): to each site let there be associated two vertex records, one for the highest vertex in its cell (if the cell is bounded above), and one for the lowest (if bounded below). With each site let there be space allocated for six edge records (three for each vertex).…”

Section: Building Vor(p ∪ Q)mentioning

confidence: 99%

A Nearly Optimal Deterministic Parallel Voronoi Diagram Algorithm

Cole¹

1996

Algorithmica

1

0

View full text Add to dashboard Cite

Abstract.We describe an n-processor, O(log(n) log log(n))-time CRCW algorithm to construct the Voronoi diagram for a set of n point-sites in the plane.Key Words. Voronoi diagram, Parallel algorithm.1. Introduction. Outline of the Algorithm. The Voronoi diagram is a geometric structure of great computational interest: see [5] for a useful survey. This paper addresses the problem of constructing the diagram in parallel, given as input a set of n points ("sites") in the plane. The Voronoi diagram for a set of sites is the locus of points equidistant from two closest sites: Figure 1 illustrates a diagram with 32 sites.The model of parallelism we assume is a CRCW PRAM, a system of independent processors accessing a shared random-access memory, where the same memory cell can be read by several processors simultaneously (concurrent read) and written by several processors simultaneously (concurrent write). Write-conflicts are resolved arbitrarily: the model of computation is an ARBITRARY CRCW PRAM.Each processor is assumed capable of exact rational and integer arithmetic in unit time.Earlier algorithms [1], [9] were presented for CREW 5 machines. The algorithm in [1] used n processors and took O(log 2 (n)) parallel time; that in [2] used n log(n) processors and took O(log(n) log log(n)) parallel time. Our algorithm reduces the overall work (parallel time × number of processors) to O(n log(n) log log(n)), while maintaining a runtime of O(log(n) log log(n)). Both of these figures are within the factor log log(n)

show abstract

Geometric Topology in Dimensions 2 and 3

Cited by 373 publications

References 1 publication

Happy fractals and some aspects of analysis on metric spaces

Happy fractals and some aspects of analysis on metric spaces

A nearly optimal deterministic parallel Voronoi diagram algorithm

A Nearly Optimal Deterministic Parallel Voronoi Diagram Algorithm

Contact Info

Product

Resources

About