Geometric separator theorems and applications

@article{Smith1998GeometricST,
  title={Geometric separator theorems and applications},
  author={Warren D. Smith and Nicholas C. Wormald},
  journal={Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)},
  year={1998},
  pages={232-243}
}
  • Warren D. Smith, N. Wormald
  • Published 8 November 1998
  • Mathematics
  • Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
We find a large number of "geometric separator theorems" such as: I: Given N disjoint isooriented squares in the plane, there exists a rectangle with /spl les/2N/3 squares inside, /spl les/2N/3 squares outside, and /spl les/(4+0(1))/spl radic/N partly in & out. II: There exists a rectangle that is crossed by the minimal spanning tree of N sites in the plane at /spl les/(4/spl middot/3/sup 1/4/+0(1))/spl radic/N points, having /spl les/2N/3 sites inside and outside. These theorems yield a large… 

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References

SHOWING 1-10 OF 44 REFERENCES
Studies in computational geometry motivated by mesh generation
TLDR
This thesis extensively generalizes the famous formula of Heron and Alexandria (75 AD), for the area of a triangle, and presents the first linear time congruence algorithm for 3-dimensional polyhedra.
Generalized nested dissection
TLDR
It is shown that sparse Gaussian elimination is efficient for any class of graphs which have good separator, and conversely that graphs without good separators are not amenable to sparse GaRussian elimination.
Euclidean spanners: short, thin, and lanky
TLDR
It is shown that it is possible to build spanners in optimal O (n log n) time and O(n) space that achieve optimal or near optimal tradeoffs between all combinations of these *Max-Planck-Institut fiir Informatik, Saarbrucken, Germany.
Improved Approximation Schemes for Geometrical Graphs Via Spanners and Banyans
TLDR
The algorithms are based on using low-weight Eu-clidean spanner graphs in conjunction with the hierarchical structure theorems that serve as the basis of Arora's work and it is shown that spanners can in principle be made orders of magnitude shorter and simpler by allowing Steiner points.
Reducing the Steiner Problem in a Normed Space
TLDR
It is proved that there always exists a Steiner minimum tree whose Steiner points are located only at points whose coordinates appear in points of P, which means that the Steiner problem for P in such a space is reduced to a Steiners problem on graphs and is solvable by any existing Steiner graph algorithms.
Loxodromic sequences of tangent spheres
In particular, if ~u/su+ 1 = x = e zt for all/~, then x n+2 1"~ 2 X 2 ( n + 2 ) 1 X n + 2 1 X -1 x 1 ) = n x 2 1 , : + 2 + l n x + l " tanh (n + 2) t = n tanh t. All such sequences of spheres (for
On the Problem of Partitioning Planar Graphs
The results in this paper are closely related to the effective use of the divide-and-conquer strategy for solving problems on planar graphs. It is shown that every planar graph can be partitioned
A Separator Theorem for Planar Graphs
Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more
A new way to weigh Malnourished Euclidean graphs
In this paper, we show that any Euclidean graph over a set V of n points in k-dimensional space that satisfies either the leapfrog property or the isolation property has small weight, i.e., has
Disk packings and planar separators
We demonstrate that the geometric separator algorithm of Miller, Teng, Thurston, and Vavasis finds a 3/4-separator of size 1.84+ for every n node planar graph. Our bound is derived from an analysis
...
1
2
3
4
5
...