# 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} }

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…

## 93 Citations

Theory and Application of Width Bounded Geometric Separator ∗

- Mathematics

We introduce the notion of width bounded geometric separator, develop the techniques for its existence as well as algorithm, and apply it to obtain a 2 √ n) time exact algorithm for the disk covering…

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- MathematicsJ. Comput. Syst. Sci.
- 2011

Theory and Application of Width Bounded Geometric Separator

- Mathematics, Computer ScienceSTACS
- 2005

The notion of thewidth bounded geometric separator is introduced and the techniques for the existence of the width bounded separator in any d-dimensional Euclidean space are developed.

A Simple Proof of the Existence of a Planar Separator

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- 2011

A simple proof of the existence of a planar separator is provided by showing that it is an easy consequence of the circle packing theorem and reprove other results on separators.

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- Mathematics, Computer ScienceWADS
- 2017

A geometric version of the graph separator theorem for the unit disk intersection graph is proved, and it is shown that no line-separator of sublinear size in $n$ exists when the authors look at disks of arbitrary radii.

Spanners for Geometric Intersection Graphs

- Mathematics, Computer ScienceWADS
- 2007

This paper presents the first algorithm for constructing spanners of ball graphs with support for fast distance queries, and shows that the spanner construction has complexity almost equivalent to the construction of a Euclidean minimum spanning tree.

A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs

- Computer Science, MathematicsSTOC
- 2018

An algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds and allows us to derive matching 2Ω(n1−1/d) lower bounds under the Exponential Time Hypothesis even in the much more restricted class of d-dimensional induced grid graphs.

Geometric Separators and the Parabolic Lift

- MathematicsCCCG
- 2013

By computing the centerpoint of the points lifted onto a paraboloid rather than using the stereographic map as in the original method, one can sample the desired sphere directly, without computing the conformal transformation.

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- Mathematics, Computer ScienceJ. Comb. Optim.
- 2008

The 3D separator designed here is the first sublinear time algorithm for finding geometric separators and is applied to derive an algorithm for the protein side-chain packing problem, which improves and simplified the previous algorithm of Xu.

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