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Separators for spherepackings and nearest neighbor graphs
 J. ACM
, 1997
"... Abstract. A collection of n balls in d dimensions forms a kply system if no point in the space is covered by more than k balls. We show that for every kply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the ..."
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Cited by 74 (7 self)
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Abstract. A collection of n balls in d dimensions forms a kply system if no point in the space is covered by more than k balls. We show that for every kply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the interior and those in the exterior of the sphere S, respectively, so that the larger part contains at most (1 � 1/(d � 2))n balls. This bound of O(k 1/d n 1�1/d) is the best possible in both n and k. We also present a simple randomized algorithm to find such a sphere in O(n) time. Our result implies that every knearest neighbor graphs of n points in d dimensions has a separator of size O(k 1/d n 1�1/d). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a diskpacking, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.
Convex Grid Drawings of 3Connected Planar Graphs
, 1994
"... We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a lineartime algorithm which, given an nvertex 3connected plane gr ..."
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Cited by 37 (7 self)
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We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a lineartime algorithm which, given an nvertex 3connected plane graph G (with n 3), finds such a straightline convex embedding of G into a (n \Gamma 2) \Theta (n \Gamma 2) grid. 1 Introduction In this paper we consider the problem of aesthetic drawing of plane graphs, that is, planar graphs that are already embedded in the plane. What is exactly an aesthetic drawing is not precisely defined and, depending on the application, different criteria have been used. In this paper we concentrate on the two following criteria: (a) edges should be represented by straightline segments, and (b) faces should be drawn as convex polygons. F'ary [6], Stein [14] and Wagner [18] showed, independently, that each planar graph can be drawn in the plane in such a way that ...
Competitive Online Routing in Geometric Graphs
 Theoretical Computer Science
, 2001
"... We consider online routing algorithms for finding paths between the vertices of plane graphs. ..."
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Cited by 34 (4 self)
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We consider online routing algorithms for finding paths between the vertices of plane graphs.
Coarsening, Sampling, And Smoothing: Elements Of The Multilevel Method
 Parallel Processing, IMA Volumes in Mathematics and its Applications, 105, Springer Verlag:247–276
, 1999
"... . The multilevel method has emerged as one of the most effective methods for solving numerical and combinatorial problems. It has been used in multigrid, domain decomposition, geometric search structures, as well as optimization algorithms for problems such as partitioning and sparsematrix ordering ..."
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Cited by 6 (0 self)
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. The multilevel method has emerged as one of the most effective methods for solving numerical and combinatorial problems. It has been used in multigrid, domain decomposition, geometric search structures, as well as optimization algorithms for problems such as partitioning and sparsematrix ordering. This paper presents a systematic treatment of the fundamental elements of the multilevel method. We illustrate, using examples from several fields, the importance and effectiveness of coarsening, sampling, and smoothing (local optimization) in the application of the multilevel method. Key words. Algorithmdesign paradigm, coarsening, combinatorial optimization, Delaunay triangulation, domain decomposition, eigenvalue problems, Gaussian elimination, geometric methods, graph partitioning, hierarchical methods, multigrid, multilevel methods, nested dissection, sampling, smoothing, spectral methods. AMS(MOS) subject classifications. Primary 1234, 5678, 9101112. 1. Introduction. The multilev...
Optimizing Over All Combinatorial Embeddings Of A Planar Graph
 INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION, PROC. 7TH INT. IPCO CONF., LNCS 1610, 361376
, 1999
"... We study the problem of optimizing over the set of all combinatorial embeddings of a given planar graph. Our objective function prefers certain cycles of G as face cycles in the embedding. The motivation for studying this problem arises in graph drawing, where the chosen embedding has an importan ..."
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Cited by 4 (2 self)
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We study the problem of optimizing over the set of all combinatorial embeddings of a given planar graph. Our objective function prefers certain cycles of G as face cycles in the embedding. The motivation for studying this problem arises in graph drawing, where the chosen embedding has an important influence on the aesthetics of the drawing. We characterize