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Spectral Partitioning Works: Planar graphs and finite element meshes
 In IEEE Symposium on Foundations of Computer Science
, 1996
"... Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extr ..."
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Cited by 153 (8 self)
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Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on boundeddegree planar graphs and finite element meshes the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for boundeddegree planar graphs and twodimensional meshes and O i n 1=d j for wellshaped ddimensional meshes. The heart of our analysis is an upper bound on the secondsmallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...
Graph partitioning for high performance scientific simulations. Computing Reviews 45(2
, 2004
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Linear Algorithms for Partitioning Embedded Graphs of Bounded Genus
 SIAM Journal of Discrete Mathematics
, 1996
"... This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " we show that any nvertex graph G of genus g can be divided in O(n + g) time into components whose sizes do not exceed "n by removing a set ..."
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Cited by 24 (4 self)
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This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " we show that any nvertex graph G of genus g can be divided in O(n + g) time into components whose sizes do not exceed "n by removing a set C of O( p (g + 1=")n) vertices. Our result improves the best previous ones with respect to the size of C and the time complexity of the algorithm. Moreover, we show that one can cut off from G a piece of no more than (1 \Gamma ")n vertices by removing a set of O( p n"(g" + 1) vertices. Both results are optimal up to a constant factor. Keywords: graph separator, graph genus, algorithm, divideandconquer, topological graph theory AMS(MOS) subject classifications: 05C10, 05C85, 68R10 1 Bulgarian Academy of Sci., CICT, G.Bonchev 25A, 1113 Sofia, Bulgaria 2 Department of Comp.Sci.,Rice University, P.O.Box 1892, Houston, Texas 77251, USA 1 Introduction Let S be a class of graphs closed under t...
Multiphase Mesh Partitioning
 APPL. MATH. MODELLING
, 1999
"... We consider the loadbalancing problems which arise from parallel scientific codes containing multiple computational phases, or loops over subsets of the data, which are separated by global synchronisation points. We motivate, derive and describe the implementation of an approach which we refer to ..."
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Cited by 23 (9 self)
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We consider the loadbalancing problems which arise from parallel scientific codes containing multiple computational phases, or loops over subsets of the data, which are separated by global synchronisation points. We motivate, derive and describe the implementation of an approach which we refer to as the multiphase mesh partitioning strategy to address such issues. The technique is tested on several examples of meshes, both real and artificial, containing multiple computational phases and it is demonstrated that our method can achieve high quality partitions where a standard mesh partitioning approach fails.
I/OEfficient Planar Separators and Applications
, 2001
"... We present a new algorithm to compute a subset S of vertices of a planar graph G whose removal partitions G into O(N/h) subgraphs of size O(h) and with boundary size O( p h) each. The size of S is O(N= p h). Computing S takes O(sort(N)) I/Os and linear space, provided that M 56hlog² B. Together with ..."
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Cited by 3 (1 self)
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We present a new algorithm to compute a subset S of vertices of a planar graph G whose removal partitions G into O(N/h) subgraphs of size O(h) and with boundary size O( p h) each. The size of S is O(N= p h). Computing S takes O(sort(N)) I/Os and linear space, provided that M 56hlog² B. Together with recent reducibility results, this leads to O(sort(N)) I/O algorithms for breadthfirst search (BFS), depthfirst search (DFS), and single source shortest paths (SSSP) on undirected embedded planar graphs. Our separator algorithm does not need a BFS tree or an embedding of G to be given as part of the input. Instead we argue that "local embeddings" of subgraphs of G are enough.
Recursive Conditioning  Anyspace conditioning method with treewidthbounded complexity
, 2000
"... We introduce an any{space algorithm for exact inference in Bayesian networks, called recursive conditioning. On one extreme, recursive conditioning takes O(n) space and O(n exp(w log n)) timewhere n is the size of a Bayesian network and w is the width of a given elimination ordertherefore, establ ..."
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We introduce an any{space algorithm for exact inference in Bayesian networks, called recursive conditioning. On one extreme, recursive conditioning takes O(n) space and O(n exp(w log n)) timewhere n is the size of a Bayesian network and w is the width of a given elimination ordertherefore, establishing a new complexity result for linear{space inference in Bayesian networks. On the other extreme, recursive conditioning takes O(n exp(w)) space and O(n exp(w)) time, therefore, matching the complexity of state{of{the{art algorithms based on clustering and elimination. In between linear and exponential space, recursive conditioning can utilize memory at increments of Xbytes, where X is the number of bytes needed to store a oating point number in a cache. Moreover, the algorithm is equipped with a formula for computing its average running time under any amount of space, hence, providing a valuable tool for time{space tradeos in demanding applications. Recursive conditioning is therefore the rst algorithm for exact inference in Bayesian networks to oer a smooth tradeo between time and space, and to explicate a smooth, quantitative relationship between these two important resources. 1
Partitioning Algorithms and Their Application to Massively Parallel Computations of MultiPhase Fluid Flows in Porous Media
"... connected in the following computational strategy. We use a mesh generator (triangle for 2D meshes and NETGEN for 3D meshes) to generate a good coarse mesh. Then the considered problem is solved sequentially on the coarse mesh (by every processor). The solution is used to compute a posteriori err ..."
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connected in the following computational strategy. We use a mesh generator (triangle for 2D meshes and NETGEN for 3D meshes) to generate a good coarse mesh. Then the considered problem is solved sequentially on the coarse mesh (by every processor). The solution is used to compute a posteriori error estimates, which are used as weights in an elementbased 1 splitting of the coarse mesh into subdomains. Such splitting ensures that the local renements that follow will produce a computational mesh with a number of triangles/tetrahedrons balanced over the subdomains. Every subdomain is \mapped" to a processor. Then, based on a posteriori error analysis, each processor renes consecutively its region independently. After every step of independent renement, there is communication between the processors to make the mesh on that level globally conforming. The last point concentrates on computer simulation and testing the developed techniques on ui
I/Oefficient algorithms for planar graphs I: Separators
"... We present I/Oefficient algorithms for computing optimal separator partitions of planar graphs. Our main result shows that, given a planar graph G with N vertices and an integer r> 0, a vertex separator of size O (N / √ r) that partitions G into O(N/r) subgraphs of size at most r and boundary s ..."
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We present I/Oefficient algorithms for computing optimal separator partitions of planar graphs. Our main result shows that, given a planar graph G with N vertices and an integer r> 0, a vertex separator of size O (N / √ r) that partitions G into O(N/r) subgraphs of size at most r and boundary size O ( √ r) can be computed in O(sort(N)) I/Os, provided that M ≥ 56r log 2 B. Together with the planar embedding algorithm presented in the companion paper [27], this result is the basis for I/Oefficient solutions to many other fundamental problems on planar graphs, including breadthfirst search and shortest paths [5, 8], depthfirst search [6, 9], strong connectivity [9], and topological sorting [8]. Our second result shows that, given I/Oefficient solutions to these problems, a general separator algorithm for graphs with costs and weights on their vertices [3] can be made I/Oefficient. Many classical separator theorems are special cases of this result. In particular, our I/Oefficient version allows the computation of a separator as produced by our first separator algorithm, but without placing any constraints on r.