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## Spectral partitioning works: planar graphs and finite element meshes, in: (1996)

Venue: | Proceedings of the 37th Annual Symposium on Foundations of Computer Science, |

Citations: | 201 - 10 self |

### Citations

1537 |
Spectral Graph Theory
- Chung
- 1997
(Show Context)
Citation Context ...ymmetric matrix are orthogonal. 288 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 • The Fiedler value, #2, of G satisfies #2 = min&x'(1,1,...,1) &xTL(G)&x &xT &x , with the minimum occurring only when &x is a Fiedler vector. • For any vector &x % Rn, we have &xTL(G)&x = " (i,j)%E (xi # xj )2. Let M be a symmetric n $ n matrix and &x be an n-dimensional vector. Then, the Rayleigh quotient of &x with respect to M is &xTM &x &xT &x . For proofs of these statements and many other fascinating facts about the eigenvalues and eigenvectors of graphs consult one of [75,56,19,16,23]. The Fiedler value, #2, of a graph is closely linked to its isoperimetric number. If G is a graph on more than three nodes, then one can show [4,57,71] (also see Theorem 2.1) #2 2 ! !(G) ! ! #2(2d # #2). 2.3. Spectral partitioning methods Let &u = (u1, . . . , un) be a Fiedler vector of the Laplacian of a graph G. The idea of spectral partitioning is to find a splitting value s and partition the vertices of G into the set of i such that ui > s and the set such that ui ! s. We call such a partition a Fiedler cut. There are several popular choices for the splitting value s: • bisection: s is th... |

1151 | A fast algorithm for particle simulations
- Greengard, Rokhlin
- 1987
(Show Context)
Citation Context ...borhood system ([MV91, MTTV96a]). ffl The computation/communication graph used in hierarchical N-body simulation methods (such as the Barnes-Hut's treecode method [BH86] and the fast-multipole method =-=[GR87]-=-) is a subgraph of an ff-overlap graph of an O(log n)-ply neighborhood system ([Ten96]). 6.3. Spherical Embeddings of Overlap Graphs In this section, we show that an ff-overlap graph is a subgraph of ... |

1037 |
Linear Algebra and its Applications
- Strang
- 1980
(Show Context)
Citation Context ...ymmetric matrix are orthogonal. 288 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 • The Fiedler value, #2, of G satisfies #2 = min&x'(1,1,...,1) &xTL(G)&x &xT &x , with the minimum occurring only when &x is a Fiedler vector. • For any vector &x % Rn, we have &xTL(G)&x = " (i,j)%E (xi # xj )2. Let M be a symmetric n $ n matrix and &x be an n-dimensional vector. Then, the Rayleigh quotient of &x with respect to M is &xTM &x &xT &x . For proofs of these statements and many other fascinating facts about the eigenvalues and eigenvectors of graphs consult one of [75,56,19,16,23]. The Fiedler value, #2, of a graph is closely linked to its isoperimetric number. If G is a graph on more than three nodes, then one can show [4,57,71] (also see Theorem 2.1) #2 2 ! !(G) ! ! #2(2d # #2). 2.3. Spectral partitioning methods Let &u = (u1, . . . , un) be a Fiedler vector of the Laplacian of a graph G. The idea of spectral partitioning is to find a splitting value s and partition the vertices of G into the set of i such that ui > s and the set such that ui ! s. We call such a partition a Fiedler cut. There are several popular choices for the splitting value s: • bisection: s is th... |

836 | Algebraic Graph Theory - Biggs - 1993 |

776 |
Algorithms in Combinatorial Geometry,
- Edelsbrunner
- 1987
(Show Context)
Citation Context ...8 Remark 5. One can prove a slightly weaker result by using a result of Miller, Teng, Thurston, and Vavasis [MTTV96a] to find a circle-preserving map that makes the center of the sphere a centerpoint =-=[Ede87]-=- of the images of particular points in the caps. If the center of the sphere is a centerpoint, then the centroid must be far away from at least a constant fraction of the centers of the caps. Thus, th... |

733 | An Analysis of the Finite Element Method - Strang, Fix - 1973 |

694 |
Computer Simulation Using Particles
- Hockney, Eastwood
- 1988
(Show Context)
Citation Context ...e.g., the N-body simulation method). However different the particular methods may be, a basic principle is common to all—accuracy of approximation is ensured by using meshes that satisfy certain numerical and geometric constraints. Meshes that satisfy these constraints are said to be well-shaped. To motivate our spectral analysis of well-shaped meshes, we review the conditions required of finite element and finite difference meshes. More detailed discussions can be found in several books and papers (for example, see [69,50,13,12,37]). Background material on the particle method can be found in [15,43,45,83]. The finite element method approximates a continuous problem by subdividing the domain (a subset of Rd ) of the problem into a mesh of polyhedral elements and then approximates the continuous function by piecewise polynomial functions on the elements. A common choice for an element is a d-dimensional simplex. Accordingly, a d-dimensional finite element mesh is a d-dimensional simplicial complex, a collection of d-dimensional simplices that meet only at shared faces [13,12,58]. The computation graph associated with each simplicial complex is often its 1-skeleton or the 1-skeleton of its geomet... |

666 | Algebraic connectivity of graphs - Fiedler - 1973 |

629 |
Partitioning sparse matrices with eigenvectors of graphs
- POTHEN, SIMON, et al.
- 1990
(Show Context)
Citation Context ... the planar separator theorem of Lipton and Tarjan [LT79] with the fact thats2 =2sOE(G). Bounds of O(1=n) on the Fiedler values of planar graphs were previously known for graphs such as regular grids =-=[PSL90]-=-, quasi-uniform graphs [GK95], and bounded-degree trees. Bounds on the Fiedler values of regular grids and quasi-uniform graphs essentially follow from the fact that the diameters of these graphs are ... |

578 |
Sphere Packings, Lattices and Groups
- Conway, Sloane
- 1993
(Show Context)
Citation Context ...f asd k-ply neighborhood system, wheresd is the kissing number in d dimensions---the maximum number of nonoverlapping unit balls in R d that can be arranged so that they all touch a central unit ball =-=[CS88]-=-. Moreover, the maximum degree of a k-nearest neighbor graph is bounded bysd k. 5.2. A Spectral Bound Theorem 13. Let G be a subgraph of an intersection graph of a k-ply neighborhood system in R d suc... |

521 |
Numerical Solutions of Partial Differential Equations by the Finite Element Method
- Johnson
- 1987
(Show Context)
Citation Context ...e element, finite difference, and finite volume methods) or particle methods (e.g., the N-body simulation method). However different the particular methods may be, a basic principle is common to all—accuracy of approximation is ensured by using meshes that satisfy certain numerical and geometric constraints. Meshes that satisfy these constraints are said to be well-shaped. To motivate our spectral analysis of well-shaped meshes, we review the conditions required of finite element and finite difference meshes. More detailed discussions can be found in several books and papers (for example, see [69,50,13,12,37]). Background material on the particle method can be found in [15,43,45,83]. The finite element method approximates a continuous problem by subdividing the domain (a subset of Rd ) of the problem into a mesh of polyhedral elements and then approximates the continuous function by piecewise polynomial functions on the elements. A common choice for an element is a d-dimensional simplex. Accordingly, a d-dimensional finite element mesh is a d-dimensional simplicial complex, a collection of d-dimensional simplices that meet only at shared faces [13,12,58]. The computation graph associated with each... |

460 | A separator theorem for planar graphs
- Lipton, Tarjan
- 1979
(Show Context)
Citation Context ...ed line of research, algorithms were developed along with proofs that they will always find small separators in various families of graphs. The seminal work in this area was that of Lipton and Tarjan =-=[LT79]-=-, who constructed a linear-time algorithm that produces a 1=3-separator of p 8n nodes in any n-node planar graph. Their result improved a theorem of Ungar [Ung51] which demonstrated that every planar ... |

400 | Eigenvalues and expanders.
- Alon
- 1986
(Show Context)
Citation Context ...near and convex programming made these techniques impractical for most applications. By recognizing a relation between the Fiedler value and the Cheeger constant [Che70] of continuous manifolds, Alon =-=[Alo86]-=- and Sinclair and Jerrum [SJ89] demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vertices in the eigenvector will produce a c... |

348 | Geometric bounds for eigenvalues of Markov chains,
- Diaconis, Stroock
- 1991
(Show Context)
Citation Context ... a graph. In a similar vein, Boppana [17] analyzed eigenvector techniques in conjunction with convex programming. However, the use of linear and convex programming made these techniques impractical for most applications. By recognizing a relation between the Fiedler value and the Cheeger constant [20] of continuous manifolds, Alon [3] and Sinclair and Jerrum [72] demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vertices in the eigenvector will produce a cut with a good ratio of cut edges to separated vertices (see also [4, 36,30,55,57]). Around the same time, improvements in algorithms for approximately computing eigenvectors, such as the Lanczos algorithm, made the computation of eigenvectors practical [67, 70]. In the next few years, a wealth of experimental work demonstrated that spectral partitioning methods work well on graphs that usually arise in practice [18,48,66,70,82]. Spectral partitioning became a standard tool for mesh partitioning in many areas [49]. Still, researchers were unable to prove that spectral partitioning techniques would work well on the graphs encountered in 286 D.A. Spielman, S.-H. Teng / Linear... |

344 | Partitioning of unstructured problems for parallel processing, in
- SIMON
- 1991
(Show Context)
Citation Context ...edu (S.-H. Teng). 1 The work was done while the author was at U.C. Berkeley. 2 This work was done while the author was at the University of Minnesota and Xerox PARC. 0024-3795/$ - see front matter ( 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2006.07.020 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 285 1. Introduction Spectral partitioning has become one of the most successful heuristics for partitioning graphs and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [70,82], solving sparse linear systems [68], and partitioning for domain decomposition [21,25]. It is also used in VLSI circuit design and simulation [26,46,2]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [18,48,49,66,70,82]. However, the quality of the partition that these methods should produce has so far eluded precise analysis. In this paper, we will prove that spectral partitioning methods give good separators for the graphs to which they are usually applied. The size of the separator produce... |

335 |
A lower bound for the smallesteigenvalue of the Laplacian
- Cheeger
- 1970
(Show Context)
Citation Context ...ex programming. However, the use of linear and convex programming made these techniques impractical for most applications. By recognizing a relation between the Fiedler value and the Cheeger constant =-=[Che70]-=- of continuous manifolds, Alon [Alo86] and Sinclair and Jerrum [SJ89] demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vertic... |

319 | A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory. - Fiedler - 1975 |

317 | Approximate counting, uniform generation and rapidly mixing markov chains.
- Sinclair, Jerrum
- 1989
(Show Context)
Citation Context ... these techniques impractical for most applications. By recognizing a relation between the Fiedler value and the Cheeger constant [Che70] of continuous manifolds, Alon [Alo86] and Sinclair and Jerrum =-=[SJ89]-=- demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vertices in the eigenvector will produce a cut with a good ratio of cut edg... |

312 | Expander flows, geometric embeddings and graph partitioning.
- Arora, Rao, et al.
- 2009
(Show Context)
Citation Context ...raphs that do not have an h-clique minor have separators of O(h3/2 " n) nodes. Plotkin et al. [65] reduced the dependency on h from h3/2 to h. Using geometric techniques, Miller, Teng, Thurston, and Vavasis [58–60,62,63,76] extended the planar separator theorem to graphs embedded in higher dimensions and showed that every well-shaped mesh in Rd has a 1/(d + 2)-separator of size O(n1#1/d). Using multicommodity flow, Leighton and Rao [53] designed a partitioning method guaranteed to return a cut whose ratio of cut size to vertices separated is within logarithmic factors of optimal. Arora et al. [8] recently improved the approximation ratio to O( ! log n). For planar graphs, a partition within a constant factor of the optimal can be found in polynomial time [38]. While spectral methods have been favored in practice, they lacked a proof of effectiveness. 1.2. Outline of paper In Section 2, we introduce the concept of a graph partition, review some facts from linear algebra that we require, and describe the class of spectral partitioning methods. In Section 3, we prove the embedding lemma, which relates the quality of geometric embeddings of a graph with its Fiedler value. We then show (us... |

307 | A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems
- Barnard, Simon
- 1992
(Show Context)
Citation Context ...a and its Applications 421 (2007) 284–305 285 1. Introduction Spectral partitioning has become one of the most successful heuristics for partitioning graphs and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [70,82], solving sparse linear systems [68], and partitioning for domain decomposition [21,25]. It is also used in VLSI circuit design and simulation [26,46,2]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [18,48,49,66,70,82]. However, the quality of the partition that these methods should produce has so far eluded precise analysis. In this paper, we will prove that spectral partitioning methods give good separators for the graphs to which they are usually applied. The size of the separator produced by spectral methods can be related to the Fiedler value—the second smallest eigenvalue of the Laplacian—of the adjacency structure to which they are applied. By showing that well-shaped meshes in d dimensions have Fiedler value at most O(1/n2/d), we show that spectral methods can be used to find bisectors of these grap... |

292 | New spectral methods for ratio cut partitioning and clustering
- Hagen, Kahng
- 2002
(Show Context)
Citation Context ...and Xerox PARC. 0024-3795/$ - see front matter ( 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2006.07.020 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 285 1. Introduction Spectral partitioning has become one of the most successful heuristics for partitioning graphs and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [70,82], solving sparse linear systems [68], and partitioning for domain decomposition [21,25]. It is also used in VLSI circuit design and simulation [26,46,2]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [18,48,49,66,70,82]. However, the quality of the partition that these methods should produce has so far eluded precise analysis. In this paper, we will prove that spectral partitioning methods give good separators for the graphs to which they are usually applied. The size of the separator produced by spectral methods can be related to the Fiedler value—the second smallest eigenvalue of the Laplacian—of the adjacency structure to which they are a... |

290 |
The geometry and topology of 3-manifolds
- Thurston
(Show Context)
Citation Context ... # (i,j)%E #n k=1(vi,k # vj,k)2#n i=1 #n k=1 v 2 i,k = #n k=1 # (i,j)%E(vi,k # vj,k)2#n k=1 #n i=1 v 2 i,k . But, for each k, # (i,j)%E(vi,k # vj,k)2#n i=1 v 2 i,k " #2, so #n k=1 # (i,j)%E(vi,k # vj,k)2#n k=1 #n i=1 v 2 i,k " #2 (this follows from the fact that # i xi/ # i yi " mini xi/yi , for xi, yi > 0). # Our method of finding a good geometric embedding of a planar graph is similar to the way in which Miller et al. [59] directly find good separators of planar graphs. We first find an embedding of the graph on the plane by using the “kissing disk” embedding of Koebe, Andreev, and Thurston [52,5,6,79]: Theorem 3.2 (Koebe–Andreev–Thurston). LetGbe a planar graph with vertex setV = {1, . . . , n} and edge set E. Then, there exists a set of disks {D1, . . . , Dn} in the plane with disjoint interiors such that Di touches Dj if and only if (i, j) % E. Such an embedding is called a kissing disk embedding of G. The analogue of a disk on the sphere is a cap. A cap is given by the intersection of a half-space with the sphere, and its boundary is a circle. We define kissing caps analogously with kissing disks. Following [59], we use stereographic projection to map the kissing disk embedding of the g... |

277 |
A partitioning strategy for nonuniform problems on multiprocessors
- Berger, Bokhari
- 1987
(Show Context)
Citation Context ...produced by inserting a uniform grid from R2 or R3 into the domain via a boundary-matching conformal mapping. Notice that, unlike a finite element mesh, a finite difference mesh need not be a collection of simplices or elements, so we cannot analyze it as we do a triangulation. In general, the derivative of the conformal transformation must vary gradually with respect to the mesh size in order to produce D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 297 good results (see, for example [80]). This means that the mesh will probably satisfy a density condition [11,63]. Let G be an undirected graph and let $ be an embedding of its nodes in Rd . We say $ is an embedding of density % if the following inequality holds for all vertices v in G: Let u be the node closest to v. Let w be the node farthest from v that is connected to v by an edge. Then ($(w) # $(v)( ($(u) # $(v)( ! %. In general, G is an %-density graph in Rd if there exists an embedding of G in Rd with density %. 6.2. Modeling well-shaped meshes We will use the overlap graph to model well-shaped meshes (Miller et al. [59]). An overlap graph is based on a k-ply neighborhood system. The neighborhood ... |

246 | An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms.
- Leighton, Rao
- 1988
(Show Context)
Citation Context ...r separator theorem to graphs embedded in higher dimensions and showed that every well-shaped mesh in R d has a 1=(d+2)-separator of size O(n 1\Gamma1=d ). Using multicommodity flow, Leighton and Rao =-=[LR88]-=- designed a partitioning method guaranteed to return a cut whose ratio of cut size to vertices separated is within logarithmic factors of optimal. While spectral methods have been favored in practice,... |

227 | The Laplacian spectrum of graphs.
- Mohar
- 1991
(Show Context)
Citation Context ...ymmetric matrix are orthogonal. 288 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 • The Fiedler value, #2, of G satisfies #2 = min&x'(1,1,...,1) &xTL(G)&x &xT &x , with the minimum occurring only when &x is a Fiedler vector. • For any vector &x % Rn, we have &xTL(G)&x = " (i,j)%E (xi # xj )2. Let M be a symmetric n $ n matrix and &x be an n-dimensional vector. Then, the Rayleigh quotient of &x with respect to M is &xTM &x &xT &x . For proofs of these statements and many other fascinating facts about the eigenvalues and eigenvectors of graphs consult one of [75,56,19,16,23]. The Fiedler value, #2, of a graph is closely linked to its isoperimetric number. If G is a graph on more than three nodes, then one can show [4,57,71] (also see Theorem 2.1) #2 2 ! !(G) ! ! #2(2d # #2). 2.3. Spectral partitioning methods Let &u = (u1, . . . , un) be a Fiedler vector of the Laplacian of a graph G. The idea of spectral partitioning is to find a splitting value s and partition the vertices of G into the set of i such that ui > s and the set such that ui ! s. We call such a partition a Fiedler cut. There are several popular choices for the splitting value s: • bisection: s is th... |

220 |
An improved spectral graph partitioning algorithm for mapping parallel computations
- Hendrickson, Leland
- 1992
(Show Context)
Citation Context ...a and its Applications 421 (2007) 284–305 285 1. Introduction Spectral partitioning has become one of the most successful heuristics for partitioning graphs and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [70,82], solving sparse linear systems [68], and partitioning for domain decomposition [21,25]. It is also used in VLSI circuit design and simulation [26,46,2]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [18,48,49,66,70,82]. However, the quality of the partition that these methods should produce has so far eluded precise analysis. In this paper, we will prove that spectral partitioning methods give good separators for the graphs to which they are usually applied. The size of the separator produced by spectral methods can be related to the Fiedler value—the second smallest eigenvalue of the Laplacian—of the adjacency structure to which they are applied. By showing that well-shaped meshes in d dimensions have Fiedler value at most O(1/n2/d), we show that spectral methods can be used to find bisectors of these grap... |

219 | Recent directions in netlist partitioning: a survey, Integration: The VLSI
- Alpert, Kahng
- 1995
(Show Context)
Citation Context ...and Xerox PARC. 0024-3795/$ - see front matter ( 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2006.07.020 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 285 1. Introduction Spectral partitioning has become one of the most successful heuristics for partitioning graphs and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [70,82], solving sparse linear systems [68], and partitioning for domain decomposition [21,25]. It is also used in VLSI circuit design and simulation [26,46,2]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [18,48,49,66,70,82]. However, the quality of the partition that these methods should produce has so far eluded precise analysis. In this paper, we will prove that spectral partitioning methods give good separators for the graphs to which they are usually applied. The size of the separator produced by spectral methods can be related to the Fiedler value—the second smallest eigenvalue of the Laplacian—of the adjacency structure to which they are a... |

217 |
Spectra of Graphs: Theory and Application
- Cvetkovic, Doob, et al.
- 1980
(Show Context)
Citation Context |

215 |
Geometry and the Imagination
- Hilbert, Cohen-Vossen
- 1952
(Show Context)
Citation Context ...ction from the plane perpendicular to S d at ff, and we let \Pi \Gamma1 ff be its inverse (so, \Pi(1) = \Gammaff). One can show that the maps \Pi ff and \Pi \Gamma1 ff are sphere-preserving maps (see =-=[HCV52]-=- or [MTTV96a] for a proof). 9 Sphere-preserving maps in the plane include rigid motions of the plane as well as dilations (and other mobius transformations). We will obtain sphere-preserving maps in t... |

214 | Provably good mesh generation
- Bern, Eppstein, et al.
- 1994
(Show Context)
Citation Context ...e element, finite difference, and finite volume methods) or particle methods (e.g., the N-body simulation method). However different the particular methods may be, a basic principle is common to all—accuracy of approximation is ensured by using meshes that satisfy certain numerical and geometric constraints. Meshes that satisfy these constraints are said to be well-shaped. To motivate our spectral analysis of well-shaped meshes, we review the conditions required of finite element and finite difference meshes. More detailed discussions can be found in several books and papers (for example, see [69,50,13,12,37]). Background material on the particle method can be found in [15,43,45,83]. The finite element method approximates a continuous problem by subdividing the domain (a subset of Rd ) of the problem into a mesh of polyhedral elements and then approximates the continuous function by piecewise polynomial functions on the elements. A common choice for an element is a d-dimensional simplex. Accordingly, a d-dimensional finite element mesh is a d-dimensional simplicial complex, a collection of d-dimensional simplices that meet only at shared faces [13,12,58]. The computation graph associated with each... |

213 | Mesh generation and optimal triangulation
- Bern, Eppstein
- 1992
(Show Context)
Citation Context ...e element, finite difference, and finite volume methods) or particle methods (e.g., the N-body simulation method). However different the particular methods may be, a basic principle is common to all—accuracy of approximation is ensured by using meshes that satisfy certain numerical and geometric constraints. Meshes that satisfy these constraints are said to be well-shaped. To motivate our spectral analysis of well-shaped meshes, we review the conditions required of finite element and finite difference meshes. More detailed discussions can be found in several books and papers (for example, see [69,50,13,12,37]). Background material on the particle method can be found in [15,43,45,83]. The finite element method approximates a continuous problem by subdividing the domain (a subset of Rd ) of the problem into a mesh of polyhedral elements and then approximates the continuous function by piecewise polynomial functions on the elements. A common choice for an element is a d-dimensional simplex. Accordingly, a d-dimensional finite element mesh is a d-dimensional simplicial complex, a collection of d-dimensional simplices that meet only at shared faces [13,12,58]. The computation graph associated with each... |

193 |
An r-dimensional quadratic placement algorithm‛,
- Hall
- 1970
(Show Context)
Citation Context ...e is a cut whose ratio of vertices separated to edges cut is O(n1/d). By removing the vertices separated by this cut, computing a Fiedler vector of the new graph, and iterating as necessary, one can find a bisector of O(n1#1/d) edges. In particular, we prove that maximum-degree ! planar graphs have Fiedler value at most 8!/n, which implies that spectral techniques can be used to find bisectors of size at most O( " n) in these graphs. These bounds are the best possible for well-shaped meshes and planar graphs. 1.1. History The spectral method of graph partitioning was born in the works of Hall [47] who applied quadratic programming to design a placement algorithm, and in the work of Donath and Hoffman [28,29] who first suggested using the eigenvectors of adjacency matrices of graphs to find partitions and point placement. Fiedler [33–35] associated the second-smallest eigenvalue of the Laplacian of a graph with its connectivity and suggested partitioning by splitting vertices according to their value in the corresponding eigenvector. Thus, we call this eigenvalue the Fiedler value and a corresponding vector a Fiedler vector. A few years later, Barnes and Hoffman [10,14] used linear prog... |

183 | Combinatorial Geometry
- Pach, Agarwal
- 1995
(Show Context)
Citation Context ... = {B .i , . . . , B .n} be the images of the balls in # under ". Let ri be the radius of B .i . Because Vdrd ! volume(B .i ), We know that n" i=1 Vdr d i ! n" i=1 volume(B .i ) ! kAd+1. By Theorem 6.1, G is a subgraph graph of the intersection graph of {($% + % + $) · B .i : 1 ! i ! n}. Thus, by Lemma 3.1, #2(L(G)) ! #n i=1 2!($% + % + $)2r2i n !(2!)($% + % + $)2 * Ad+1 Vd +2/d * k n +2/d . Given the bound on the Fiedler value, the ratio achievable by a Fiedler cut follows immediately from Theorem 2.1 and the corresponding bisector size follows Lemma A.1. # Remark. Recently, Agarwal and Pach [1] and, independently, Spielman and Teng [73] gave an elementary proof of the sphere separator theorem of Miller et al. [60] on planar graphs and intersection graphs. However, these proofs do not directly extend to overlap graphs. The relation between overlap graphs and intersection graphs established by Theorem 6.1 enables us to prove the overlap graph separator theorem using the intersection graph separator theorem. The same reduction also extends the deterministic linear time algorithm for finding a good sphere separator from intersection graphs to overlap graphs [32]. 7. Final remarks The ge... |

181 |
Lower bounds for the partitioning of graphs.
- Donath, Hoffman
- 1973
(Show Context)
Citation Context ...is cut, computing a Fiedler vector of the new graph, and iterating as necessary, one can find a bisector of O(n1#1/d) edges. In particular, we prove that maximum-degree ! planar graphs have Fiedler value at most 8!/n, which implies that spectral techniques can be used to find bisectors of size at most O( " n) in these graphs. These bounds are the best possible for well-shaped meshes and planar graphs. 1.1. History The spectral method of graph partitioning was born in the works of Hall [47] who applied quadratic programming to design a placement algorithm, and in the work of Donath and Hoffman [28,29] who first suggested using the eigenvectors of adjacency matrices of graphs to find partitions and point placement. Fiedler [33–35] associated the second-smallest eigenvalue of the Laplacian of a graph with its connectivity and suggested partitioning by splitting vertices according to their value in the corresponding eigenvector. Thus, we call this eigenvalue the Fiedler value and a corresponding vector a Fiedler vector. A few years later, Barnes and Hoffman [10,14] used linear programming in combination with an examination of the eigenvectors of the adjacency matrix of a graph. In a similar v... |

174 | Performance of dynamic load balancing algorithms for unstructured mesh calculations. Concurrency, Practice and Experience
- Williams
- 1991
(Show Context)
Citation Context ...edu (S.-H. Teng). 1 The work was done while the author was at U.C. Berkeley. 2 This work was done while the author was at the University of Minnesota and Xerox PARC. 0024-3795/$ - see front matter ( 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2006.07.020 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 285 1. Introduction Spectral partitioning has become one of the most successful heuristics for partitioning graphs and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [70,82], solving sparse linear systems [68], and partitioning for domain decomposition [21,25]. It is also used in VLSI circuit design and simulation [26,46,2]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [18,48,49,66,70,82]. However, the quality of the partition that these methods should produce has so far eluded precise analysis. In this paper, we will prove that spectral partitioning methods give good separators for the graphs to which they are usually applied. The size of the separator produce... |

173 |
Spectral k-way ratio-cut partitioning and clustering,”
- Chan, Schlag, et al.
- 1993
(Show Context)
Citation Context ...and Xerox PARC. 0024-3795/$ - see front matter ( 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2006.07.020 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 285 1. Introduction Spectral partitioning has become one of the most successful heuristics for partitioning graphs and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [70,82], solving sparse linear systems [68], and partitioning for domain decomposition [21,25]. It is also used in VLSI circuit design and simulation [26,46,2]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [18,48,49,66,70,82]. However, the quality of the partition that these methods should produce has so far eluded precise analysis. In this paper, we will prove that spectral partitioning methods give good separators for the graphs to which they are usually applied. The size of the separator produced by spectral methods can be related to the Fiedler value—the second smallest eigenvalue of the Laplacian—of the adjacency structure to which they are a... |

163 |
A hierarchical O(N log N) force calculation algorithm,” The Institute for Advanced Study
- Barnes, Hut
- 1986
(Show Context)
Citation Context ... an ff-overlap graph of a 1-ply neighborhood system ([MV91, MTTV96a]). ffl The computation/communication graph used in hierarchical N-body simulation methods (such as the Barnes-Hut's treecode method =-=[BH86]-=- and the fast-multipole method [GR87]) is a subgraph of an ff-overlap graph of an O(log n)-ply neighborhood system ([Ten96]). 6.3. Spherical Embeddings of Overlap Graphs In this section, we show that ... |

154 |
On the angle condition in the finite element method,
- Babuska, Aziz
- 1976
(Show Context)
Citation Context ...thod, a linear system is defined over a mesh, with variables representing physical quantities at the nodes. The nonzero structure of the coefficient matrix of such a linear system is exactly the adjacency structure of the 1-skeleton of the simplicial complex. To ensure accuracy, in addition to the conditions that a mesh must conform to the boundaries of the region and be fine enough, each individual element of the mesh must be well-shaped. A common shape criterion for the finite element method is that the angles of each element are not too small, or the aspect ratio of each element is bounded [9,13,37]. Other numerical formulations require slightly different conditions. For example, the controlled volume formulation [64,61] using a Voronoi diagram requires that the radius aspect ratio (the ratio of the circumscribed radius to the shortest edge length of an element in the dual Delaunay diagram) is bounded. The finite difference method also uses a discrete structure, a finite difference mesh, to approximate a continuous problem. Finite difference meshes are often produced by inserting a uniform grid from R2 or R3 into the domain via a boundary-matching conformal mapping. Notice that, unlike a... |

126 |
Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process
- Fill
- 1991
(Show Context)
Citation Context ... a graph. In a similar vein, Boppana [17] analyzed eigenvector techniques in conjunction with convex programming. However, the use of linear and convex programming made these techniques impractical for most applications. By recognizing a relation between the Fiedler value and the Cheeger constant [20] of continuous manifolds, Alon [3] and Sinclair and Jerrum [72] demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vertices in the eigenvector will produce a cut with a good ratio of cut edges to separated vertices (see also [4, 36,30,55,57]). Around the same time, improvements in algorithms for approximately computing eigenvectors, such as the Lanczos algorithm, made the computation of eigenvectors practical [67, 70]. In the next few years, a wealth of experimental work demonstrated that spectral partitioning methods work well on graphs that usually arise in practice [18,48,66,70,82]. Spectral partitioning became a standard tool for mesh partitioning in many areas [49]. Still, researchers were unable to prove that spectral partitioning techniques would work well on the graphs encountered in 286 D.A. Spielman, S.-H. Teng / Linear... |

125 |
Kontaktprobleme der konformen Abbildung,
- Koebe
- 1936
(Show Context)
Citation Context ... # (i,j)%E #n k=1(vi,k # vj,k)2#n i=1 #n k=1 v 2 i,k = #n k=1 # (i,j)%E(vi,k # vj,k)2#n k=1 #n i=1 v 2 i,k . But, for each k, # (i,j)%E(vi,k # vj,k)2#n i=1 v 2 i,k " #2, so #n k=1 # (i,j)%E(vi,k # vj,k)2#n k=1 #n i=1 v 2 i,k " #2 (this follows from the fact that # i xi/ # i yi " mini xi/yi , for xi, yi > 0). # Our method of finding a good geometric embedding of a planar graph is similar to the way in which Miller et al. [59] directly find good separators of planar graphs. We first find an embedding of the graph on the plane by using the “kissing disk” embedding of Koebe, Andreev, and Thurston [52,5,6,79]: Theorem 3.2 (Koebe–Andreev–Thurston). LetGbe a planar graph with vertex setV = {1, . . . , n} and edge set E. Then, there exists a set of disks {D1, . . . , Dn} in the plane with disjoint interiors such that Di touches Dj if and only if (i, j) % E. Such an embedding is called a kissing disk embedding of G. The analogue of a disk on the sphere is a cap. A cap is given by the intersection of a half-space with the sphere, and its boundary is a circle. We define kissing caps analogously with kissing disks. Following [59], we use stereographic projection to map the kissing disk embedding of the g... |

110 |
Eigenvalues and Graph Bisection: An Average-Case Analysis.
- Boppana
- 1987
(Show Context)
Citation Context ...ctor. A few years later, Barnes and Hoffman [Bar82, BH84] used linear programming in combination with an examination of the eigenvectors of the adjacency matrix of a graph. In a similar vein, Boppana =-=[Bop87]-=- analyzed eigenvector techniques in conjunction with convex programming. However, the use of linear and convex programming made these techniques impractical for most applications. By recognizing a rel... |

107 |
Conductance and the rapid mixing property for markov chains: the approximation of permanent resolved.
- Jerrum, Sinclair
- 1988
(Show Context)
Citation Context ...on also extends the deterministic linear time algorithm for finding a good sphere separator from intersection graphs to overlap graphs [EMT95]. 7. Good ratio cuts Alon [Alo86] and Sinclair and Jerrum =-=[SJ88]-=- proved that graphs with small Fiedler eigenvalue have a good ratio cut (Alon's theorem actually demonstrates the existence of a small vertex separator). A corollary of an extension of their work by M... |

103 | A separator theorem for graphs with an excluded minor and its applications,
- Alon, Seymour, et al.
- 1990
(Show Context)
Citation Context ...Hutchinson, and Tarjan [GHT84] extended these results to show that every graph of genus at most g has a separator of size O i p gn j . Another generalization was obtained by Alon, Seymour, and Thomas =-=[AST90]-=-, who showed that graphs that do not have an h-clique minor have separators of O(h 3=2 p n) nodes. Plotkin, Rao, and Smith [PRS94] reduced the dependency on h from h 3=2 to h. Using geometric techniqu... |

103 |
Isoperimetric number of graphs,
- Mohar
- 1989
(Show Context)
Citation Context ... perpendicular to the all-ones's vector (although this is not explicitly stated in her work). In this section, we will present a new proof of Mihail's theorem (see also [AM85, Fil91, DS91, Mih89] and =-=[Moh89]-=- for a tighter bound). Theorem 21 (Mihail). Let G = (V; E) be a graph on n nodes of maximum degree d, let Q be its Laplacian matrix, and let OE be its isoperimetric number. For any vector ~x 2 R n suc... |

100 | Separators for spherepackings and nearest neighbor graphs.
- Miller, Teng, et al.
- 1997
(Show Context)
Citation Context ...ng bisector size follows Lemma 22. 2 Remark 20. Recently, Agarwal and Pach [AP95] and, independently, Spielman and Teng [ST96] gave an elementary proof of the sphere separator theorem of Miller et al =-=[MTTV96b]-=- on planar graphs and intersection graphs. However, these proofs do not directly extend to overlap graphs. The relation between overlap graphs and intersection graphs established by Theorem 16 enables... |

94 | A unified geometric approach to graph separators. - Miller, Teng, et al. - 1991 |

93 |
A separation theorem for graphs of bounded genus,
- Gilbert, Hutchinson, et al.
- 1984
(Show Context)
Citation Context ...8n nodes in any n-node planar graph. Their result improved a theorem of Ungar [Ung51] which demonstrated that every planar graph has a separator of size O( p n log n). Gilbert, Hutchinson, and Tarjan =-=[GHT84]-=- extended these results to show that every graph of genus at most g has a separator of size O i p gn j . Another generalization was obtained by Alon, Seymour, and Thomas [AST90], who showed that graph... |

89 |
Numerical Grid Generation: Foundation and Applications,
- Thompson, Warsi, et al.
- 1985
(Show Context)
Citation Context ..., to approximate a continuous problem. Finite difference meshes are often produced by inserting a uniform grid from R2 or R3 into the domain via a boundary-matching conformal mapping. Notice that, unlike a finite element mesh, a finite difference mesh need not be a collection of simplices or elements, so we cannot analyze it as we do a triangulation. In general, the derivative of the conformal transformation must vary gradually with respect to the mesh size in order to produce D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 297 good results (see, for example [80]). This means that the mesh will probably satisfy a density condition [11,63]. Let G be an undirected graph and let $ be an embedding of its nodes in Rd . We say $ is an embedding of density % if the following inequality holds for all vertices v in G: Let u be the node closest to v. Let w be the node farthest from v that is connected to v by an edge. Then ($(w) # $(v)( ($(u) # $(v)( ! %. In general, G is an %-density graph in Rd if there exists an embedding of G in Rd with density %. 6.2. Modeling well-shaped meshes We will use the overlap graph to model well-shaped meshes (Miller et al. [59])... |

85 |
On convex polyhedra in lobacevskii space,
- Andreev
- 1970
(Show Context)
Citation Context ... # (i,j)%E #n k=1(vi,k # vj,k)2#n i=1 #n k=1 v 2 i,k = #n k=1 # (i,j)%E(vi,k # vj,k)2#n k=1 #n i=1 v 2 i,k . But, for each k, # (i,j)%E(vi,k # vj,k)2#n i=1 v 2 i,k " #2, so #n k=1 # (i,j)%E(vi,k # vj,k)2#n k=1 #n i=1 v 2 i,k " #2 (this follows from the fact that # i xi/ # i yi " mini xi/yi , for xi, yi > 0). # Our method of finding a good geometric embedding of a planar graph is similar to the way in which Miller et al. [59] directly find good separators of planar graphs. We first find an embedding of the graph on the plane by using the “kissing disk” embedding of Koebe, Andreev, and Thurston [52,5,6,79]: Theorem 3.2 (Koebe–Andreev–Thurston). LetGbe a planar graph with vertex setV = {1, . . . , n} and edge set E. Then, there exists a set of disks {D1, . . . , Dn} in the plane with disjoint interiors such that Di touches Dj if and only if (i, j) % E. Such an embedding is called a kissing disk embedding of G. The analogue of a disk on the sphere is a cap. A cap is given by the intersection of a half-space with the sphere, and its boundary is a circle. We define kissing caps analogously with kissing disks. Following [59], we use stereographic projection to map the kissing disk embedding of the g... |

82 |
An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cli s,
- Strang, Fix
- 1973
(Show Context)
Citation Context |

79 |
Partitioning the nodes of a graph.
- Barnes
- 1985
(Show Context)
Citation Context ...n the works of Hall [47] who applied quadratic programming to design a placement algorithm, and in the work of Donath and Hoffman [28,29] who first suggested using the eigenvectors of adjacency matrices of graphs to find partitions and point placement. Fiedler [33–35] associated the second-smallest eigenvalue of the Laplacian of a graph with its connectivity and suggested partitioning by splitting vertices according to their value in the corresponding eigenvector. Thus, we call this eigenvalue the Fiedler value and a corresponding vector a Fiedler vector. A few years later, Barnes and Hoffman [10,14] used linear programming in combination with an examination of the eigenvectors of the adjacency matrix of a graph. In a similar vein, Boppana [17] analyzed eigenvector techniques in conjunction with convex programming. However, the use of linear and convex programming made these techniques impractical for most applications. By recognizing a relation between the Fiedler value and the Cheeger constant [20] of continuous manifolds, Alon [3] and Sinclair and Jerrum [72] demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vert... |

79 | A Delaunay Based Numerical Method for Three Dimensions: generation, formulation and partition. 683-692
- Millar, Talmor, et al.
- 1995
(Show Context)
Citation Context ...) in R d in which the radius aspect ratio of its dual Delaunay diagram is bounded by a. Then there is a constant ff depending only on d and a so that the dual Delaunay diagram is an ff-density graph (=-=[MTTW95]-=-). 15 ffl If G is an ff-density graph in R d , then the maximum degree of G is bounded by a constant depending only on ff and d; and, G is a subgraph of an ff-overlap graph of a 1-ply neighborhood sys... |

76 |
0.879-Approximation Algorithms for MAX CUT and MAX 2SAT.
- Goemans, Williamson
- 1994
(Show Context)
Citation Context ...d cut. The integer program would be min X (i;j)2E (x i \Gamma x j ) 2 s:t: n X i=1 x i = 0; and x i 2 f\Sigma1g : However, unlike the semi-definite relaxation of MaxCut used by Goemans and Williamson =-=[GW94]-=-, we do not know if our relaxation provides a constant factor approximation of the optimum. 4. Sphere-preserving maps Let B d be the unit ball in d dimensions: f(x 1 ; : : : ; x d )j P n i=1 x 2 is1g.... |

73 |
The Chaco User's Guide, Version 1.0,"
- Hendrickson, Leland
- 1993
(Show Context)
Citation Context ...at spectral partitioning methods work well on graphs that usually arise in practice [BS92, HL92, PSL90, Sim91, Wil90]. Spectral partitioning became a standard tool for mesh partitioning in many areas =-=[HL93]-=-. Still, researchers were unable to prove that spectral partitioning techniques would work well on the graphs encountered in practice. This failure is partially explained by results of Guattery and Mi... |

72 | Numerical Solution of Partial Di®erential Equations by the Finite Element Method - Johnson - 1995 |

70 |
1, isoperimetric inequalities for graphs, and superconcentrators,
- Alon, Milman
- 1985
(Show Context)
Citation Context ... a graph. In a similar vein, Boppana [17] analyzed eigenvector techniques in conjunction with convex programming. However, the use of linear and convex programming made these techniques impractical for most applications. By recognizing a relation between the Fiedler value and the Cheeger constant [20] of continuous manifolds, Alon [3] and Sinclair and Jerrum [72] demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vertices in the eigenvector will produce a cut with a good ratio of cut edges to separated vertices (see also [4, 36,30,55,57]). Around the same time, improvements in algorithms for approximately computing eigenvectors, such as the Lanczos algorithm, made the computation of eigenvectors practical [67, 70]. In the next few years, a wealth of experimental work demonstrated that spectral partitioning methods work well on graphs that usually arise in practice [18,48,66,70,82]. Spectral partitioning became a standard tool for mesh partitioning in many areas [49]. Still, researchers were unable to prove that spectral partitioning techniques would work well on the graphs encountered in 286 D.A. Spielman, S.-H. Teng / Linear... |

67 |
On convex polyhedra of finite volume in lobacevskii space,
- Andreev
- 1970
(Show Context)
Citation Context |

62 |
Conductance and convergence of Markov chains: a combinatorial treatment of expanders,
- Mihail
- 1989
(Show Context)
Citation Context ...d that graphs with small Fiedler eigenvalue have a good ratio cut (Alon's theorem actually demonstrates the existence of a small vertex separator). A corollary of an extension of their work by Mihail =-=[Mih89]-=- demonstrates that one can obtain a good ratio cut from any vector with small Rayleigh quotient that is perpendicular to the all-ones's vector (although this is not explicitly stated in her work). In ... |

61 |
Direct discretization of planar div–curl problems,
- Nicolaides
- 1992
(Show Context)
Citation Context ...ure of the coefficient matrix of such a linear system is exactly the adjacency structure of the 1-skeleton of the simplicial complex. To ensure accuracy, in addition to the conditions that a mesh must conform to the boundaries of the region and be fine enough, each individual element of the mesh must be well-shaped. A common shape criterion for the finite element method is that the angles of each element are not too small, or the aspect ratio of each element is bounded [9,13,37]. Other numerical formulations require slightly different conditions. For example, the controlled volume formulation [64,61] using a Voronoi diagram requires that the radius aspect ratio (the ratio of the circumscribed radius to the shortest edge length of an element in the dual Delaunay diagram) is bounded. The finite difference method also uses a discrete structure, a finite difference mesh, to approximate a continuous problem. Finite difference meshes are often produced by inserting a uniform grid from R2 or R3 into the domain via a boundary-matching conformal mapping. Notice that, unlike a finite element mesh, a finite difference mesh need not be a collection of simplices or elements, so we cannot analyze it as... |

60 |
Diameters and eigenvalues,
- Chung
- 1989
(Show Context)
Citation Context ...form graphs [GK95], and bounded-degree trees. Bounds on the Fiedler values of regular grids and quasi-uniform graphs essentially follow from the fact that the diameters of these graphs are large (see =-=[Chu89]-=-). Bounds on trees can be obtained from the fact that every boundeddegree tree has a ffi-separator of size 1 for some constant ffi in the range 0 ! ffi ! 1=2 that depends only on the degree. However, ... |

59 | Eigenvectors of acyclic matrices. - Fiedler - 1975 |

51 |
An O(n) algorithm for three-dimensional n-body simulation,
- Zhao
- 1987
(Show Context)
Citation Context ...e.g., the N-body simulation method). However different the particular methods may be, a basic principle is common to all—accuracy of approximation is ensured by using meshes that satisfy certain numerical and geometric constraints. Meshes that satisfy these constraints are said to be well-shaped. To motivate our spectral analysis of well-shaped meshes, we review the conditions required of finite element and finite difference meshes. More detailed discussions can be found in several books and papers (for example, see [69,50,13,12,37]). Background material on the particle method can be found in [15,43,45,83]. The finite element method approximates a continuous problem by subdividing the domain (a subset of Rd ) of the problem into a mesh of polyhedral elements and then approximates the continuous function by piecewise polynomial functions on the elements. A common choice for an element is a d-dimensional simplex. Accordingly, a d-dimensional finite element mesh is a d-dimensional simplicial complex, a collection of d-dimensional simplices that meet only at shared faces [13,12,58]. The computation graph associated with each simplicial complex is often its 1-skeleton or the 1-skeleton of its geomet... |

49 | Shallow excluded minors and improved graph decomposition, in:
- Plotkin, Rao, et al.
- 1994
(Show Context)
Citation Context ... j . Another generalization was obtained by Alon, Seymour, and Thomas [AST90], who showed that graphs that do not have an h-clique minor have separators of O(h 3=2 p n) nodes. Plotkin, Rao, and Smith =-=[PRS94]-=- reduced the dependency on h from h 3=2 to h. Using geometric techniques, Miller, Teng, Thurston, and Vavasis [MT90, MTTV96a, MTTV96b, MTV91, MV91, Ten91] extended the planar separator theorem to grap... |

48 |
On estimating the largest eigenvalue with the Lanczos algorithm,
- Parlett, Simon, et al.
- 1982
(Show Context)
Citation Context ...practical for most applications. By recognizing a relation between the Fiedler value and the Cheeger constant [20] of continuous manifolds, Alon [3] and Sinclair and Jerrum [72] demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vertices in the eigenvector will produce a cut with a good ratio of cut edges to separated vertices (see also [4, 36,30,55,57]). Around the same time, improvements in algorithms for approximately computing eigenvectors, such as the Lanczos algorithm, made the computation of eigenvectors practical [67, 70]. In the next few years, a wealth of experimental work demonstrated that spectral partitioning methods work well on graphs that usually arise in practice [18,48,66,70,82]. Spectral partitioning became a standard tool for mesh partitioning in many areas [49]. Still, researchers were unable to prove that spectral partitioning techniques would work well on the graphs encountered in 286 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 practice. This failure is partially explained by results of Guattery and Miller [42] demonstrating that naive applications of spect... |

48 |
Conductance and convergence of Markov chains!a combinatorial treatment of expanders,
- Mihail
- 1989
(Show Context)
Citation Context |

47 |
Separators in two and three dimensions,
- Miller, Thurston
- 1990
(Show Context)
Citation Context ...al books and papers (for example, see [69,50,13,12,37]). Background material on the particle method can be found in [15,43,45,83]. The finite element method approximates a continuous problem by subdividing the domain (a subset of Rd ) of the problem into a mesh of polyhedral elements and then approximates the continuous function by piecewise polynomial functions on the elements. A common choice for an element is a d-dimensional simplex. Accordingly, a d-dimensional finite element mesh is a d-dimensional simplicial complex, a collection of d-dimensional simplices that meet only at shared faces [13,12,58]. The computation graph associated with each simplicial complex is often its 1-skeleton or the 1-skeleton of its geometric dual (as used in the finite volume method). In the finite element method, a linear system is defined over a mesh, with variables representing physical quantities at the nodes. The nonzero structure of the coefficient matrix of such a linear system is exactly the adjacency structure of the 1-skeleton of the simplicial complex. To ensure accuracy, in addition to the conditions that a mesh must conform to the boundaries of the region and be fine enough, each individual elemen... |

46 |
Algorithms for partitioning of graphs an,d computer logic based on eigenvectors of con,nection matrices,
- DONATH, HOFFMAN
- 1972
(Show Context)
Citation Context ...is cut, computing a Fiedler vector of the new graph, and iterating as necessary, one can find a bisector of O(n1#1/d) edges. In particular, we prove that maximum-degree ! planar graphs have Fiedler value at most 8!/n, which implies that spectral techniques can be used to find bisectors of size at most O( " n) in these graphs. These bounds are the best possible for well-shaped meshes and planar graphs. 1.1. History The spectral method of graph partitioning was born in the works of Hall [47] who applied quadratic programming to design a placement algorithm, and in the work of Donath and Hoffman [28,29] who first suggested using the eigenvectors of adjacency matrices of graphs to find partitions and point placement. Fiedler [33–35] associated the second-smallest eigenvalue of the Laplacian of a graph with its connectivity and suggested partitioning by splitting vertices according to their value in the corresponding eigenvector. Thus, we call this eigenvalue the Fiedler value and a corresponding vector a Fiedler vector. A few years later, Barnes and Hoffman [10,14] used linear programming in combination with an examination of the eigenvectors of the adjacency matrix of a graph. In a similar v... |

44 |
Condition of finite element matrices generated from nonuniform meshes
- Fried
- 1972
(Show Context)
Citation Context |

41 | Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes,
- Chan, Smith
- 1993
(Show Context)
Citation Context ...k was done while the author was at the University of Minnesota and Xerox PARC. 0024-3795/$ - see front matter ( 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2006.07.020 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 285 1. Introduction Spectral partitioning has become one of the most successful heuristics for partitioning graphs and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [70,82], solving sparse linear systems [68], and partitioning for domain decomposition [21,25]. It is also used in VLSI circuit design and simulation [26,46,2]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [18,48,49,66,70,82]. However, the quality of the partition that these methods should produce has so far eluded precise analysis. In this paper, we will prove that spectral partitioning methods give good separators for the graphs to which they are usually applied. The size of the separator produced by spectral methods can be related to the Fiedler value—the second smallest eigenvalu... |

37 | Spheres, and Separators: A Unified Geometric Approach to Graph Partitioning. - Points - 1992 |

36 | Provably good partitioning and load balancing algorithms for parallel adaptive n-body simulation,
- Teng
- 1998
(Show Context)
Citation Context ...rarchical N-body simulation methods (such as the Barnes-Hut's treecode method [BH86] and the fast-multipole method [GR87]) is a subgraph of an ff-overlap graph of an O(log n)-ply neighborhood system (=-=[Ten96]-=-). 6.3. Spherical Embeddings of Overlap Graphs In this section, we show that an ff-overlap graph is a subgraph of the intersection graph obtained by projecting its neighborhoods onto the sphere and th... |

35 | On the performance of the spectral graph partitioning methods, in:
- Guattery, Miller
- 1995
(Show Context)
Citation Context ...ll, researchers were unable to prove that spectral partitioning techniques would work well on the graphs encountered in practice. This failure is partially explained by results of Guattery and Miller =-=[GM95]-=- demonstrating that naive applications of spectral partitioning, such as spectral bisection, will fail miserably on some graphs that could conceivably arise in practice. By bounding the Fiedler values... |

29 | Spectral nested dissection,
- Pothen, Simon, et al.
- 1992
(Show Context)
Citation Context ... partitioning graphs and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [Sim91, Wil90], solving sparse linear systems =-=[PSW92]-=-, and partitioning for domain decomposition [CR87, CS93]. It is also used in VLSI circuit design and simulation [CSZ93, HK92, AK95]. Substantial experimental work has demonstrated that spectral method... |

26 |
A deterministic linear time algorithm for geometric separators and its applications,
- Eppstein, Miller, et al.
- 1995
(Show Context)
Citation Context ... using the intersection graph separator theorem. The same reduction also extends the deterministic linear time algorithm for finding a good sphere separator from intersection graphs to overlap graphs =-=[EMT95]-=-. 7. Good ratio cuts Alon [Alo86] and Sinclair and Jerrum [SJ88] proved that graphs with small Fiedler eigenvalue have a good ratio cut (Alon's theorem actually demonstrates the existence of a small v... |

24 | Spectral partitioning, eigenvalue Bounds, and circle packings for graphs of bounded genus, in:
- Kelner
- 2004
(Show Context)
Citation Context ...action. The following are two natural extensions of Theorem 3.3. Conjecture 1 (Graphs with bounded genus). Let G be a graph of n nodes and genus g " 1. If the maximum vertex degree of G is !, then, the Fiedler value of G is at most O ,!g n - . Conjecture 2 (Graphs with bounded forbidden minors). Let G be a graph of n nodes. If the maximum vertex degree of G is ! and G does not have Kh, the clique of h vertices, as a minor, then the Fiedler value of G is at most O ,!hk n - , for some small constant k. D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 301 Kelner [51] recently proved both of these conjectures. The following conjecture remains open. Suppose G is an undirected graph. Let G1, . . . , Gh be h disjoint connected subgraphs of G. Let H be a graph minor defined by G1, . . . , Gh. The depth of H is the maximum diameter of G1, . . . , Gh. Conjecture 3 (Graphs with bounded shallow excluded minors). Let G be a graph of n nodes. If the maximum vertex degree of G is ! and there exists d " 1 such that for every L, G does not have Ld -clique minor of depth L, then the Fiedler value of G is at most O(!/n2/d). Plotkin et al. [65] proved that any graph that ... |

20 |
Disk packings and planar separators.
- Spielman, Teng
- 1996
(Show Context)
Citation Context ...ievable by a Fiedler cut follows immediately from Theorem 21 and the corresponding bisector size follows Lemma 22. 2 Remark 20. Recently, Agarwal and Pach [AP95] and, independently, Spielman and Teng =-=[ST96]-=- gave an elementary proof of the sphere separator theorem of Miller et al [MTTV96b] on planar graphs and intersection graphs. However, these proofs do not directly extend to overlap graphs. The relati... |

18 |
Density graphs and separators, in:
- Miller, Vavasis
- 1991
(Show Context)
Citation Context ...produced by inserting a uniform grid from R2 or R3 into the domain via a boundary-matching conformal mapping. Notice that, unlike a finite element mesh, a finite difference mesh need not be a collection of simplices or elements, so we cannot analyze it as we do a triangulation. In general, the derivative of the conformal transformation must vary gradually with respect to the mesh size in order to produce D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 297 good results (see, for example [80]). This means that the mesh will probably satisfy a density condition [11,63]. Let G be an undirected graph and let $ be an embedding of its nodes in Rd . We say $ is an embedding of density % if the following inequality holds for all vertices v in G: Let u be the node closest to v. Let w be the node farthest from v that is connected to v by an edge. Then ($(w) # $(v)( ($(u) # $(v)( ! %. In general, G is an %-density graph in Rd if there exists an embedding of G in Rd with density %. 6.2. Modeling well-shaped meshes We will use the overlap graph to model well-shaped meshes (Miller et al. [59]). An overlap graph is based on a k-ply neighborhood system. The neighborhood ... |

17 | On the angle condition in the - nite element method - Babuska, Aziz - 1976 |

17 | Finding separator cuts in planar graphs within twice the optimal.
- Garg, Saran, et al.
- 1999
(Show Context)
Citation Context ...ques, Miller, Teng, Thurston, and Vavasis [58–60,62,63,76] extended the planar separator theorem to graphs embedded in higher dimensions and showed that every well-shaped mesh in Rd has a 1/(d + 2)-separator of size O(n1#1/d). Using multicommodity flow, Leighton and Rao [53] designed a partitioning method guaranteed to return a cut whose ratio of cut size to vertices separated is within logarithmic factors of optimal. Arora et al. [8] recently improved the approximation ratio to O( ! log n). For planar graphs, a partition within a constant factor of the optimal can be found in polynomial time [38]. While spectral methods have been favored in practice, they lacked a proof of effectiveness. 1.2. Outline of paper In Section 2, we introduce the concept of a graph partition, review some facts from linear algebra that we require, and describe the class of spectral partitioning methods. In Section 3, we prove the embedding lemma, which relates the quality of geometric embeddings of a graph with its Fiedler value. We then show (using the main result of Section 4) that every planar graph has a “nice” embedding as a collection of spherical caps on the surface of a unit sphere in three dimensions... |

16 |
Graph Separator theorems and sparse gaussian elimination,
- Gilbert
- 1980
(Show Context)
Citation Context ...amma 1 (n 1\Gamma1=d \Gamma 1): Lipton and Tarjan [LT79] showed that by repeatedly applying an ff-separator of size fi p n, one can obtain a bisection of size fi=(1 \Gamma p 1 \Gamma ff) p n. Gilbert =-=[Gil80]-=- extended this result to graphs with positive vertex weights at the expense of a 1=(1 \Gamma p 2) factor in the bisection bound. Djidjev and Gilbert [DG92] further generalized this result to graphs wi... |

14 |
Finite element meshes and geometric separators
- Miller, Teng, et al.
- 1995
(Show Context)
Citation Context ...act that P i x i = P i y ismin i x i =y i , for x i ; y i ? 0). 2 Our method of finding a good geometric embedding of a planar graph is similar to the way in which Miller, Teng, Thurston, and Vavasis =-=[MTTV96a] directly -=-find good separators of planar graphs. We first find an embedding of the graph on the plane by using the "kissing disk" embedding of Koebe, Andreev, and Thurston [Koe36, And70a, And70b, Thu8... |

12 | Separators in graphs with negative and multiple vertex weights,
- Djidjev, Gilbert
- 1992
(Show Context)
Citation Context ...ze fi=(1 \Gamma p 1 \Gamma ff) p n. Gilbert [Gil80] extended this result to graphs with positive vertex weights at the expense of a 1=(1 \Gamma p 2) factor in the bisection bound. Djidjev and Gilbert =-=[DG92]-=- further generalized this result to graphs with arbitrary weights. Leighton and Rao [LR88] showed that one can obtain an O(ff)-approximation to a 1/3-separator from an ff-approximation to a ratio cut.... |

11 |
A framework for the analysis and construction of domain decomposition preconditioners,
- Chan, Resasco
- 1987
(Show Context)
Citation Context ...k was done while the author was at the University of Minnesota and Xerox PARC. 0024-3795/$ - see front matter ( 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2006.07.020 D.A. Spielman, S.-H. Teng / Linear Algebra and its Applications 421 (2007) 284–305 285 1. Introduction Spectral partitioning has become one of the most successful heuristics for partitioning graphs and matrices. It is used in many scientific numerical applications, such as mapping finite element calculations on parallel machines [70,82], solving sparse linear systems [68], and partitioning for domain decomposition [21,25]. It is also used in VLSI circuit design and simulation [26,46,2]. Substantial experimental work has demonstrated that spectral methods find good partitions of the graphs and matrices that arise in many applications [18,48,49,66,70,82]. However, the quality of the partition that these methods should produce has so far eluded precise analysis. In this paper, we will prove that spectral partitioning methods give good separators for the graphs to which they are usually applied. The size of the separator produced by spectral methods can be related to the Fiedler value—the second smallest eigenvalu... |

11 |
An approximate max- ow min-cut theorem for uniform multicommodity ow problems with applications to approximation algorithms
- Leighton, Rao
- 1988
(Show Context)
Citation Context ...e planar separator theorem to graphs embedded in higher dimensions and showed that every well-shaped mesh in R d has a 1=(d+2)-separator of size O(n 1,1=d ). Using multicommodity ow, Leighton and Rao =-=[LR88]-=- designed a partitioning method guaranteed to return a cut whose ratio of cut size to vertices separated is within logarithmic factors of optimal. While spectral methods have been favored in practice,... |

10 |
Combinatorial aspects of geometric graphs,
- Teng
- 1998
(Show Context)
Citation Context ...or defined by G1, . . . , Gh. The depth of H is the maximum diameter of G1, . . . , Gh. Conjecture 3 (Graphs with bounded shallow excluded minors). Let G be a graph of n nodes. If the maximum vertex degree of G is ! and there exists d " 1 such that for every L, G does not have Ld -clique minor of depth L, then the Fiedler value of G is at most O(!/n2/d). Plotkin et al. [65] proved that any graph that excludes Kh as a depth L minor has a separator of size O(lh2 log n + n/L). In addition, they showed that for any well-shaped meshes G in Rd , G excludes Kh minors of depth L for h = $(Ld#1). Teng [78] extended the latter result to knearest neighborhood graphs in Rd . The above conjecture is a natural extension of Theorem 6.4 on well-shaped meshes and Corollary 5.2 on k-nearest neighbor graphs. To prove Conjecture 3, we may need to develop new techniques. The proofs of this paper as well as of Kelner [51] critically used the fact that every graph considered has a geometric realization (as the intersection of a disk-packings). So far, there is no similar embedding results for graphs with bounded shallow excluded minors. One approach is to develop a combinatorial characterization of graph eig... |

8 |
A theorem on planar graphs,
- Ungar
- 1951
(Show Context)
Citation Context ...his area was that of Lipton and Tarjan [LT79], who constructed a linear-time algorithm that produces a 1=3-separator of p 8n nodes in any n-node planar graph. Their result improved a theorem of Ungar =-=[Ung51]-=- which demonstrated that every planar graph has a separator of size O( p n log n). Gilbert, Hutchinson, and Tarjan [GHT84] extended these results to show that every graph of genus at most g has a sepa... |

6 | Condition of #nite element matrices generated from nonuniform meshes - Fried - 1982 |

6 | A uni ed geometric approach to graph separators - Miller, Teng, et al. - 1991 |

6 |
Algebraic Graph Theory, second ed.,
- Biggs
- 1993
(Show Context)
Citation Context |

5 |
Partitioning, spectra and linear programming,
- Barnes, Hoffman
- 1984
(Show Context)
Citation Context ...n the works of Hall [47] who applied quadratic programming to design a placement algorithm, and in the work of Donath and Hoffman [28,29] who first suggested using the eigenvectors of adjacency matrices of graphs to find partitions and point placement. Fiedler [33–35] associated the second-smallest eigenvalue of the Laplacian of a graph with its connectivity and suggested partitioning by splitting vertices according to their value in the corresponding eigenvector. Thus, we call this eigenvalue the Fiedler value and a corresponding vector a Fiedler vector. A few years later, Barnes and Hoffman [10,14] used linear programming in combination with an examination of the eigenvectors of the adjacency matrix of a graph. In a similar vein, Boppana [17] analyzed eigenvector techniques in conjunction with convex programming. However, the use of linear and convex programming made these techniques impractical for most applications. By recognizing a relation between the Fiedler value and the Cheeger constant [20] of continuous manifolds, Alon [3] and Sinclair and Jerrum [72] demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vert... |

3 | Algebraic connectibity of graphs - Fiedler - 1973 |

3 | Eigenvectors of acyclic matices - Fiedler - 1975 |

2 | Separators: a uni ed geometric approach to graph partitioning - Points - 1991 |

2 | Points, spheres, and separators: a unified geometric approach to graph partitioning, - Teng - 1991 |

1 | On convex polyhedra of nite volume in lobacevskii space - Andreev - 1970 |

1 | Spectra ofGraphs : Theory and Application - Cvetkovic, Doob, et al. - 1990 |

1 |
A hierarchical O(nlog n) force calculation algorithm,
- Barnes, Hut
- 1986
(Show Context)
Citation Context ...e.g., the N-body simulation method). However different the particular methods may be, a basic principle is common to all—accuracy of approximation is ensured by using meshes that satisfy certain numerical and geometric constraints. Meshes that satisfy these constraints are said to be well-shaped. To motivate our spectral analysis of well-shaped meshes, we review the conditions required of finite element and finite difference meshes. More detailed discussions can be found in several books and papers (for example, see [69,50,13,12,37]). Background material on the particle method can be found in [15,43,45,83]. The finite element method approximates a continuous problem by subdividing the domain (a subset of Rd ) of the problem into a mesh of polyhedral elements and then approximates the continuous function by piecewise polynomial functions on the elements. A common choice for an element is a d-dimensional simplex. Accordingly, a d-dimensional finite element mesh is a d-dimensional simplicial complex, a collection of d-dimensional simplices that meet only at shared faces [13,12,58]. The computation graph associated with each simplicial complex is often its 1-skeleton or the 1-skeleton of its geomet... |

1 |
Finite element meshes and geometric separators,SIAM
- Miller, Teng, et al.
- 1998
(Show Context)
Citation Context ...nent-wise application of this fact. Write &vi as (vi,1, . . . , vi,n). Then, for all {&v1, . . . , &vn} such that #n i=1 &vi = &0, we have # (i,j)%E (&vi # &vj(2#n i=1 (&vi(2 = # (i,j)%E #n k=1(vi,k # vj,k)2#n i=1 #n k=1 v 2 i,k = #n k=1 # (i,j)%E(vi,k # vj,k)2#n k=1 #n i=1 v 2 i,k . But, for each k, # (i,j)%E(vi,k # vj,k)2#n i=1 v 2 i,k " #2, so #n k=1 # (i,j)%E(vi,k # vj,k)2#n k=1 #n i=1 v 2 i,k " #2 (this follows from the fact that # i xi/ # i yi " mini xi/yi , for xi, yi > 0). # Our method of finding a good geometric embedding of a planar graph is similar to the way in which Miller et al. [59] directly find good separators of planar graphs. We first find an embedding of the graph on the plane by using the “kissing disk” embedding of Koebe, Andreev, and Thurston [52,5,6,79]: Theorem 3.2 (Koebe–Andreev–Thurston). LetGbe a planar graph with vertex setV = {1, . . . , n} and edge set E. Then, there exists a set of disks {D1, . . . , Dn} in the plane with disjoint interiors such that Di touches Dj if and only if (i, j) % E. Such an embedding is called a kissing disk embedding of G. The analogue of a disk on the sphere is a cap. A cap is given by the intersection of a half-space with the ... |

1 |
Dafna Talmor, Shang-Hua Teng, Noel Walkington, A Delaunay based numerical method for three dimensions: generation, formulation, and partition,
- Miller
- 1995
(Show Context)
Citation Context ...ure of the coefficient matrix of such a linear system is exactly the adjacency structure of the 1-skeleton of the simplicial complex. To ensure accuracy, in addition to the conditions that a mesh must conform to the boundaries of the region and be fine enough, each individual element of the mesh must be well-shaped. A common shape criterion for the finite element method is that the angles of each element are not too small, or the aspect ratio of each element is bounded [9,13,37]. Other numerical formulations require slightly different conditions. For example, the controlled volume formulation [64,61] using a Voronoi diagram requires that the radius aspect ratio (the ratio of the circumscribed radius to the shortest edge length of an element in the dual Delaunay diagram) is bounded. The finite difference method also uses a discrete structure, a finite difference mesh, to approximate a continuous problem. Finite difference meshes are often produced by inserting a uniform grid from R2 or R3 into the domain via a boundary-matching conformal mapping. Notice that, unlike a finite element mesh, a finite difference mesh need not be a collection of simplices or elements, so we cannot analyze it as... |