D. A. Spielman and S.-H. Teng. Spectral partitioning works: Planar graphs and finite element meshes. In IEEE Symposium on Foundations of Computer Science, pages 96--105, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....guaranteed bounds on the error better than the naive value of 3 4 . Semidefinite programming methods involve a lot of machinery, and in practice, their efficacy is sometimes questioned [14] Efficient Solution Rather than using semidefinite approaches, we will resort to spectral partitioning [27] for computational efficiency reasons. In doing so, we exploit particularly two additional facts that hold in our situation: 1. Rather than being a general graph, we have an instance which is largely a bipartite graph with some noise edges added. 2. Neither side of the bipartite graph is much ....

D. Spielman and S. Teng. Spectral partitioning works: Planar graphs and finite element meshes. In 37th Annual Symposium on Foundations of Computer Science, 1996.

....and Juvan and Mohar [J M1, J M2] in the labeling problems. Spectral partitioning which is based on eigenvectors of Laplace eigenvalues of graphs has proved to be one of the most successful heuristic approaches in the design of partition algorithms [B S, H L, H M P R] in parallel computation [S T, Sim, Wil], in solving sparse linear systems [P S W] clustering [H K, C S Z] and in ranking [J M2, H M P R] Similar kind of applications is based on the properties of the Perron Frobenius eigenvector of a nonnegative matrix. This technique is suitable for the ranking problems. We refer to [M P2] for ....

D. A. Spielman, S.-H. Teng, Spectral partitioning works: Planar graphs and finite element meshes, preprint, 1996.

....uses up to n # eigenvectors for 0 # 1 4 , there exist graphs for which the algorithm fails to find a separator with a cut quotient within a factor of n 1 4 # 1 times the isoperimetric number. Finally, we provide a summary of some important subsequent results by Spielman and Teng [27], and relate our results to them. This paper makes an additional contribution: While the counterexamples have simple structures and intuitively might be expected to cause problems for spectral separator algorithms, the challenge is to provide good bounds on # 2 for these graphs. For this purpose ....

.... included a stronger lower bound on the minimum bisection size [6] work by Rendl, Wolkowicz, and others using optimization approaches [24, 10] and the particular bisection and graph partitioning methods considered in this paper [18] 23] 25] Since our work first appeared [17] Spielman and Teng [27] have extended the latter methods to include recursion. It is worth noting that spectral methods have not been limited to graph partitioning; work has been done using the spectrum of the adjacency matrix in graph coloring [4] and using the Laplacian spectrum to prove theorems about expander graph ....

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D. A. Spielman and S. H. Teng, Spectral partitioning works: Planar graphs and finite element meshes, Tech. Report UCB CSD-96-898, U.C. Berkeley, 1996. An extended abstract appeared in the 37th Annual Symposium on Foundations of Computer Science.

.... [18] and to analyze the quality of preconditioners [4, 13] Bounds on # 2 are useful in the analysis of spectral partitioning, both because # 2 occurs in bounds on cut quality [24] and because they can be used in isolating structural properties of the eigenvectors used in making the cuts [16, 28]. The eigenvalue # 2 has been related to expansion properties of graphs and can be used in determining if a graph is an expander [1, 2] One common class of techniques for computing such lower bounds uses properties of graph embeddings [9, 15, 20, 22, 26, 27] In such methods, a graph H is ....

.... of the connection between Laplacian spectra (particularly with respect to # 2 ) and properties of the associated graphs dates back to Fiedler s work in the 1970s (see, e.g. 10] and [11] These properties have been used in graph algorithms, particularly algorithms for finding small separators [17, 25, 28]. The relationship between graph embeddings and matrix representations has been GRAPH EMBEDDINGS AND LAPLACIAN EIGENVALUES 705 the subject of much interesting research. A large proportion of this work has been aimed at bounding the second largest eigenvalues of time reversible Markov chains in ....

D. A. Spielman and S. H. Teng, Spectral partitioning works: Planar graphs and finite element meshes, Techical report UCB CSD-96-898, University of California at Berkeley, Berkeley, CA, 1996.

.... School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213 (glmiller cs.cmu.edu) This work supported in part by NSF Grant CCR 9505472, DARPA contract N00014 95 1246, and Army Contract DAAH04 95 1 0607. 1 structural properties of the eigenvectors used in making the cuts [16, 28]. The eigenvalue # 2 has been related to expansion properties of graphs, and can be used in determining if a graph is an expander [1, 2] One common class of techniques for computing such lower bounds uses properties of graph embeddings [9, 15, 20, 22, 26, 27] In such methods, a graph H is ....

.... of the connection between Laplacian spectra (particularly with respect to # 2 ) and properties of the associated graphs dates back to Fiedler s work in the 1970 s (see, e.g. 10] and [11] These properties have been used in graph algorithms, particularly algorithms for finding small separators [17, 25, 28]. The relationship between graph embeddings and matrix representations has been the subject of much interesting research. A large proportion of this work has been aimed at bounding the second largest eigenvalues of time reversible Markov chains in order to bound the mixing time for random walks. ....

D. A. Spielman and S. H. Teng, Spectral partitioning works: Planar graphs and finite element meshes, Tech. Report UCB CSD-96-898, U.C. Berkeley, 1996. An extended abstract appeared in the 37th Annual Symposium on Foundations of Computer Science.

.... value s are: bisection cut, in which s is the median of fu 21 , u 22 , u 2n g; ratio cut, in which s is the value that gives the best conductance; sign cut, in which s is equal to 0; and gap cut, in which s is a value in the largest gap in the sorted list of Fiedler vector components [33]. 1 The eigenvector, u 2 = u 21 , u 22 , u 2n ) of the second smallest eigenvalue, # 2 2 The Laplacian matrix of a graph is derived from its weight matrix: l i j = a i j , l ii = j a i j 39 u u 2 1 S L Figure 3 5: The vector u 2 is the longest vector on S L u 1 (This ....

D. Spielman and S. Teng. Spectral partitioning works: planar graphs and finite element meshes. In Proc. 37nd IEEE Symp. on Foundations of Comp. Science, 1996.

....vertices of S with vertices of S. In [6, 14] it is shown that i 2 (G) 2 dmax(G) 2(G) 2 i(G) with maximal degree dmax (G) Additionally, it is 2 = O( 1 n ) for bounded degree planar graphs and two dimensional meshes and 2 = O( 1 n 2=d ) for well shaped d dimensional meshes [18]. If the lower bound of equation (1) is not strict, either all vertices are still incident to cut edges but not to exactly 2 2 of them, or there are vertices which are not incident to any cut edge. We will present a new method to determine an upper bound for 2 . This method makes use of the ....

D. A. Spielman and S.-H. Teng. Spectral partitioning works: Planar graphs and finite element meshes. In Proc. 37th Conf. on Foundations of Computer Science, pages 96--105, 1996.

....value. Pothen, Simon, and Liou [1] explored the usage of the Fiedler vector for partitioning graphs. Hendrickson and Leland [15] generalized spectral bisection to perform quadrasection and octasection by using the third and fourth smallest eigenpairs of the graph Laplacian. Spielman and Teng [29] present an upper bound on the Fiedler value of bounded degree d dimensional graphs, and they relate this upper bound to the number of edges cut by a Fiedler cut. In contrast, Guattery and Miller [14] present graphs for which spectral partitioning yields poor separators. Simon and Teng [28] show ....

D. Spielman and S. H. Teng. Spectral partitioning works: planar graphs and finite element meshes. In 37th Annual Symposium Foundations of Computer Science, Burlington, Vermont, October 1996. IEEE, IEEE Press.

....mesh in a suitable sense (see [14] The proof of Lemma 2.2 and the statements of Theorem 2.1 and 2.2 imply that the graph G of ( GammaA=A) A 11 ) B 22 is also the graph of a well shaped mesh. Namely, it is a bounded degree subgraph of some overlap graph (see [13] 14] It was shown in [20] that the upper bound on the second smallest eigenvalue of the Laplacian matrix of G (the Fiedler value) is of the order O(1=NIF 2=3 ) Then using the techniques from [20] we obtain that there exists a O(NIF 2=3 ) size bisector of G. Therefore, G satisfies the so called NIF 2=3 separator ....

....graph of a well shaped mesh. Namely, it is a bounded degree subgraph of some overlap graph (see [13] 14] It was shown in [20] that the upper bound on the second smallest eigenvalue of the Laplacian matrix of G (the Fiedler value) is of the order O(1=NIF 2=3 ) Then using the techniques from [20] we obtain that there exists a O(NIF 2=3 ) size bisector of G. Therefore, G satisfies the so called NIF 2=3 separator theorem: there exist constants 1=2 ff 1; fi 0 such that the vertices of G can be partitioned into sets GA ; GB and the vertex separator GC such that jGA j; jGB j ffNIF ....

D.A. Spielman and S.H. Teng. Spectral partitioning works: Planar graphs and finite element meshes. Technical Report UCB//CSD-96-898, University of Berkeley, 1996.

....p, since the second Laplacian eigenvalue satisfies [23] # 2 = pn #( p(1 p)n log n] 1 2 ) More interesting are the implications of these bounds for degree bounded finite element meshes in two and three dimensions. We will employ the following result proved recently by Spielman and Teng [38]. Theorem 7.2. The second Laplacian eigenvalue of an overlap graph embedded in d dimensions is bounded by O(n 2 d ) # Planar graphs are overlap graphs in two dimensions and well shaped meshes in three dimensions are also overlap graphs with d =3. Table 7.1 summarizes the asymptotic lower ....

D. A. Spielman and S.-H. Teng, Spectral partitioning works: Planar graphs and finite element meshes, manuscript, Computer Science Department, University of Minnesota, Minneapolis, MN, 1996.

....field describes a location and the temperature at that location. The spatial objects are governed by interaction rules relating objects to each other, along with constraints on feature values. 2 To be precise, this gives the smallest positive eigenvalue of the Laplacian graph of the field [14]. 4 (a) b) Figure 3: A field modeled by a network of local spatial objects: a) Finite difference grid; b) Finite element mesh. The grid or mesh defines spatial adjacency between a spatial object (solid circle) and its neighbors (empty circles) The shaded region in the mesh represents ....

D. Spielman and S. Teng. Spectral partitioning works: Planar graphs and finite element meshes. Technical Report UCB CSD-96-898, UC Berkeley, 1996.

.... and to analyze the quality of preconditioners [Axe92, GMZ95] Bounds on # 2 are useful in the analysis of spectral partitioning, both because # 2 occurs in bounds on cut quality [Moh89] and because they can be used in isolating structural properties of the eigenvectors used in making the cuts [GM95, ST96]. The eigenvalue # 2 has been related to expansion properties of graphs, and can be used in determining if a graph is an expander [AM85a, Alo86] Related work involving graph embeddings has been used to bound the mixing time for random walks [JS89, SJ89, DS91, Sin92, Kah96] In this case, the ....

....show bounds for classes of graphs in order to state results in terms of asymptotic algorithm behavior. For example, in the analysis of spectral partitioning, GM95] uses eigenvalue bounds on a family of bounded degree graphs to prove facts about the structure of eigenvectors used in partitioning; [ST96] gives an upper bound on planar graph eigenvalues that can be applied in bounds on the cut quotient of the resulting cut. The embedding techniques we present are well suited to producing such general results, and can be used with known results about embeddings. We have used them to generate lower ....

[Article contains additional citation context not shown here]

D. A. Spielman and S. H. Teng. Spectral partitioning works: Planar graphs and finite element meshes. Technical Report UCB CSD-96-898, U.C. Berkeley, 1996. An extended abstract appeared in the 37th Annual Symposium on Foundations of Computer Science.

....the second Laplacian eigenvalue satisfies [23] 2 = pn Gamma Theta( p(1 Gamma p)n log n] 1=2 ) More interesting are the implications of these bounds for degree bounded finite element meshes in two and three dimensions. We will employ the following result proved recently by Spielman and Teng [38]. Theorem 7.2. The second Laplacian eigenvalue of an overlap graph embedded in d dimensions is bounded by O(n Gamma2=d ) 2 problem separator 2 Esize(A) Ework(A) size LB UB LB UB d dim. O(n 1 Gamma1=d ) Theta(n Gamma2=d ) Omega Gamma n 2 Gamma2=d ) O(n 2 Gamma1=d ) ....

D. A. Spielman. and S-H. Teng, Spectral partitioning works: Planar graphs and finite element meshes, Manuscript, 1996. Available on the Web at the URL http://cs.berkeley.edu/~spielman/spect.html.

....1970 s. Spectral methods relate good partitions of an undirected graph to the eigenvalues and eigenvectors of certain matrices derived from the graph. These methods have been found to exhibit good performance in many contexts that are difficult to attack by purely combinatorial means (see e.g. [4, 12, 26, 36]) and they has been successfully applied in areas such as finite element mesh decomposition [36] and the analysis of Monte Carlo simulation methods [26] Recently the heuristic intution underlying spectral partitioning has been used in the context of information retrieval [14] and in methods for ....

....of certain matrices derived from the graph. These methods have been found to exhibit good performance in many contexts that are difficult to attack by purely combinatorial means (see e.g. 4, 12, 26, 36] and they has been successfully applied in areas such as finite element mesh decomposition [36] and the analysis of Monte Carlo simulation methods [26] Recently the heuristic intution underlying spectral partitioning has been used in the context of information retrieval [14] and in methods for identifying densely connected hypertextual regions of the World Wide Web [28, 9] Our new method ....

D. Spielman, S. Teng, "Spectral partitioning works: Planar graphs and finite-element meshes," Proc. IEEE Symp. on Foundations of Computer Science, 1996.

....vector. Pothen, Simon, and Liou [26] explored the usage of the Fiedler vector for partitioning graphs. Hendrickson and Leland [17] generalized spectral bisection to perform quadrasection and octasection by using the third and fourth smallest eigenpairs of the graph Laplacian. Spielman and Teng [30] present an upper bound on the Fiedler value of bounded degree d dimensional graphs, and they relate this upper bound to the number of edges cut by a Fiedler cut. In this paper we apply graph theory to improve the computation of the Fiedler vector. In our new approaches the eigensolver is the ....

D. Spielman and S. H. Teng. Spectral partitioning works: planar graphs and finite element meshes. In 37th Annual Symposium Foundations of Computer Science, Burlington, Vermont, October 1996. IEEE, IEEE Press.

....1970 s. Spectral methods relate good partitions of an undirected graph to the eigenvalues and eigenvectors of certain matrices derived from the graph. These methods have been found to exhibit good performance in many contexts that are difficult to attack by purely combinatorial means (see e.g. [4, 12, 26, 36]) and they has been successfully applied in areas such as finite element mesh decomposition [36] and the analysis of Monte Carlo simulation methods [26] Recently the heuristic intuition underlying spectral partitioning has been used in the context of information retrieval [14] and in methods for ....

....of certain matrices derived from the graph. These methods have been found to exhibit good performance in many contexts that are difficult to attack by purely combinatorial means (see e.g. 4, 12, 26, 36] and they has been successfully applied in areas such as finite element mesh decomposition [36] and the analysis of Monte Carlo simulation methods [26] Recently the heuristic intuition underlying spectral partitioning has been used in the context of information retrieval [14] and in methods for identifying densely connected hypertextual regions of the World Wide Web [9, 28] Our new method ....

D. Spielman, S. Teng, "Spectral partitioning works: Planar graphs and finite-element meshes," Proc. IEEE Symp. on Foundations of Computer Science, 1996.

.... 2 1 Introduction The graph partitioning problem is an NP hard combinatorial optimization problem (Garey and Johnson, 1979) that arises in many applications such as parallel and distributed computing, VLSI circuit design and simulation, transportation management, and data mining (Pothen, 1997; Spielman and Teng, 1996). In this article, we consider a special case, the graph bi partitioning problem (GBP) which can be stated as follows. Given an undirected graph G = V; E) the GBP is to find a partition of the nodes in two equally sized sets denoted V 1 and V 2 so that the number of edges between nodes in the ....

Spielman, D. A. and Teng, S.-H. (1996). Spectral Partitioning Works: Planar Graphs and Finite Element Meshes. In 37th Annual Symposium on Foundations of Computer Science, pages 96-- 105, Burlington, Vermont.

....along those directions and chooses as candidates those with small gradient vector magnitudes. In the example of Figure 5, an implementation of the algorithm chooses directions of weak coupling 3 To be precise, this gives the smallest positive eigenvalue of the Laplacian graph of the field (Spielman Teng 1996). Figure 9: Decomposition of a complex thermal domain. that yield the disaggregated source structure shown in Figure 9. The use of physical structures (gradient trajectory N graph) allows spatial aggregation to design a control structure that decomposes a field into components that can be ....

Spielman, D., and Teng, S. 1996. Spectral partitioning works: Planar graphs and finite element meshes. Technical Report UCB CSD-96-898, UC Berkeley.

....n 1 Gamma1=d ) boxes. The proof of the above theorem given in [Ten96] further shows that a partitioning of this quality can be found by the geometric partitioning algorithm (GEO) of Miller, Teng, Thurston, and Vavasis [MTTV97] and a variant of spectral partitioning given by Spielman and Teng [ST96] 5.4.2 Load Balancing and Partitioning Heuristics Orthogonal recursive bisection (ORB) and Morton and Peano Hilbert ordering have been used to partition particles in the Barnes Hut method [Sal90, WS93] and to partition boxes in an adaptive fast multipole method [SHHG93] We have developed a ....

....parallel implementations of these heuristics: the partitioning algorithms takes only a few lines of arithmetic array operations plus a subroutine call to the HPF sort intrinsic. 5.4. 3 Provably Good Partitioning Algorithms Geometric partitioning (GEO) MTTV97] and a variant (by Spielman and Teng [ST96] of recursive spectral bisection (RSB) PSL90] both offer guarantees on the quality of the partitions. GEO lends itself to efficient data parallel implementations, as shown in Chapter 6. Extensive use of sampling can be used to reduce the computational complexity without a significant ....

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D. A. Spielman and S.-H. Teng. Spectral partitioning works: planar graphs and finite element meshes. In Proceedings of the 37th Annual Symposium on Foundation of Computer Science, pages 96--107, 1996.

....coordinates of large absolute value in non principal eigenvectors. For this reason, we will at times refer to them as non principal communities. We discuss this issue in Section 3. The heuristic intuition behind this approach is analogous to the spectral partitioning of undirected graphs (e.g. [7, 11, 34]) however, it is important to note that what we are doing here is not simply a spectral partitioning of the Web graph. In particular, we are studying non principal eigenvectors of symmetric matrices derived from the (asymmetric) adjacency matrix of the base set; and the structures we find are ....

....are denser as subgraphs in the link structure; we will sometimes refer to this notion as the strength of a community. Another interesting feature of the communities derived from non principal eigenvectors is the following. Drawing on the heuristic intuition underlying spectral graph partitioning [7, 11, 34], one expects pairs of communities (X i ; Y i ) and (X Gamma i ; Y Gamma i ) associated with the same eigenvector to be very sparsely connected in the underlying graph. In some cases, this sparse linkage can have meaning in the context of the query topic. Basic Results We now give ....

D. Spielman, S. Teng, "Spectral partitioning works: Planar graphs and finite-element meshes," Processedings of the 37th IEEE Symposium on Foundations of Computer Science, 1996.

.... and finite element models commonly used in engineering and science for numerically approximating a field (Vichnevetsky 1981) We note that SA supports many important reasoning and controlling tasks 2 To be precise, this gives the smallest positive eigenvalue of the Laplacian graph of the field (Spielman Teng 1996). ffl Spatial Objects Geometric description Feature description Examples: temperature; pressure; velocity ffl Constitutive Laws Examples: Fourier s law: heat flux = Gammak dT dx Ohm s law: charge flux = Gammafl dV dx Hooke s law: stress = E du dx ffl Spatial Neighborhood ....

Spielman, D., and Teng, S. 1996. Spectral partitioning works: Planar graphs and finite element meshes.

....spectral methods. Theory exists to show the effectiveness of this approach. The work of Guattery and Miller [7] present examples of graphs for which spectral partitioning does not work well. Nevertheless, spectral partition methods work well on boundeddegree planar graphs and finite element meshes [17]. Chan et al. 3] show that spectral partitioning is optimal in the sense that the partition vector induced by it is the closest partition vector to the second eigenvector, in any l s norm where s 1. Nested dissection is predicated upon finding a good partition of a graph. Simon and Teng [16] ....

D. Spielman and S. H. Teng. Spectral partitioning works: planar graphs and finite element meshes. In 37th Annual Symposium Foundations of Computer Science, Burlington, Vermont, October 1996. IEEE, IEEE Press.

....small. In particular, for a family of r regular graphs fG p g with r = r(p) and Gp =r 0 as p 1 the right hand side of (18) asymptotically equals p 4 Delta r r Gamma1 . In general, some other approaches for estimating the bisection width and cutwidth of graphs are known (see e.g. 80] and [107] for application of spectral methods) Further information on this topic can be found in the surveys [40, 41] If the graph H is not a path, just a few exact results on estimating the edge congestion are known. A general lower bound is due to the isoperimetric approach [20] econ(G; H) max m ....

Spielman D.A., Teng S.-H. : Spectral partitioning works: Planar graphs and finite elements meshes, In: Proc. 37th Annual Symposium on Foundations of Comp. Sci., Oct. 14-16, 1996, Burlington, IEEE Computer Society Press, 1996, 96--105.

.... Min cut graph partitioning is known to be NP complete, and many heuristic methods have been proposed (see [4] for a survey) Spectral methods are well established [7] 16] 18] 27] 36] and have also been the subject of extensive study in recent years [2] 5] 6] 10] 19] 23] 25] 29] 37] [40]. These methods use eigenvectors of the Laplacian or adjacency matrix to construct various types of geometric representations of the graph. We loosely categorize previous methods according to five basic representations: ffl Linear orderings (spectral bipartitioning) The classic works of Hall [27] ....

....in both the VLSI and scientific computing communities. SB takes the linear ordering induced by the second eigenvector of the Laplacian (also called the Fiedler vector) and splits it to yield a 2 way partitioning. Subsequent works have proposed variations of this algorithm [6] 36] 25] and [23] [40] analyzed its performance. ffl Multiple linear orderings: Barnes [7] proposed a multiple eigenvector extension to spectral bipartitioning. Barnes tries to map each cluster in a k way partitioning to one of k eigenvectors while minimizing the rounding error according to a transportation ....

D. A. Spielman and S.-H. Teng, Spectral partitioning works: planar graphs and finite element meshes, Manuscript (1996).

....than or equal to the median are assigned to another part. This strategy is often referred as spectral bisection. Spectral bisection will fail miserably on some graphs that could conceivably arise in practice as shown by Guattery and Miller [29] Fortunately, a recent result of Spielman and Teng [48] showed that a careful use of Fiedler vector can produce provably good partition for planar graphs, finite element meshes, and N body simulation graphs. For any vector x 2 R n , we have x T L(G) x = X (i;j)2E (x i Gamma x j ) 2 : Moreover, the Fiedler value, 2 , of G is given by ....

....effectiveness of local optimization in the multilevel paradigm. Another example on local optimization will be given in Section 6. In Section 3.3, we present a multilevel method for the approximation of the Fiedler vector of a graph. Fiedler vector can be used to find a good partition of a graph [48]. In Section 3.5, we give two provably good multilevel geometric partitioning algorithms for finding a k way partition. We now discuss some other multilevel partitioning methods. 16 Shang Hua Teng Bui and Jones [14] were among the first to propose the multilevel method for graph partitioning. ....

D. A. Spielman and S.-H. Teng. Spectral partitioning works: planar graphs and finite element meshes. UCB//CSD-96-898, U.C. Berkeley, 1996.

....an n ff separator theorem or G is n ff separable if there is a constant c such that every n node graph in G has a bisection of cut size at most cn ff . Moreover, we refer to the latter type of bisections as n ff separators. More information concerning small separators can be found in [12, 8, 18]. In the reminder of this paper we denote by G(ff) a family of graphs that is n ff separable and closed under the subgraph operation. Several such families are known. For example, Gilbert, Hutchinson, and Tarjan [10] showed that bounded degree graphs with bounded genus are n 1=2 ....

....Gilbert, Hutchinson, and Tarjan [10] showed that bounded degree graphs with bounded genus are n 1=2 separable; Alon, Seymour, and Thomas [1] proved that bounded degree graphs that do not have an h clique minor for a constant h are n 1=2 separable. Miller, Teng, Thurston, and Vavasis [14, 15, 18] showed that well shaped meshes in R and nearest neighbor graphs in R are (n 1 Gamma1=d ) separable; The min total boundary exact partitioning problem has been addressed in the literature. The following lemma has been shown in [17] Lemma 3.1 Let k be an integer such that k 1. Then, for ....

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D. A. Spielman and S.-H. Teng. Spectral partitioning works: planar graphs and finite element meshes. In Proceedings of the 37-th Annual Symposium on Foundation of Computer Science, pages 96--107, IEEE, 1996,

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D. A. Spielman and S.-H. Teng. Spectral partitioning works: Planar graphs and finite element meshes. In IEEE Symposium on Foundations of Computer Science, pages 96--105, 1996.

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Spielman, D.A., Teng, S.H.: Spectral partitioning works: Planar graphs and finite element meshes. In: IEEE Symposium on Foundations of Computer Science. (1996) 96--105

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D. Spielman, S.H. Teng. Spectral Partitioning works: Planar graphs and finite element meshes, FOCS 1996.

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D. A. Spielman and S.-H. Teng, "Spectral Partitioning Works: Planar Graphs and Finite Element Meshes," in 37th Annual Symposium on Foundations of Computer Science, (Burlington, Vermont), pp. 96--105, 1996.

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D. A. Spielman and S.-H. Teng. Spectral partitioning works: Planar graphs and finite element meshes. In IEEE Symposium on Foundations of Computer Science, pages 96--105, 1996.

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Spielman, D.A., Teng, S.H.: Spectral partitioning works: Planar graphs and finite element meshes. In: IEEE Symposium on Foundations of Computer Science. (1996) 96--105

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D. A. Spielman and S.-H. Teng. Spectral partitioning works: planar graphs and finite element meshes. In Proceedings of the 37th IEEE Symposium on Foundations of Computing (FOCS'96), pages 96--105, Los Alamitos, CA, 1996. IEEE Computer Society.

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D. Spielman and S. H. Teng, Spectral Partitioning Works: Planar Graphs and Finite Element Meshes, in 37th Annual Symposium on Foundations of Computer Science (FOCS'96), Burlington, VT, Oct. 1996, pp. 96--105.

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D. A. Spielman, S.-H. Teng, Spectral partitioning works: Planar graphs and finite element meshes, preprint, 1996.

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D. Spielman and S. Teng. "Spectral Partitioning Works: Planar Graphs and finite element meshes." In Proc. of 37th FOCS, 1996.

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