Earl R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. Alg. Disc. Meth., 3(4):541--550, December 1982. 27  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... for the graph is used to find a good partition; c) multilevel algorithms, such as those in [13] 14] 30] and [32] that first coarsen the graph, partition the smaller graph, then uncoarsen to obtain a partition for the original graph; d) optimization based methods, such as those in [5], 6] 7] 18] and [45] where approximations to the best partitions are obtained by solving optimization problems. See [3] for a survey of results in this area prior to 1995. Here we focus on optimization based formulations. Much of the earlier work in this area involves relaxations in which ....

.... Much of the earlier work in this area involves relaxations in which constraints are dropped in an optimization problem to obtain a tractable problem whose optimal solution is a lower bound for the optimal partition (see, for example, 6] 17] 41] We also mention the work of Barnes [5] in which a spectral decomposition of the adjacency matrix is used with the solution of a related transportation problem (linear cost function and linear constraints) to approximate the best partition. In [7] a diagonal perturbation of the adjacency matrix is used to make it positive definite, and ....

E. R. Barnes, An algorithm for partitioning the nodes of a graph, SIAM J. Alg. Discrete Methods, 3 (1982), pp. 541--550.

....diagonal degree matrix D defined by D ii = d(v i ) The eigenvalues and eigenvectors of such matrices are the subject of the relatively recent subfield of graph theory dealing with graph spectra. Early theoretical work connecting graph spectra and partitioning is due to Barnes, Donath and Hoffman [1] [6] 7] More recent eigenvector and eigenvalue methods have dealt with both module placement (Frankle and Karp [11] and Tsay and Kuh [41] and graph min cut bisection (Boppana [3] and Blanks [2] In general, these previous works formulate the partitioning problem as the assignment or placement ....

E. R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. Alg. Disc. Meth., 3(4):541-- 550, 1982.

....A = a ij ) find the k cluster that minimises P k h=1 E h where E h = v i 2C h v j = 2C h a ij . In other words, one tries to find the k clusters with a minimal number of inter cluster edges. One class of heuristics that proves successful as an approximation to this are spectral methods [1,15]. The Recursive Spectral Bisection [26] used in our structuring algorithm, applies the median cut procedure, described below, recursively in a divide and conquer fashion until the required number of subsets is obtained. For describing the median cut procedure we first need to define the ....

E. R. Barnes. An algorithm for Partitioning the Nodes of a Graph. Siam J. Algorithms and Discrete Methods, 3(4):541--549, 1992.

....are quite often the basis for load balancing in parallel computing. Hendrickson and Leland [18] provide a good overview on static graph partitioning algorithms. They include greedy algorithms like the famous one by Kernighan and Lin [23] and extensions of it [5] so called spectral methods [2, 4, 1] and other hybrid approaches. Several of these algorithms have been implemented and integrated into tools that can be used as off line pre processors to partitioning a central computing problem. Among these tools are Chaco [19] METIS [22] and JOSTLE [40] Recent research also tackles the problem ....

E. R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM Journal on Algebraic and Discrete Methods, 3(4):541-550, 1982.

....between the next two groups to be combined is no longer positive [12] These rule grouping algorithms are not applicable to expert systems running on real time architectures because there is no provision for balancing the process level and rule level overheads. Another class of algorithms [3, 13, 15] deals with a more general graph partitioning problem as follows: given a graph G with costs on its nodes and edges, partition the nodes of G into k subsets of specified sizes, s 1 ; s 2 ; Delta Delta Delta ; s k , so as to minimize the total cost of the edges cut. The problem can be related to ....

E.R. Barnes, "An algorithm for partitioning the nodes of a graph" SIAM Journal on Algebraic and Discrete Methods, Vol. 3, No. 4, Dec. 1982. pp. 541-550.

....go back to Ho#mann and Donath [9] who proved a lower bound on the size of the minimum bisection of a graph, and Fiedler [11] 12] who explored the properties of # 2 and its associated eigenvector for the Laplacian. There has been much subsequent work, including Barnes s partitioning algorithm [5], Boppana s work that 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 ....

E. R. Barnes, An algorithm for partitioning the nodes of a graph, SIAM Journal on Algebraic and Discrete Methods, 3 (1982), pp. 541--550.

....edge twice; we do this to avoid 1 2 terms throughout this work. Min cut graph partitioning is known to be NP complete, so heuristic methods must be invoked. Previous approaches have included seeded epitaxial growth, iterative improvement [16] genetic algorithms [6] etc. Spectral methods [1] 2] [4] [7] 8] 11] 13] 15] have been successful in recent years and are of particular interest for our present work. These works share a common trait of using eigenvectors to construct some type of geometric representation of the graph. We note four such representations: ffl Linear ordering or ....

....to construct some type of geometric representation of the graph. We note four such representations: ffl Linear ordering or 1 dimensional placement: Hall [13] showed that the second eigenvector of the Laplacian yields an optimum 1 dimensional placement in terms of squared wirelength. Barnes [4] proposed a method using multiple eigenvectors that reduces to sorting the coordinates of the adjacency matrix s largest eigenvector when k = 2. Hagen and Kahng [11] extended Barnes idea to ratio cut partitioning by considering all possible splits of the linear ordering induced by the Laplacian s ....

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E. R. Barnes, "An Algorithm for Partitioning the Nodes of a Graph," Siam J. Alg. Disc. Methods Vol. 3, No. 4, 1992, pp. 541-549.

....processor loads. An application of graph partitioning to parallel molecular dynamics simulations is given in [24] The maximum clique problem is another graph problem that has been given a quadratic programming formulation [11, 21] Work related to other approaches to graph partitioning includes [1, 2, 3, 4, 8, 10, 15, 16, 17, 18, 20, 22, 23, 25]. 2. Optimality conditions. For any matrix Y, the support is de ned by supp Y = f(i; j) y ij 6= 0g: Given a set f(i; j) 1 i n; 1 j kg; let P( denote the set of n by k matrices Y with supp Y and with each row or column of Y either completely zero, or containing single 1 and ....

E. R. Barnes, An algorithm for partitioning the nodes of a graph, SIAM Journal on Algebraic and Discrete Mathematics, 3 (1982), pp. 541-550.

....4 5 ) 2 3 : 5 ) 2 3 A 5 : A 9 : 4 9 4 (1) s.t. i) Module placement constraints: 3 4 5 9 : 4J7 #LK : #EM # N#E (ii) Block size constraints ) 2 3 : 5 9 : 4PO #QK 4 # MG# R# 9 4PS T # U # O ;V OXW O Barnes [11] developed a polynomial time heuristic for approximating the above 0 1 quadratic transportation problem. The heuristic is based on approximating the netlist or hypergraph by a weighted graph G [12, 13] that tightly estimates the number of cut nets in any netlist partition. The numerical ....

....first finds the largest eigenvalues of the connection matrix A of the graph and their corresponding orthonormal eigenvectors #E # (# ) Let : be the Y Z component of the eigenvector corresponding to the kth largest eigenvalue of the adjacency matrix of G. Barnes [11] shows that the solution of the following linear transportation problem gives an approximate solution to the graph partitioning problem: 9 ) 2 3 : 5 ) 3 4 5 : 4 [ 4 : 4 (2) s.t. i) Position constraints: 3 465 9 : 4 7 # K : 7 #EM # ]#E (ii) ....

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E.R. Barnes, "An Algorithm for Partitioning The Nodes of a Graph", SIAM Journal of Algebraic and Discrete Methods, vol. 3, n. 4, pp. 541--550, December 1982.

....of the graph Laplacian matrix. A drawback of these methods is that they cannot be directly applied to hypergraphs, so a hyperedge model is needed to approximate the hypergraph by a graph. The relation between the spectrum of a graph and other graph properties has been an area of active research [4, 8, 15, 25, 26, 43], but only recently spectrum based methods have been successfully applied to graph partitioning [1, 2, 21, 31, 46, 47, 53] We present below a spectral graph K partitioning formulation that extends Rend and Wolkowicz s to graphs containing weighted vertices. Other Global Methods. Simulated ....

....columns consist of the eigenvectors of L corresponding to L s (K 1) smallest positive eigenvalues in nondecreasing order (so S N Y = Z) then the lower bound is attained for X = 1 K u N u t K N K Z R t K . Solutions that relax all but the third constraint in (P2) e.g. due to Barnes [4], Bolla [7] and Chan et al. 12] result in the geometric representation X = u N N : Z] instead. The most efficient algorithm for computing the eigenvalues and eigenvectors of a large, sparse, and symmetric N N matrix is the Lanczos algorithm with O(N 1.4 ) runtime [14, 45] Thus, O(N ....

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E. Barnes, "An Algorithm for Partitioning the Nodes of a Graph," SIAM Journal on Algebraic Discrete Methods, Vol. 3, No. 4, pp. 541--550, 1982.

.... with look ahead scheme [24] and the simulated annealing approach [23, 17] The analytical methods include both the use of a linear placement formulation with the quadratic objective function, which is solved by computing the second smallest eigenvector of the Laplacian matrix of the given network [14, 2, 4, 18], and the use of the linear placement formulation with a linear objective function, which is solved by an iterative method in [26] The min cut based method uses the maximum flow algorithm to compute a series of minimum cuts in the given circuit in 2order to obtain an area balanced cut with ....

E. R. Barnes, "An Algorithm for Partitioning the Nodes of a Graph," SIAM J. Alg. Disc. Math., Vol. 3, pp. 541-550, 1982.

....cost for aggregate queries on networks. Contributions: We propose a new access method, CCAM, to efficiently support aggregate queries over general networks such as road maps. We adapt the heuristic graph partitioning approach 2 to cluster the 2 The literature in the area of graph partitioning [4, 8, 15, 5] albeit in a different context and has only focused on partitioning static graphs, without considering dynamic updates. Proc. of the IEEE Eleventh Intl. Conference on Data Engineering, March 1995 3 nodes of a given network into file pages by the connectivity relationship. Ideally, the clustering ....

E.R. Barnes. "An Algorithm for Partitioning the Nodes of a Graph". SIAM Journal Alg. Disc. Meth., 3(4):541--550, December 1982.

....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 [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 ....

E. R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. on Algebraic and Discrete Methods, 4(3):541--550, 1982.

....presented in [9, 10, 11] Two approaches are simulated annealing and recursive bisection. Simulated annealing changes the work distribution randomly among processors [11, 12] This method requires iterations to achieve optimal balance of load which can be expensive. Eigenvalue recursive bisection [9, 11, 13, 14] requires solution of eigenvalue problems and is also quite expensive for dynamic load balancing. Orthogonal recursive bisection uses cutting planes to partition the computational grid based upon centroidal co ordinates of the cells. This approach is the least expensive among the above methods. ....

E. R. Barnes, "An Algorithm for Partitioning the Nodes of a Graph", SIAM J. Alg. Disc. Meth., Vol. 3, pp 541, 1982.

....number of elements on the partition interfaces can be large. Moreover, the method cannot handle complex 3 D grids easily. Eigenvalue recursive bisection requires the solution of an eigenvalue problem and is quite expensive but this technique reduces the number of elements on partition interfaces [2, 4]. One of the most effective eigenvalue techniques is recursive spectral bisection (RSB) which partitions on the basis of graph connectivity. This technique has been used for partitioning of unstructured meshes to obtain high quality subdomains [2, 3] A second key issue in the development of ....

E. R. Barnes, "An Algorithm for Partitioning the Nodes of a Graph," SIAM Journal of Alg. Disc. Methods, Vol. 3, p. 541, March 1982.

....a pair of connected nodes that are more likely to be accessed together are allocated to a common page of the file. Analysis and experiments show that the proposed method leads to reduced I O costs and a higher WCRR for many interesting networks. The literature in the area of graph partitioning [6, 7, 13, 25] has only focused on partitioning static graphs without considering dynamic updates. We address the following two issues. First, the static graph partitioning approach is not efficient when the entire network cannot fit into main memory. In general, road maps are very large databases [3, 26] and ....

....The graph partitioning problem is to partition the nodes of a graph with costs on its edges into subsets of given sizes, so as to minimize the sum of the costs on all the cut edges. Although the graph partitioning problem is NP complete [14] many good heuristics based on spectral partitioning [6] and iterative approaches [7, 13, 25] have been proposed to solve this problem efficiently. The implementation of CCAM operations takes advantage of these heuristics. The WCRR model is proposed on the basis of the available database statistics on access frequencies. One way to gather such ....

E.R. Barnes. "An Algorithm for Partitioning the Nodes of a Graph". SIAM Journal Alg. Disc. Meth., 3(4):541--550, December 1982.

....the minimum spanning trees algorithm can also be used as a simple heuristic for the graphpartitioning approach. However, the ratio cut [5] KL [22] and other methods [21] are superior [5, 21, 22] These heuristics are based on spectral partitioning and iterative approaches. In spectral techniques [1], eigenvalues and eigenvectors of the matrix representation of the graph are computed. Partition results are derived by mapping the information provided by eigenvectors onto actual partitions. Iterative approaches [5, 13, 22, 35] start from an initial partition, then iteratively apply pairwise ....

E.R. Barnes. "An Algorithm for Partitioning the Nodes of a Graph". SIAM Journal Alg. Disc. Meth., 3(4):541--550, December 1982.

....to intelligently enforce the balanced partition and the minimum edge cut requirements. A partition of G k can be obtained using various algorithms such as (a) spectral bisection [32, 31, 1, 18] b) geometric bisection [28, 27] if coordinates are available) and (c) combinatorial methods [24, 2, 9, 10, 12, 4]. Since the size of the coarser graph G k is small (i.e. V k 100) this step takes a small amount. METIS implements four different schemes for partitioning the coarsest graph, that are evaluated in [22] Three of these algorithms are based on graph growing heuristics, and the other one ....

....If options[0] 0, then the default parameters are used (like the simple interface of METIS) If options[0] 1, then the remaining four elements of options are interpreted as follows (see Section 3. 1 for the allowable values) options[1] The number of vertices to coarsen down to (CoarsenTo) options[2] The matching type (MType) options[3] The initial partitioning algorithm (IPType) options[4] The refinement type (RType) In the case of KMETIS options[3] is unused. numbering If numbering=0, then C style numbering is assumed that starts from 0, if numbering=1, the Fortran style numbering is ....

Earl R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. Algebraic Discrete Methods, 3(4):541--550, December 1984.

....are quite often the basis for load balancing in parallel computing. Hendrickson and Leland [18] provide a good overview on static graph partitioning algorithms. They include greedy algorithms like the famous one by Kernighan and Lin [23] and extensions of it [5] so called spectral methods [2, 4, 1] and other hybrid approaches. Several of these algorithms have been implemented and integrated into tools that can be used as o line pre processors to partitioning a central computing problem. Among these tools are Chaco [19] METIS [22] and JOSTLE [40] Recent research also tackles the problem ....

E. R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM Journal on Algebraic and Discrete Methods, 3(4):541-550, 1982.

....for j = 1; Delta Delta Delta ; n 0 U i I for i = 1; Delta Delta Delta ; k ; 73) Barnes and Hoffman in [BH84] describe how to use the eigenvectors associated with the k largest eigenvalues of the optimal matrix A Diag x to generate a partition of the nodes of the graph. See also Barnes [Bar82a] and [Bar82b] An important special case of the graph partitioning problem is the case when all m i s are equal. In that case the graph partitioning problem simplifies to: min (k=n)1 T x trace V s:t: V Diag x A V 0 max A ffl Y s:t: Y ii = k n for i = 1; Delta Delta Delta ; n ....

E. R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. Alg. and Disc. Meth., 3, 1982.

.... The relation between the properties of a graph and its spectrum (the eigenvalues eigenvectors of its associated matrices) has been an area of active research for several years [13, 8, 16, 20] The spectra of the adjacency matrix A or the Laplacian Q of a graph are the basis for both partitioning [3, 9, 12] and placement techniques [11] III B Spectral ratio cut partitioning using the Laplacian Q Spectral partitioning forms clusters of vertices based on the embedding implied by the eigenvectors V of a graph matrix, which can be the Laplacian Q, or the adjacency matrix A of the graph. To minimize ....

E. R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM Journal on Algorithm and Discrete Method, 3(4):541--550, Dec. 1982.

....it could be regarded as part of the same algorithmic approach to graph bisection as seed growth, since they both consider moving vertices based on edge crossings. We also implemented another standard technique that operates on a very different principle. This method, known as spectral bisection [1, 17, 15], uses an eigenvector computation to embed a graph along a line such that the sum of the squared distances between embedded connected vertices is minimized. Then we join the ends of the line to form a circle, and 22 consider each of the jV j=2 bisections resulting from breaking the ring into two ....

E. R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM Journal of Algebraic and Discrete Methods, 3(4):541--550, 1982.

....However, the Kernighan Lin algorithm has found its niche in a local post processing phase, to further reduce the number of edges cut, after another global algorithm is used to compute a good initial partition. The use of the Kernighan Lin algorithm for this purpose has been advocated in [9, 89], and this is one of the post processing options in CHACO [53] Bui and Jones have implemented a multilevel Kernighan Lin algorithm to improve the quality of the partitions computed [15, 62] The Kernighan Lin algorithm has been used to partition the coarsest level and to refine initial partitions ....

....that has a small isoperimetric ratio jffi(A; B)j=jAj, where A is the smaller of the two subsets. Donath and Hoffman [29] were the earliest to use spectral methods to partition graphs in the context of circuit layout. Spectral partitioning algorithms were considered and analyzed by Barnes [9], Boppana [12] Alon, Galil, and Milman [2] Mohar [80] and many others. Aspvall and Gilbert [6] used eigenvectors of the adjacency matrix to color the vertices of a graph. Pothen, Simon, and Liou [89] used the spectral partitioning algorithm to compute separators for parallel computing, proved ....

E. R. Barnes, An algorithm for partitioning the nodes of a graph, SIAM J. Alg. Disc. Meth., 3 (1982), pp. 541--550.

....the minimum spanning trees algorithm can also be used as a simple heuristic for the graph partitioning approach. However, the ratio cut [5] KL [22] and other methods [21] are superior [5, 21, 22] These heuristics are based on spectral partitioning and iterative approaches. In spectral techniques [1], eigenvalues and eigenvectors of the matrix representation of the graph are computed. Partition results are derived by mapping the information provided by eigenvectors onto actual partitions. Iterative approaches [5, 13, 22, 34] start from an initial partition, then iteratively apply pairwise ....

E.R. Barnes. "An Algorithm for Partitioning the Nodes of a Graph". SIAM Journal Alg. Disc. Meth., 3(4):541--550, December 1982. 22

....if all objects are of equal size and all edge weights equal 1. The problem is in P if the CG is a tree and all objects have the same size [GJ79] or all edge weights equal 1 [Had74] For a given CG and a given number of partitions r a lower bound for the optimum external costs was found by Barnes [Bar82]: EC opt EC max 0 1 2 r X i=1 B 3 i (C) where EC opt and EC max denote the optimum and maximum external costs, respectively, C is the weighted adjacency matrix of the CG, and B is the maximum number of ob 1 The external costs are often called the cut size of the partitioning. jects ....

E. R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM Journal of Algebraic and Discrete Mathematics, 3(4):541--550, 1982.

....best result is selected as the final 1 the work of the second author was partially supported by a NSERC graduate scholarship. partition [7] Meta heuristics are used to guide interchange heuristics out of locally optimal partitions [8] Another technique is to start from good initial partitions [7, 8, 9]. Unlike partitioning heuristics, clustering heuristics identify and merge strongly connected cells into clusters, condensing the circuit. Clustering and partitioning may be implemented as a two phase partitioning heuristic. The circuit is initially clustered. Cell interchanges are applied to the ....

E. R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. of Algebraic and Discrete Methods, 3(4):541--550, December 1982.

....time algorithms for finding optimal bisections are known only for trees, bounded width planar graphs, and solid grid graphs [28] Furthermore, no approximation algorithms are known even for planar graphs. The current state of the art consists of heuristics that often perform well in practice [2, 4, 11, 25] and algorithms that have been shown to have good average case behavior over certain probability distributions on graphs [7, 9, 11, 14] The next section briefly surveys some of these algorithms. The rest of this paper focuses on the spectral methods and is organized as follows. Section 3 covers ....

....This section concludes with two theorems that play a central role in the two spectral algorithms that we will study. One of these two theorems is a slight generalization of a theorem due to Fan [15] that forms the basis for Section 4 s simplified exposition of an algorithm due to Barnes [2]. The other theorem fills in a major hole in Boppana s extended abstract [7] Boppana s algorithm is the topic of Section 5. We point out a problem with the original algorithm, discuss a remedy proposed by Boppana [8] and then demonstrate the problem with the remedy. Fortunately, these problems ....

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Earl R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM Journal on Algebraic and Discrete Methods, 3(4):541--550, December 1982.

....be formulated as the search for a projection of n dimensional space onto a k dimensional subspace, mapping the n unit vector basis (the n nodes of a graph) into k distinct points (the partitions) to minimize the weighted quadratic displacement. This is essentially the formulation given by Barnes [2, 3]. However, unlike Barnes formulation we do not assume any pre determined partition sizes. By Hall s result [4] using the k eigenvectors of the graph s Laplacian, corresponding to the smallest k eigenvalues provides a projection which minimizes the weighted quadratic displacement under the ....

....Handshaking Lemma (the sum of the degrees of all nodes in an undirected graph is twice the number of edges) and the trace of a symmetric matrix (the sum of its diagonal entries) is the sum of its eigenvalues. 1 aka Kirchoff matrix aka admittance matrix 2 This was defined in Barnes SIAM paper [2]. Chan, Schlag, Zien UCSC May, 1992 4 Observation 2 If Y is an n Theta k matrix then Y T Q(G)Y is a k Theta k matrix whose gh th component is 1 2 n X i=1 n X j=1 a ij (y ig Gamma y jg ) y ih Gamma y jh ) Proof: Since Y T Q(G)Y = Y T (D(G) Gamma A(G) Y = Y T D(G)Y Gamma ....

E. R. Barnes, "An algorithm for partitioning the nodes of a graph," SIAM Journal on Algorithm and Discrete Method, vol. 3, pp. 541--550, Dec. 1982.

....algorithms which have been developed for the min cut graph partitioning problem can also be applied to efficiently solve the max cut graph partitioning problem. These heuristics Proc. of the IEEE Eleventh Intl. Conference on Data Engineering, March 1995 4 are based on spectral partitioning [1], and iterative approaches [14, 3] 3 Implementation and Analysis of Max Cut Declustering In this section, we present the implementation and analysis of the max cut declustering approach. The analysis will demonstrate in forthcoming theorem 1 that the max cut declustering approach is capable of ....

E.R. Barnes. "An Algorithm for Partitioning the Nodes of a Graph". SIAM Journal Alg. Disc. Meth., 3(4):541--550, December 1982.

....: n. If we let d(v i ) denote the degree of node v i (i.e. the sum of the weights of all edges incident to v i ) we obtain the n Theta n diagonal degree matrix D defined by D ii = d(v i ) Early theoretical work connecting graph spectra and partitioning is due to Barnes, Donath and Hoffman [1] [6] 7] Most recent eigenvector and eigenvalue methods have dealt with both module placement (Frankle and Karp [9] Kleinhans et al. 17] and Tsay and Kuh [21] and graph min cut bisection (Blanks [2] and Boppana [3] In general, these previous works formulate the partitioning problem as the ....

E. R. Barnes, "An Algorithm for Partitioning the Nodes of a Graph", SIAM J. Alg. Disc. Meth. 3(4) (1982), pp. 541-550.

....of the graph Laplacian matrix. A drawback of these methods is that they cannot be directly applied to hypergraphs, so a hyperedge model is needed to approximate the hypergraph by a graph. The relation between the spectrum of a graph and other graph properties has been an area of active research [4, 8, 15, 25, 26, 43], but only recently spectrum based methods have been successfully applied to graph partitioning [1, 2, 21, 30, 31, 46, 47, 53] We present below a spectral graph K partitioning formulation that extends Rend and Wolkowicz s to graphs containing weighted vertices. Other Global Methods. Simulated ....

....of the eigenvectors of L corresponding to L s (K 1) smallest positive eigenvalues in nondecreasing order (so S N Y = Z) then the lower bound is attained for X = 1 K u N u t K N K Z R t K . Solutions that relax all but the third constraint in (P2) e.g. due to Barnes [4], Bolla [7] and Chan et al. 12] result in the geometric representation X = u N N : Z] instead. The most efficient algorithm for computing the eigenvalues and eigenvectors of a large, sparse, and symmetric N N matrix is the Lanczos algorithm with O(N 1.4 ) runtime [14, 45] Thus, ....

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E. Barnes, "An Algorithm for Partitioning the Nodes of a Graph," SIAM Journal on Algebraic Discrete Methods, Vol. 3, No. 4, pp. 541--550, 1982. A HYPERGRAPH FRAMEWORK FOR OPTIMAL MODEL-BASED DECOMPOSITION OF DESIGN PROBLEMS 32

.... scheme [Kr83] and the simulated annealing approach [KiGV83, GrSu84] The analytical methods use a linear placement formulation with either the quadratic wirelength objective function, which is solved by computing the second smallest eigenvector of the Laplacian matrix of the given circuit [Ba82, Bo87, DoHo73, HaKa92, AlYa95], or a linear wirelength objective function, which is solved by an iterative method in [RiDJ94, LiLC95] The min cut based method uses the maximum flow algorithm to compute a series of minimum cuts in the given circuit in order to obtain an area balanced cut with small cut size [YaWo94] The ....

E. R. Barnes, "An Algorithm for Partitioning the Nodes of a Graph", SIAM J. Alg. Disc. Math., vol. 3, pp 541-550, 1982

....and Donath [DH73] who proved a lower bound on the size of the minimum bisection of a graph, and Fiedler [Fie73] Fie75] who explored the properties of 2 and its associated eigenvector for the Laplacian of a graph. There has been much subsequent work, including Barnes s partitioning algorithm [Bar82], Boppana s work that included a stronger lower bound on the minimum bisection size [Bop87] and the particular bisection and graph partitioning problems that we are considering in this paper [HK92] PSL90] Sim91] We note that spectral methods have not been limited to graph partitioning; work ....

Earl R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. Alg. Disc. Meth., 3(4):541-- 550, December 1982.

....then each subgraph is divided into two, and the process goes on recursively. Among them, eigenvalue computations [2] 17] or multilevel techniques [19] Many other heuristics have been designed to deal directly with the general p edge partition problem, such as linear programming transportation [3], greedy algorithms [12] 10] nodes interchanging methods [4] or geometrical approaches [22] 1] Note that some of these methods can be utilized jointly. Particularly, nodes interchanging methods require some initial partition. On the other hand, these methods are often used as a ....

....problem for large graphs: from the matrix point of view, i.e. using the Laplacian matrix [14] associated to the graph, or directly from the graph itself. The first point of view is based on spectral properties of the Laplacian matrix. Among the different works, let us mention those of Barnes [3], Simon [25] and Hendrickson and Leland [19] The other way is to work directly on the graph by applying greedy methods: the heuristic presented below is based on such a greedy algorithm. Let us mention that the first greedy algorithm for partitioning finite element meshes was proposed by Farhat ....

E.R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. Alg. Disc. Meth., vol 3, n o 4, pp 541-550, 1982.

....function is thus Max n b X k=1 nm X i=1 nm X j=1 a ij x ik x jk (1.2) s.t. i) Module placement constraints: n b X k=1 x ik = 1; 8 i = 1; 2; nm (ii) Block size constraints nm X i=1 x ik nm n b ; 8 k = 1; 2; n b x ik 2 f0; 1g; 1 i nm ; 1 k n b Barnes [3] developed a polynomial time heuristic for approximating the above 0 1 quadratic transportation problem. The heuristic is based on approximating the netlist or hypergraph by a weighted graph G, that tightly underestimates the number of cut nets in any netlist partition. The numerical optimization ....

....algorithm first finds the n b largest eigenvalues of the connection matrix A of the graph and their corresponding orthonormal eigenvectors u 1 ; u 2 ; un b . Let u ik be the ith component of the eigenvector corresponding to the jth largest eigenvalue of the adjacency matrix of G. Barnes [3] shows that 4 SHAWKI AREIBI AND ANTHONY VANNELLI the solution of the following linear transportation problem gives an approximate solution to the graph partitioning problem: Max nm X i=1 n b X k=1 u ik p m k x ik (1.3) s.t. n b X k=1 x ik = 1; 8 i = 1; 2; nm nm X i=1 x ....

E.R. Barnes, An Algorithm For Partitioning The Nodes Of A Graph, SIAM Journal Of Algebraic And Discrete Methods 3 (December 1982), no. 4, 541--550.

....but we do not pursue this any further. 3.3 Lower Bounds The upper bounding techniques find approximate solution matrices X which in general are not feasible because they are not integer. However, in [21] it was observed that X can also be used to obtain lower bounds on w(E uncut ) see also [4, 5]. maxftr X t Y : Y satisfies (2:3) 2:4)g (3.11) produces a partition Y that is closest to X in Frobenius norm. Alternatively, the problem maxftr X t AY : Y satisfies (2:3) 2:4)g (3.12) amounts to a linearization of the graph partitioning cost function at X . The optimal Y from this problem ....

E.R. BARNES. An algorithm for partitioning the nodes of a graph. SIAM J. Algebraic and Discrete Mathematics, 3:541--550, 1982.

....and Bound approaches to solve the partitioning problem in the case k = 2 and for general weighted graphs. Both methods seem to work only for extremely thin graphs ( average degree not more than 4) Several articles are devoted to finding good partitions using spectral information from A. In [11] the formulation (F) is used and a transportation problem is proposed to find a feasible X . The transportation costs are determined by the (pairwise orthogonal and normed) eigenvectors of A, corresponding to the k largest eigenvalues. The formulation (P) is used in [2] Therein A 0 is shifted by ....

....variable Y 2 n Thetak . minf GammatrX t Y : Y 2 Fg Since the sum of the elements of Y is n, note that the objective function is equivalent to tr( 1 2 u n u t k Gamma X) t Y: This latter function has an l 1 norm quality. We point out that this idea is also (implicitly) used by Barnes [11] to derive feasible solutions. Barnes uses the appropriately normalized eigenvectors corresponding to the largest eigenvalues of A for X . It is clear that the above model works for any X , as long as X t X = M . This approximation model has the disadvantage however, that the structure of the ....

E.R. BARNES. An algorithm for partitioning the nodes of a graph. SIAM J. Algebraic and Discrete Mathematics, 3:541--550, 1982.

....of the graph Laplacian matrix. A drawback of these methods is that they cannot be directly applied to hypergraphs, so a hyperedge model is needed to approximate the hypergraph by a graph. The relation between the spectrum of a graph and other graph properties has been an area of active research (Barnes, 1982; Boppana, 1987; Donath and Hoffman, 1973) but only recently spectrum based methods has been successfully applied to graph partitioning (Alpert and Kahng, 1993; Alpert and Yao, 1995; Hagen and Kahng, 1992; Hendrickson and Leland, 1992; Pothen et al. 1990; Simon, 1991) Michelena and Papalambros ....

Barnes, E., 1982, "An Algorithm for Partitioning the Nodes of a Graph," SIAM Journal on Algebraic Discrete Methods, Vol. 3, No. 4, pp. 541--550.

....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 [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 ....

E. R. Barnes. An algorithm for partitioning of nodes of a graph. SIAM J. on Algebraic and Discrete Methods, 4(3):541--550, 1982.

....= 1; Delta Delta Delta ; n 0 U i I for i = 1; Delta Delta Delta ; k: 5.13b) Barnes and Hoffman in [5] describe a method that uses eigenvectors associated with the k largest eigenvalues of the optimal matrix A Diagx to generate a partition of the nodes of the graph. See also Barnes [6, 7]. An important special case of the graph partitioning problem is the case when all m i s are equal. In that case the graph partitioning problem simplifies to: min (k=n)1 T x trace V s:t: V Diag x A V 0 max A ffl Y s:t: Y ii = k n for i = 1; Delta Delta Delta ; n 0 Y I: ....

E. R. Barnes, An algorithm for partitioning the nodes of a graph, SIAM J. Alg. and Disc. Meth., 3 (1982).

.... scheme [Kri84] and the simulated annealing based approach [KGV83, GS84] The analytical method uses a linear placement formulation with either (i) the quadratic wire length objective function solved by computing the second smallest eigenvector of the Laplacian matrix of the given circuit [Bar82, Bop87, DH73, HK92, AY95] or (ii) the linear wire length objective function solved by an iterative method [RDJ94, LLC95] The mincut based method uses the maximum flow algorithm to compute a series of minimum cuts in the given circuit in order to obtain an area balanced cut with the smallest ....

E. R. Barnes. "An algorithm for partitioning the nodes of a graph". SIAM Journal Alg. Disc. Math., pages 541--550, 1982.

....to intelligently enforce the balanced partition and the small edge cut requirements. A partition of Gm can be obtained using various algorithms such as (a) spectral bisection [47, 46, 2, 24] b) geometric bisection [37, 36] if coordinates are available 2 ) and (c) combinatorial methods [31, 3, 11, 12, 17, 5, 33, 21]. Since the size of the coarser graph Gm is small (i.e. V m 100) this step takes a small amount of time. We implemented four different algorithms for partitioning the coarse graph. The first algorithm uses the spectral bisection. The other three algorithms are combinatorial in nature, ....

Earl R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. Algebraic Discrete Methods, 3(4):541--550, December 1984.

....a circuit into portions that are implemented on separate physical components, such as printed circuit boards or chips. For the circuit partitioning problem, three different classes of algorithms were used [Arei93a] to generate good partitions. The techniques are, Numerical Optimization Methods [Barn82], Iterative Improvement algorithms [Arei93a] and Simulated Annealing [Whit84] The above mentioned heuristics have many disadvantages that limit their use. The numerical eigenvector approach by Barnes [Barn82] has a worst case time complexity of O(n 3 ) for multiblock partitions. Simulated ....

....to generate good partitions. The techniques are, Numerical Optimization Methods [Barn82] Iterative Improvement algorithms [Arei93a] and Simulated Annealing [Whit84] The above mentioned heuristics have many disadvantages that limit their use. The numerical eigenvector approach by Barnes [Barn82] has a worst case time complexity of O(n 3 ) for multiblock partitions. Simulated Annealing major disadvantage is the long computation time, on the order of hours or days for certain realistic problems such as VLSI placement and routing. The Interchange methods are the most successful ....

E.R. Barnes, "An Algorithm For Partitioning The Nodes Of A Graph," SIAM Journal Of Algebraic And Discrete Methods, vol. 3, No. 4, pp. 541--550, December 1982.

....code for each module v i according to the signs of the i th entries in a set of d eigenvectors. Then, modules with the same code are assigned to the same cluster. Despite its simplicity, this new clustering algorithm is strongly motivated by theoretical results for both spectral bipartitioning [6] and multi dimensional vector partitioning [4] The algorithm also has linear time complexity (not including the eigenvector computation) and is at least as effective as previous clustering algorithms in terms of two phase Fiduccia Mattheyses bipartitioning. 1. INTRODUCTION Clustering of netlist ....

....same orthant of the embedding, they are assigned to the same cluster. In contrast to complicated geometric clustering techniques, our algorithm is completely obvious and, in addition, has strong theoretical motivation. We show that this algorithm is a natural extension of spectral bipartitioning [6] and also follows the recent result of [4] which establishes the equivalence of min cut graph partitioning and an eigenvector based vector partitioning formulation. We have tested the quality of these clusterings via two phase FM; our experiments show that simple eigenvector based clustering ....

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E. R. Barnes, "An Algorithm for Partitioning the Nodes of a Graph," Siam J. Algorithms and Discrete Methods (3)4, 1992, pp. 541-549.

....and Donath [DH73] who proved a lower bound on the size of the minimum bisection of a graph, and Fiedler [Fie73] Fie75] who explored the properties of 2 and its associated eigenvector for the Laplacian of a graph. There has been much subsequent work, including Barnes s partitioning algorithm [Bar82], Boppana s work that included a stronger lower bound on the minimum bisection size [Bop87] and the particular bisection and graph partitioning problems that we are considering in this paper [HK92] PSL90] Sim91] We note that spectral methods have not been limited to graph partitioning; work ....

Earl R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. Alg. Disc. Meth., 3(4):541--550, December 1982.

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Earl R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. Alg. Disc. Meth., 3(4):541--550, December 1982. 27

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E. R. Barnes. An algorithm for partitioning of nodes of a graph. SIAM J. on Algebraic and Discrete Methods, 4(3):541--550, 1982.

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E.R. Barnes, An algorithm for partitioning the nodes of a graph, SIAM J. Alg. Dic. Math. 3 (1982) 541--549.

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E. R. Barnes. An algorithm for partitioning the nodes of a graph. SIAM J. ALG. DISC. METH., 3(4):541--50, 82.

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E. R. Barnes, An Algorithm for Partitioning the Nodes of a Graph, SIAM J. Alg. Disc. Meth., 3 (1982) 541.

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