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 ....

[Article contains additional citation context not shown here]

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) ....

[Article contains additional citation context not shown here]

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 ....

[Article contains additional citation context not shown here]

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.

No context found.

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

No context found.

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

No context found.

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

No context found.

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

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC