Cuthill, E. and McKee, J. (1969), Reducing the bandwidth of sparse symmetric matrices, in: Proceedings of the ACM National Conference, Association for Computing Machinery, New York, NY, pp. 157--172. 4  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by reordering the data based on the order in which it is referenced in the loop. Consecutive packing (CPACK [6] and graph partitioning (Gpart [9] are two example data reordering transformations discussed in this paper. Other runtime data reordering transformations include Reverse Cuthill McKee [5, 2] and space filling curves [25, 17] A CPACK inspector traverses the data mappings for the loop with indirect memory references in lexicographical order of the loop iteration space. The first time the loop touches a piece of data, that data is packed into the next location for the new data ....

E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In Proc. 24th National Conf. ACM, pages 157--172, New York, 1969.

....has found a dynamic programming algorithm which can determine if a graph has bandwidth k in time O(n k 1 ) for any fixed value of k. Monien and Sudborough [8] showed how to reduce the time bound to O(n ) One of the most successful heuristic algorithms is one discovered by Cuthill and McKee [5] which is a member of a class Computer Science Department, Campus Box 1045, Washington University, St. Louis, Missouri 63130, UUCP: fihnp4,seismog wucs jst, CSNET:wucs jst seismo.ARPA of algorithms which are referred to here as the level algorithms. An algorithm is classified as a level ....

....of these gives a running time of O(n ) The procedure make mod levels can be implemented to run in O(jEj) O(n ) time, and the sorting steps in lines [6] and [7] require at most O(n log n) There are other possible strategies for arranging the vertices within each level. Cuthill and McKee [5], who first suggested the level algorithms, arranged the vertices within levels according to the order in which they were visited by a breadth first search algorithm. This results in an arbitrary ordering of the first level and arranges each vertex in subsequent levels based on the position of its ....

Cuthill, E., J. McKee. "Reducing the Bandwidth of Sparse Symmetric Matrices". In ACM National Conference Proceedings 24, 157-172, 1969.

....can obtain significantly smaller envelope sizes compared to other currently used algorithms. All previous envelope reduction algorithms (known to us) e.g. the reverse Cuthill McKee (RCM) algorithm, the Gibbs Poole Stockmeyer (GPS) algorithm, the Gibbs King (GK) algorithm, and the Sloan algorithm [3, 14, 15, 23, 36], are combinatorial in nature, employing some variant of breadth first search to compute the ordering. In contrast, the spectral algorithm is an algebraic algorithm whose good envelope reduction properties are somewhat intriguing and poorly understood. In this paper we attempt to provide a raison ....

....j )j jffi(V j )j= Delta; substituting the lower bound for jffi(V j )j, and summing this latter expression over all j, we obtain the lower bound on the envelope size. The upper bound is obtained by using the inequality c j (A) jffi(V j )j with the upper bound in Lemma 3.1. 2 Cuthill and McKee [3] proposed one of the earliest ordering algorithms for reducing the envelope size of a sparse matrix. George [12] discovered that reversing this ordering often leads to a significant reduction in envelope size and work. Since then the reverse CuthillMcKee (RCM) ordering has become one of the most ....

E. H. Cuthill and J. McKee, Reducing the bandwidth of sparse symmetric matrices, in Proceedings of the 24th Nat. Conf. ACM, ACM Publications, 1969, pp. 157--172.

....at run time if necessary. They perform a hybrid (static and dynamic) data dependence analysis inter procedurally. As we have described in this paper, traversing data dependences at run time is necessary for some run time reordering transformations. Many run time data reordering transformations [4, 2, 21, 7, 12] fit within our framework. Space filling curves and register tiling for sparse matrix vector multiply are two types of data reordering transformations that are more specialized. Data reorderings generated from space filling curves [28, 20] traverse data mappings and mappings of data to spatial ....

E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In Proceedings of the 24th National Conference ACM, pages 157--172, 1969.

....of 18 elements and drop from there. Both the P4 and EV6 exceed VIRAM performance for this reason. CRS banded uses the same format and algorithm as CRS, but reflects a different nonzero structure that would likely result from bandwidth reduction orderings, such as reverse Cuthill McKee (RCM) [7]. This has little effect on IRAM, but improves the cache hit rate on some of the other machines. The Ellpack (or Itpack) format [13] forces all rows to have the same length by padding them with zeros. It still has indexed memory operations, but increases available data parallelism through ....

E.Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. Proc. ACM Natl. Conf., 1969, 157-192.

....diagonal dominance is needed. It is usually better to err on the side of being too well conditioned. While this will slow convergence somewhat, the linear solves are much more robust. To help reduce the effect of entries dropped from the preconditioner, reverse Cuthill McKee reordering is used [6]. Good reordering is especially important in the multiblock case, due to the increased number of far off diagonal entries resulting from the block boundaries. The GMRES algorithm only requires matrixvector multiplies, and does not explicitly require the matrix, except in forming the ....

Cuthill E. H and McKee J. M. Reducing the bandwidth of sparse symmetric matrices. Proc Proceedings of the 24th National Conference of the Association for Computing Machinery, pp 157--172. Brondon Press, 1969.

....The bandwidth of G is the minimum bandwidth over all layouts of G, namely, bw(G) minfbwL (G) j L is a layout of Gg. The BANDWIDTH problem is to decide for a given graph G and integer k, if bw(G) k. This problem has been studied intensely because of its application to sparse matrix algebra [9]. It is known to be NP complete even for binary trees [16] and for caterpillars with hair length at most three [34] On the other hand, it is solvable in O(n ) for arbitrary k [23] and in linear time for k = 2 [16] Sandwich Problems: Given two graphs G ) such that G a supergraph ....

E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In Proc. 24th Nat. Conf. ACM, pages 157--172, 1969.

.... adjacent, ii) the removal of all V I from G disconnects the graph into multiple subgraphs G I , and (iii) any resulting subgraph G I is adjacent to at most two separators (V I 1 and V I ) The traditional approach to nding a one way dissection is to rst perform a Reverse Cuthill McKee ordering [1] to nd a dominant axis for the graph, then perform a levelset ordering on this axis, and then to choose a set of levels to remove. Each removed level becomes a vertex separator V I . Given this one way dissection, we can then label the subgraphs G I with color 1, the separators V I with color 2, ....

E. Cuthill and J. McKee, Reducing the bandwidth of sparse symmetric matrices, in Proceedings of the 24th National Conference of the ACM, New York, 1969.

....order elements in an unstructured mesh to improve locality. This reordering technique was developed by George [26] for a different purpose: bandwidth and profile minimization of sparse matrices. George s strategy was a refinement of a breadth first ordering technique developed by Cuthill and McKee [27]. The Cuthill McKee and Reverse Cuthill McKee orderings use an adjacency list representation of an undirected graph and renumber graph nodes using a breadth first traversal in which all unnumbered neighbors of a node x are added to a FIFO queue of nodes to be numbered by order of increasing ....

E. Cuthill and J. McKee, "Reducing the Bandwidth of Sparse Symmetric Matrices," Proc. ACM National Conference, Association of Computing Machinery, (1969).

....essentially a simplified minimum degree code. Since our experiments use matrices whose graphs are regular meshes in 2 and 3 dimensions, we also run incomplete Cholesky with the natural ordering of the mesh. We expect that for unstructured meshes, envelope minimizing orderings such as Cuthill McKee [9] or Sloan [19] would produce results similar to natural orderings of regular meshes [11] We used both the natural row by row ordering of the mesh and METIS s ordering for the incomplete factorization. We have found, however, that the modified factorization always break down when the matrix is ....

E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In Proceedings of the 24th National Conference of the Association for Computing Machinery, pages 157--172, 1969.

....our experiments use matrices whose graphs are regular meshes in 2 and 3 dimensions, we also run IC and MIC with the natural ordering of the mesh. Unrelaxed MIC breaks down with METIS and GENMMD orderings. We expect that for unstructured meshes, envelope minimizing orderings such as Cuthill McKee [10] or Sloan [18] would produce results similar to natural orderings of regular meshes [12] The experiments use two sparse Cholesky factorization algorithms that we have implemented. One code is a supernodal multifrontal sparse Cholesky code [11, 19] 8 CHEN AND TOLEDO This code can only perform ....

E. Cuthill and J. McKee, Reducing the bandwidth of sparse symmetric matrices, in Proceedings of the 24th National Conference of the Association for Computing Machinery, 1969, pp. 157--172.

....linear layouts of graphs. Pathwidth is a graph parameter closely associated with interval graphs, of importance to both algorithmic and structural graph theory, and with applications to VLSI layout [6, 11] Bandwidth is another widely studied graph parameter, with applications to sparse matrices [1], and notorious for the difficulty of its computation even for trees [3] Recently, in a study of problems motivated by molecular biology, Kaplan and Shamir [5] showed a somewhat surprising connection between bandwidth and pathwidth, using the well known notion of proper interval graphs, or ....

E. Cuthill and J. McKee, Reducing the bandwidth of sparse symmetric matrices, Proc. Nat. Conf. ACM 157-172, 1969.

....in [8] In Table 9, we display the number of iterations with SQMR, selecting the same density parameters as those used for the experiments reported in Table 9, but using di#erent orderings to permute the original pattern of MSym Frob . More precisely we consider the reverse Cuthil MacKee ordering [13] (RCM) the minimum degree [18, 31] ordering (MD) the spectral nested dissection ordering [28] SND) and lastly we reorder the matrix by putting the denser rows and columns first (DF) It can be seen that MSym Frob is not too sensitive to the ordering and none of the tested orderings appears ....

E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In Proceedings 24th National Conference of the Association for Computing Machinery, Brandon Press, New Jersey, pages 157--172. Brandon Press, New Jersey, 1969.

....proposed by Barnard, Pothen and Simon [3] which derives a reordering of the matrix by computing an eigenvector of a discrete Laplacian matrix associated with the given matrix. This method is therefore deeply di erent from the other reordering schemes such as the reverse Cuthill McKee (RCM [5]) Gibbs Pole Stockmayer (GPS[13] 14] Gibbs King (GK [18] and the more recent WBRA [9] 10] The Spectral Method is especially designed to reduce the work bound of the matrix and the bandwidth of the reordered matrix can be rather Dipartimento di Informatica, Universit a di Pisa, Corso Italia ....

....= 10 6 and an initial band of 5 10 5 has been added to the list. The pattern of TREE is shown in Figure 8. In order to acquire reference values of bw and wb we applied to the test matrices the original Spectral Method (SM in the following) and the well known bandwidth reduction algorithm RCM [5]. 13 The computation of these quantities has been carried out with MATLAB using the standard sparse matrix eigenvalue routines to get the permutation returned by SM and the routine symrcm which implements RCM. The SM method could not be applied to reducible matrices and to the 6 larger ones. In ....

E. H. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In Proc. 24th Nat. Conf. Assoc. Comp. Mach., pages 157-172. ACM Publications, 1969.

....when the rows and columns of M are ordered so that the profile of the associated G is reduced. A number of heuristic algorithms for bandwidth and profile reduction are available. The reverse Cuthill McKee (RCM) algorithm, a modification by George [8] of the algorithm defined by Cuthill and McKee [4], is perhaps the most widely used. This algorithm was originally developed for bandwidth reduction. The Gibbs Poole Stockmeyer (GPS) algorithm [10] was developed to address both bandwidth and profile reduction, whereas profile reduction is the primary goal of the Gibbs King (GK) 9] and Snay [18] ....

....approach and then offering insights into the ordering process based on examples. Assume the two step generalstrategy described in Section 1 is used. A variety of methods can be used for selecting a starting node in Step 1. For now assume the use of a simple strategy suggested by Cuthill and McKee [4], namely to select a node with minimum degree. The degree of node v is denoted as d(v) defined as the number of nodes adjacent to v. Node u is adjacent to v iff uvE. Other alternatives for starting node selection are considered in Section 4. FIM is one possible strategy for node placement in ....

E. CUTHILL AND J. MCKEE, Reducing the bandwidth of sparse symmetric matrices, in ACM National Conference Proceedings, 24, 1969, pp. 157-172.

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Cuthill, E. and McKee, J. (1969), Reducing the bandwidth of sparse symmetric matrices, in: Proceedings of the ACM National Conference, Association for Computing Machinery, New York, NY, pp. 157--172. 4

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E. Cuthill and J. McKee, \Reducing the Bandwidth of Sparse Symmetric Matrices", ACM National Conference Proceedings 24 (1969) 157-172.

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E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In In Proc. 24th Nat. Conf. ACM, pages 157--172, 1969.

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E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In In Proc. 24th Nat. Conf. ACM, pages 157--172, 1969.

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E. Cuthill and J. McKee, Reducing the bandwidth of sparse symmetric matrices, in Proceedings of ACM National Conference, Association of Computing Machinery, New York, 1969.

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E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In Proc. 24th Nat. Conf. of the ACM, pages 157--172, 1969. 24

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E. Cuthill and J. McKee, Reducing the band width of sparse symmetric matrices, in Proc. ACM Nat. Conference, 1969, pp. 157--172.

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E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. Proceedings of the 24 th National Conference of the ACM, pages 157--172, 1969.

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Cuthill, E., and McKee, J.,"Reducing the Band Width of Sparse Symmetric Matrices ", Proc. ACM Nat. Conf., 1969, pp. 157-172.

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E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices. In Proceedings of the 24th National Conference of the Association for Computing Machinery, pages 157#172, New Jersey, 1969. Brandon Press.

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