A spectral algorithm for envelope reduction of sparse matrices

Barnard, Stephen T.; Pothen, Alex; Simon, Horst D.

doi:10.1145/169627.169790

Cited by 91 publications

(123 citation statements)

References 27 publications

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“…Indeed, the results reported in [4] and [15] indicate that RCMM is also superior to the well-known GPS [7] algorithm. In addition, we compared our method with the spectral analysis approach described in [1], summarised in Sect. 1 and used to set up the variable X in Sect.…”

Section: Resultsmentioning

confidence: 99%

“…In addition, the values for X, Y and Z for each fitness case are carefully chosen during the initialisation of the system in such a way as to maximise the chance that they represent good permutations. In particular, following the ideas of [1] (see Sect. 1) the X values were set to be the components of the eigenvector associated with the first non-zero eigenvalue of the Laplacian matrix associated with the matrix to be optimised.…”

Section: The Generation Of Permutationsmentioning

confidence: 99%

“…However, the algorithm is significantly more complex to implement. More recently, Barnard et al [1] have proposed the use of spectral analysis of the Laplacian matrix associated with the graph representing the nonzero elements in a sparse matrix as an effective method for the reduction of the envelope of a graph. In particular, the method permutes a matrix based on the eigenvector associated with the first non-zero eigenvalue of the Laplacian matrix.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, a Genetic Programming (GP) [11,20] approach is proposed and tested against a high-performance version of the Cuthill and McKee algorithm [3] and a version of the spectral analysis proposed by Barnard et al [1]. To the best of our knowledge, no prior attempt to use GP to solve BMP has been reported in the literature.…”

Section: Introductionmentioning

confidence: 99%

“…Instead, we artificially increase the chance of selecting the variables X, Y and Z ensuring that every tree contains at least one variable. 1 Tournament selection was used to choose individual program(s) from the population based on fitness to participate in genetic operations. New individual programs were created by applying the genetic operations of reproduction, sub-tree crossover and point mutation with specified probabilities.…”

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confidence: 99%

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A Genetic Programming Approach to the Matrix Bandwidth-Minimization Problem

Koohestani

Poli

2010

Parallel Problem Solving From Nature, PPSN XI

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Abstract. The bandwidth of a sparse matrix is the distance from the main diagonal beyond which all elements of the matrix are zero. The bandwidth minimisation problem for a matrix consists of finding the permutation of rows and columns of the matrix which ensures that the non-zero elements are located in as narrow a band as possible along the main diagonal. This problem, which is known to be NP-complete, can also be formulated as a vertex labelling problem for a graph whose edges represent the non-zero elements of the matrix. In this paper, a Genetic Programming approach is proposed and tested against two of the best-known and widely used bandwidth reduction algorithms. Results have been extremely encouraging.Keywords: Bandwidth Minimization Problem; Genetic Programming; Graph Labelling; Sparse Matrices; Combinatorial Optimisation. BackgroundThe Bandwidth Minimization Problem (BMP) is a very well-known problem, familiar to applied mathematicians and arising in many applications in science and engineering [18]. BMP consists of finding the permutation of rows and columns of a matrix which ensures that the non-zero elements are located in as narrow a band as possible along the main diagonal. One of the most common applications of bandwidth-minimisation algorithms arises from the need to efficiently solve large systems of equations [19]. In such a scenario, more efficient solutions are obtained if the rows and columns of the matrix representing the set of equations can be permuted in such a way that the bandwidth of the matrix is minimized [19]. BMP has also connections with a wide range of other problems, including: finite element analysis of mechanical systems, large scale power transmission systems, circuit design, VLSI design, data storage, chemical kinetics, network survivability, numerical geophysics, industrial electromagnetics, saving large hypertext media and topology compression of road networks.The BMP is NP-complete [18] and, hence, it is highly unlikely that there exists an algorithm which finds the minimum bandwidth of a matrix in polynomial time. It has also been proved that the BMP is NP-complete even for trees with

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Section: Resultsmentioning

confidence: 99%

Section: The Generation Of Permutationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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A Genetic Programming Approach to the Matrix Bandwidth-Minimization Problem

Koohestani

Poli

2010

Parallel Problem Solving From Nature, PPSN XI

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Ordering symmetric sparse matrices for small profile and wavefront

Reid

Scott

1999

Int. J. Numer. Meth. Engng.

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The ordering of large sparse symmetric matrices for small pro"le and wavefront or for small bandwidth is important for the e$ciency of frontal and variable-band solvers. In this paper, we look at the computation of pseudoperipheral nodes and compare the e!ectiveness of using an algorithm based on level-set structures with using the spectral method as the basis of the Reverse Cuthill}McKee algorithm for bandwidth reduction. We also consider a number of ways of improving the performance and e$ciency of Sloan's algorithm for pro"le and wavefront reduction, including the use of di!erent weights, the use of supervariables, and implementing the priority queue as a binary heap. We also examine the use of the spectral ordering in combination with Sloan's algorithm. The design of software to implement the reverse Cuthill}McKee algorithm and a modi"ed Sloan's algorithm is discussed. Extensive numerical experiments that justify our choice of algorithm are reported on.

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