Average Bandwidth Reduction in Sparse Matrices Using Hybrid Heuristics

This paper proposes a hybrid heuristic aiming the reduction of the average bandwidth of sparse matrices. The proposed heuristic combines a greedy selection of rows and columns to be interchanged with an approach based on a genetic-inspired crossover. Several ideas of limiting the search tree are also investigated. Preliminary numerical experiments illustrate the fact that the proposed heuristic leads to better results with respect to the average bandwidth than the classical Cuthill-McKee algorithm. 1. The Matrix Bandwidth Reduction Problem Reducing the bandwidth of sparse matrices is important in solving large linear systems of equations and is especially useful in designing efficient parallel implementation of the solving procedures. The bandwidth of a matrix is related to the concentration of nonzero elements around the main diagonal and can be measured in different ways. The most frequently used bandwidth measure is the maximum of the absolute value of the difference between the row and column indices of nonzero elements, i.e. bw = max{|i−j|; aij 6= 0, i = 1, n, j = 1, n} for a square matrix A of size n. However, the bw measure does not provide too much information about the grouping pattern of non-zero elements around the main diagonal, which is particularly important in balancing the load on different processors in the case of parallel implementations [1]. This motivates the interest in defining and using other measures of the compactness of the non-zero elements around the main diagonal. In this work we use the average bandwidth proposed in [8] and defined by Eq. (1), where m Received by the editors: April 10, 2011. 2010 Mathematics Subject Classification. 68T20, 65F30. 1998 CR Categories and Descriptors. I.2.8 [Artificial Intelligence]: Problem Solving, Control Methods, and Search – Heuristic Methods; G.1.3 [Numerical Analysis]: Numerical Linear Algebra – Sparse, structured, and very large systems (direct and iterative methods).