Cuthill–McKee algorithm

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Cuthill-McKee ordering of the same matrix
RCM ordering of the same matrix

In the mathematical subfield of matrix theory, the Cuthill–McKee algorithm (CM), named for Elizabeth Cuthill and J. McKee ,[1] is an algorithm to permute a sparse matrix that has a symmetric sparsity pattern into a band matrix form with a small bandwidth. The reverse Cuthill–McKee algorithm (RCM) due to Alan George is the same algorithm but with the resulting index numbers reversed. In practice this generally results in less fill-in than the CM ordering when Gaussian elimination is applied.[2]

The Cuthill McKee algorithm is a variant of the standard breadth-first search algorithm used in graph algorithms. It starts with a peripheral node and then generates levels R_i for i=1, 2,.. until all nodes are exhausted. The set  R_{i+1} is created from set  R_i by listing all vertices adjacent to all nodes in  R_i . These nodes are listed in increasing degree. This last detail is the only difference with the breadth-first search algorithm.

Algorithm[edit]

Given a symmetric n\times n matrix we visualize the matrix as the adjacency matrix of a graph. The Cuthill–McKee algorithm is then a relabeling of the vertices of the graph to reduce the bandwidth of the adjacency matrix.

The algorithm produces an ordered n-tuple R of vertices which is the new order of the vertices.

First we choose a peripheral vertex (the vertex with the lowest degree) x and set R := ({x}).

Then for i = 1,2,\dots we iterate the following steps while |R| < n

  • Construct the adjacency set A_i of R_i (with R_i the i-th component of R) and exclude the vertices we already have in R
A_i := \operatorname{Adj}(R_i) \setminus R
  • Sort A_i with ascending vertex order (vertex degree).
  • Append A_i to the Result set R.

In other words, number the vertices according to a particular breadth-first traversal where neighboring vertices are visited in order from lowest to highest vertex order.

See also[edit]

References[edit]

  1. ^ E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices In Proc. 24th Nat. Conf. ACM, pages 157–172, 1969.
  2. ^ J. A. George and J. W-H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, 1981