Cuthill–McKee algorithm

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]

Cuthill-McKee ordering of a matrix

RCM ordering of the same matrix

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 ${\displaystyle R_{i}}$ for ${\displaystyle i=1,2,..}$ until all nodes are exhausted. The set ${\displaystyle R_{i+1}}$ is created from set ${\displaystyle R_{i}}$ by listing all vertices adjacent to all nodes in ${\displaystyle R_{i}}$. These nodes are listed in increasing degree. This last detail is the only difference with the breadth-first search algorithm.

Algorithm

Given a symmetric ${\displaystyle n\times n}$  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 ${\displaystyle i=1,2,\dots }$  we iterate the following steps while |R| < n

• Construct the adjacency set ${\displaystyle A_{i}}$  of ${\displaystyle R_{i}}$  (with ${\displaystyle R_{i}}$  the i-th component of R) and exclude the vertices we already have in R

${\displaystyle A_{i}:=\operatorname {Adj} (R_{i})\setminus R}$ 

• Sort ${\displaystyle A_{i}}$  with ascending vertex order (vertex degree).
• Append ${\displaystyle A_{i}}$  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

• Graph bandwidth
• Sparse matrix

References

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

• Cuthill–McKee documentation for the Boost C++ Libraries.
• A detailed description of the Cuthill–McKee algorithm.
• symrcm MATLAB's implementation of RCM.
• reverse_cuthill_mckee RCM routine from SciPy written in Cython.