Cuthill–McKee algorithm

In the mathematical subfield of matrix theory, the Cuthill–McKee algorithm (named for Elizabeth Cuthill and J. McKee) is an algorithm to permute a matrix into a banded form with a small bandwidth of sparse symmetric matrices. 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.

Algorithm

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 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.
• 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

• Graph bandwidth
• Sparse matrix

References

• Cuthill–McKee documentation for the Boost C++ Libraries.
• A detailed description of the Cuthill–McKee algorithm.