# Cuthill–McKee algorithm

Cuthill-McKee ordering of a 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 ${\displaystyle R_{i}}$ 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.