Cuthill–McKee algorithm

In numerical linear algebra, the Cuthill–McKee algorithm (CM), named after Elizabeth Cuthill and James McKee, 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 and Joseph Liu 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.

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 ordered according to predecessors and degree.