Editing Cuthill–McKee algorithm (section)

==Algorithm==

Given a symmetric <math>n\times n</math> matrix we visualize the matrix as the [[adjacency matrix]] of a [[Graph (discrete mathematics)|graph]]. The Cuthill–McKee algorithm is then a relabeling of the [[vertex (graph theory)|vertices]] of the graph to reduce the bandwidth of the adjacency matrix.

The algorithm produces an ordered [[n-tuple|''n''-tuple]] <math>R</math> of vertices which is the new order of the vertices.

First we choose a [[peripheral vertex]] (the vertex with the lowest [[Degree (graph theory)|degree]]) <math>x</math> and set <math>R := ( \{ x \})</math>.

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

*Construct the adjacency set <math>A_i</math> of <math>R_i</math> (with <math>R_i</math> the ''i''-th component of <math>R</math>) and exclude the vertices we already have in <math>R</math>
:<math>A_i := \operatorname{Adj}(R_i) \setminus R</math>
*Sort <math>A_i</math> ascending by minimum predecessor (the already-visited neighbor with the earliest position in R), and as a tiebreak ascending by [[Degree (graph theory)|vertex degree]].<ref>The Reverse Cuthill-McKee Algorithm in Distributed-Memory [https://archive.siam.org/meetings/csc16/slides/ariful_azad_CSC16.pdf], slide 8, 2016</ref>
*Append <math>A_i</math> to the Result set <math>R</math>.

In other words, number the vertices according to a particular [[level structure]] (computed by [[breadth-first search]]) where the vertices in each level are visited in order of their predecessor's numbering from lowest to highest. Where the predecessors are the same, vertices are distinguished by degree (again ordered from lowest to highest).