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[[File:can 73 cm.pdf|thumb|Cuthill-McKee ordering of the same matrix]]
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[[File:can 73 rcm.pdf|thumb|RCM ordering of the same matrix]]
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In the [[mathematics|mathematical]] subfield of [[Matrix (mathematics)|matrix theory]], the '''Cuthill–McKee algorithm''' ('''CM'''), named for Elizabeth Cuthill  and J. McKee
,<ref name="cm">E. Cuthill and J. McKee. [http://portal.acm.org/citation.cfm?id=805928''Reducing the bandwidth of sparse symmetric matrices''] In Proc. 24th Nat. Conf. [[Association for Computing Machinery|ACM]], pages 157–172, 1969.</ref> is an [[algorithm]] to permute a [[sparse matrix]] that has a [[symmetric matrix|symmetric]] sparsity pattern into a  [[band matrix]] form with a small [[bandwidth (matrix theory)|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 [[Sparse_matrix#Reducing_fill-in|fill-in]]  than the CM ordering when Gaussian elimination is applied.<ref name="gl">J. A.  George and J. W-H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, 1981</ref>
 
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 [[Level structure|levels]] <math>R_i</math> for <math>i=1, 2,..</math> until all nodes
are exhausted. The set <math> R_{i+1} </math> is created from set <math> R_i</math>
by listing all vertices adjacent to all nodes in <math> R_i </math>. These
nodes are listed in increasing degree. This last detail is the only difference
with the breadth-first search algorithm.
 
==Algorithm==
 
Given a symmetric <math>n\times n</math> matrix we visualize the matrix as the [[adjacency matrix]] of a [[graph (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]] ''R'' 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]]) ''x'' and set ''R'' := ({''x''}).
 
Then for <math>i = 1,2,\dots</math> we iterate the following steps while |''R''| < ''n''
 
*Construct the adjacency set <math>A_i</math> of <math>R_i</math> (with <math>R_i</math> the ''i''-th component of ''R'') and exclude the vertices we already have in ''R''
:<math>A_i := \operatorname{Adj}(R_i) \setminus R</math>
*Sort <math>A_i</math> with ascending vertex order ([[Degree (graph theory)|vertex degree]]).
*Append <math>A_i</math> to the Result set ''R''.
 
In other words, number the vertices according to a particular [[breadth-first search|breadth-first traversal]] where neighboring vertices are visited in order from lowest to highest vertex order.
 
==See also==
*[[Graph bandwidth]]
*[[Sparse matrix]]
 
==References==
<references />
* [http://www.boost.org/doc/libs/1_37_0/libs/graph/doc/cuthill_mckee_ordering.html Cuthill–McKee documentation] for the [[Boost C++ Libraries]].
* [http://ciprian-zavoianu.blogspot.com/2009/01/project-bandwidth-reduction.html A detailed description of the Cuthill–McKee algorithm].
* [http://www.mathworks.com/help/matlab/ref/symrcm.html symrcm] MATLAB's implementation of RCM.
 
{{DEFAULTSORT:Cuthill-McKee algorithm}}
[[Category:Matrix theory]]
[[Category:Graph algorithms]]
[[Category:Sparse matrices]]

Latest revision as of 01:18, 13 September 2014

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