# RCM Reverse Cuthill McKee Ordering

RCM is a FORTRAN90 library which computes the Reverse Cuthill McKee ("RCM") ordering of the nodes of a graph.

The RCM ordering is frequently used when a matrix is to be generated whose rows and columns are numbered according to the numbering of the nodes. By an appropriate renumbering of the nodes, it is often possible to produce a matrix with a much smaller bandwidth.

The bandwidth of a matrix is computed as the maximum bandwidth of each row of the matrix. The bandwidth of a row of the matrix is essentially the number of matrix entries between the first and last nonzero entries in the row, with the proviso that the diagonal entry is always treated as though it were nonzero.

This library includes a few routines to handle the common case where the connectivity can be described in terms of a triangulation of the nodes, that is, a grouping of the nodes into sets of 3-node or 6-node triangles. The natural description of a triangulation is simply a listing of the nodes that make up each triangle. The library includes routines for determining the adjacency structure associated with a triangulation, and the test problems include examples of how the nodes in a triangulation can be relabeled with the RCM permutation.

Here is a simple example of how reordering can reduce the bandwidth. In our first picture, we have nine nodes:

```           5--7--6
|  | /
4--8--2
|  |  |
9--1--3
```
```        * . 1 . . . . 1 1
. * 1 . . 1 1 1 .
1 1 * . . . . . .
. . . * . . . 1 1
. . . . * . 1 1 .
. 1 . . . * 1 . .
. 1 . . 1 1 * . .
1 1 . 1 1 . . * .
1 . . 1 . . . . *
```
which has a disastrous bandwidth of 17 (lower and upper bandwidths of 8, and 1 for the diagonal.)

If we keep the same connectivity graph, but relabel the nodes, we could get a picture like this:

```           7--8--9
|  | /
3--5--6
|  |  |
1--2--4
```
```        * 1 1 . . . . . .
1 * . 1 1 . . . .
1 . * . 1 . . . .
. 1 . * . 1 . . .
. 1 1 . * 1 1 . .
. . . 1 1 * . 1 1
. . . . 1 . * 1 .
. . . . . 1 1 * 1
. . . . . 1 . 1 *
```
which has a bandwidth of 7 (lower and upper bandwidths of 3, and 1 for the diagonal.)

### Languages:

RCM is available in a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

MESH_BANDWIDTH, a FORTRAN90 program which returns the geometric bandwidth associated with a mesh of elements of any order and in a space of arbitrary dimension.

SPARSE_CC, a data directory which contains a description and examples of the CC format, ("compressed column") for storing a sparse matrix, including a way to write the matrix as a set of three files.

SPARSE_CR, a data directory which contains a description and examples of the CR format, ("compressed row") for storing a sparse matrix, including a way to write the matrix as a set of three files.

SPARSEPAK, a FORTRAN90 library which solves sparse linear systems using the Reverse Cuthill-McKee reordering scheme.

TET_MESH_RCM, a FORTRAN90 program which reads files describing a tetrahedral mesh of nodes in 3D, and applies the RCM algorithm to produce a renumbering of the tet mesh with a reduced bandwidth.

TRIANGULATION ORDER3, a directory which contains a description and examples of order 3 triangulations.

TRIANGULATION ORDER6, a directory which contains a description and examples of order 6 triangulations.

TRIANGULATION_RCM, a FORTRAN90 program which reads files describing a triangulation of nodes in 2D, and applies the RCM algorithm to produce a renumbering of the triangulation with a reduced bandwidth.

### Reference:

1. HL Crane, Norman Gibbs, William Poole, Paul Stockmeyer,
Algorithm 508: Matrix Bandwidth and Profile Reduction,
ACM Transactions on Mathematical Software,
Volume 2, Number 4, December 1976, pages 375-377.
2. Alan George, Joseph Liu,
Computer Solution of Large Sparse Positive Definite Matrices,
Prentice Hall, 1981,
ISBN: 0131652745,
LC: QA188.G46
3. Norman Gibbs,
Algorithm 509: A Hybrid Profile Reduction Algorithm,
ACM Transactions on Mathematical Software,
Volume 2, Number 4, December 1976, pages 378-387.
4. Norman Gibbs, William Poole, Paul Stockmeyer,
An Algorithm for Reducing the Bandwidth and Profile of a Sparse Matrix,
SIAM Journal on Numerical Analysis,
Volume 13, Number 2, April 1976, pages 236-250.
5. John Lewis,
Algorithm 582: The Gibbs-Poole-Stockmeyer and Gibbs-King Algorithms for Reordering Sparse Matrices,
ACM Transactions on Mathematical Software,
Volume 8, Number 2, June 1982, pages 190-194.

### List of Routines:

• DEGREE computes the degrees of the nodes in the connected component.
• GENRCM finds the reverse Cuthill-Mckee ordering for a general graph.
• GRAPH_01_LABEL returns the labels for graph 1.
• I4_SWAP swaps two I4's.
• I4_UNIFORM returns a scaled pseudorandom I4.
• I4MAT_PRINT_SOME prints some of an I4MAT.
• I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of the transpose of an I4MAT.
• I4ROW_COMPARE compares two rows of an I4ROW.
• I4ROW_SORT_A ascending sorts an I4ROW.
• I4ROW_SWAP swaps two rows of an I4ROW.
• I4VEC_HEAP_D reorders an I4VEC into an descending heap.
• I4VEC_INDICATOR sets an I4VEC to the vector A(I)=I.
• I4VEC_PRINT prints an I4VEC.
• I4VEC_REVERSE reverses the elements of an I4VEC.
• I4VEC_SORT_HEAP_A ascending sorts an I4VEC using heap sort.
• LEVEL_SET generates the connected level structure rooted at a given node.
• LEVEL_SET_PRINT prints level set information.
• PERM_CHECK checks that a vector represents a permutation.
• PERM_INVERSE produces the inverse of a given permutation.
• PERM_UNIFORM selects a random permutation of N objects.
• R82VEC_PERMUTE permutes an R82VEC in place.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• RCM renumbers a connected component by the reverse Cuthill McKee algorithm.
• ROOT_FIND finds a pseudo-peripheral node.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• TIMESTAMP prints the current YMDHMS date as a time stamp.