TRIANGULATION_RCM
Reverse Cuthill-McKee Node Reordering

TRIANGULATION_RCM is a C++ program which computes the reverse Cuthill-McKee (RCM) reordering for nodes in a triangulation composed of 3-node or 6-node triangles.

The user supplies a node file and a triangle file, containing the coordinates of the nodes, and the indices of the nodes that make up each triangle. Either 3-node or 6-node triangles may be used.

The program reads the data, computes the adjacency information, carries out the RCM algorithm to get the permutation, applies the permutation to the nodes and triangles, and writes out new node and triangle files that correspond to the RCM permutation.

Note that the node file would normally contain exactly 2 values on each line, namely the X and Y coordinates of the nodes. However, this is not necessary. Extra information can be included on each line, for instance, a "Z" coordinate. Each line should include the same number of items, but all will be permuted correctly together. The program does not actually need to know the coordinates of the nodes, so in fact, ANY data (as long as it is real numeric data) associated with the nodes can be listed in the node file, and will be correctly permuted.

Usage:

triangulation_rcm prefix

• prefix_nodes.txt contains the node coordinates,
• prefix_elements.txt contains the element definitions.
• prefix_rcm_nodes.txt will contain the RCM node coordinates,
• prefix_rcm_elements.txt will contain the RCM element definitions.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

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

QUAD_MESH_RCM, a C++ program which computes the reverse Cuthill-McKee (RCM) reordering for nodes in a mesh of 4-node quadrilaterals.

RCM, a C++ library which carries out reverse Cuthill-McKee computations.

TABLE_DELAUNAY, a FORTRAN90 program which computes the triangulation of a set of nodes whose coordinates are stored in a file.

TET_MESH_RCM, a FORTRAN90 library which applies the reverse Cuthill-McKee reordering to a tetrahedral mesh of nodes in 3D.

TRIANGLE, a C program which computes a triangulation of a geometric region.

TRIANGULATION, a C++ library which carries out various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.

TRIANGULATION_BOUNDARY_NODES, a C++ program which reads data defining a triangulation, determines which nodes lie on the boundary, and writes their coordinates to a file.

TRIANGULATION_CORNER, a C++ program which patches triangulations so that no triangle has two sides on the boundary.

TRIANGULATION_DELAUNAY_DISCREPANCY, a C++ program which measures the amount by which a triangulation fails the local Delaunay test;

TRIANGULATION_DISPLAY_OPENGL, a C++ program which reads files defining a triangulation and displays an image using Open GL.

TRIANGULATION_HISTOGRAM, a C++ program which computes histograms of data over a triangulation.

TRIANGULATION_L2Q, a C++ program which reads data defining a 3-node triangulation and generates midside nodes and writes out the corresponding 6-node triangulation.

TRIANGULATION_MASK, a C++ program which takes an existing triangulation and deletes triangles and their corresponding nodes as requested by the user.

TRIANGULATION_NODE_TO_ELEMENT, a C++ program which reads files describing a set of nodes, their triangulation, and the value of one or more quantities at each node, and outputs a file that averages the quantities for each element. This operation in effect creates an "order1" finite element model of the data.

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_ORIENT, a C++ program which reads data defining a triangulation, makes sure that every triangle has positive orientation, and if not, writes a corrected triangle file.

TRIANGULATION_PLOT, a C++ program which reads data defining a triangulation and creates a PostScript image of the nodes and triangles.

TRIANGULATION_Q2L, a C++ program which reads data defining a 6-node triangulation, and subdivides each triangle into 4 3-node triangles, writing the resulting triangulation to a file.

TRIANGULATION_QUAD, a C++ program which estimates the integral of a function over a triangulated region.

TRIANGULATION_QUALITY, a C++ program which reads data defining a triangulation and computes a number of quality measures.

TRIANGULATION_REFINE, a C++ program which reads data defining a triangulation, replaces each triangle by four congruent smaller ones, and writes the new triangulation information to a file.

TRIANGULATION_TRIANGLE_NEIGHBORS, a C++ program which reads data defining a triangulation, determines the neighboring triangles of each triangle, and writes that information to a file.

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. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
   Computational Geometry,
   Springer, 2000,
   ISBN: 3-540-65620-0.
3. Alan George, Joseph Liu,
   Computer Solution of Large Sparse Positive Definite Matrices,
   Prentice Hall, 1981,
   ISBN: 0131652745,
   LC: QA188.G46
4. Norman Gibbs,
   Algorithm 509: A Hybrid Profile Reduction Algorithm,
   ACM Transactions on Mathematical Software,
   Volume 2, Number 4, December 1976, pages 378-387.
5. 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, 1976, pages 236-250.
6. Joseph ORourke,
   Computational Geometry,
   Second Edition,
   Cambridge, 1998,
   ISBN: 0521649765,
   LC: QA448.D38.

Source Code:

• triangulation_rcm.cpp, the source code.

Examples and Tests:

TEST3 works with a 3-node triangulation:

• test3_output.txt, the printed output from the command
  "triangulation_rcm test3"
• test3_nodes.txt, the node file.
• test3_nodes.png, a PNG image of the nodes created by TRIANGULATION_PLOT.
• test3_elements.txt, the triangle file.
• test3_elements.png, an image of the triangles before the reordering.
• test3_rcm_nodes.txt, the RCM reordered node file.
• test3_rcm_nodes.png, a PNG image of the reordered nodes created by TRIANGULATION_PLOT.
• test3_rcm_elements.txt, the RCM reordered triangle file.
• test3_rcm_elements.png, an image of the triangles after the reordering.

TEST6 works with a 6-node triangulation:

• test6_output.txt, the printed output from the command
  "triangulation_rcm test6"
• test6_nodes.txt, the node file.
• test6_nodes.png, a PNG image of the nodes created by TRIANGULATION_PLOT.
• test6_elements.txt, the triangle file.
• test6_elements.png, an image of the triangles before the reordering.
• test6_rcm_nodes.txt, the RCM reordered node file.
• test6_rcm_nodes.png, a PNG image of the reordered nodes created by TRIANGULATION_PLOT.
• test6_rcm_elements.txt, the RCM reordered triangle file.
• test6_rcm_elements.png, an image of the triangles after the reordering.

List of Routines:

• MAIN is the main program for TRIANGULATION_RCM.
• ADJ_BANDWIDTH computes the bandwidth of an adjacency matrix.
• ADJ_PERM_BANDWIDTH computes the bandwidth of a permuted adjacency matrix.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DEGREE computes the degrees of the nodes in the connected component.
• FILE_COLUMN_COUNT counts the columns in the first line of a file.
• FILE_EXIST reports whether a file exists.
• FILE_ROW_COUNT counts the number of row records in a file.
• GENRCM finds the reverse Cuthill-Mckee ordering for a general graph.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• I4_MODP returns the nonnegative remainder of integer division.
• I4_SWAP switches two I4's.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• I4MAT_DATA_READ reads data from an I4MAT file.
• I4MAT_HEADER_READ reads the header from an I4MAT file.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
• I4MAT_WRITE writes an I4MAT file with no header.
• I4ROW_COMPARE compares two rows of an I4ROW.
• I4ROW_SORT_A ascending sorts the rows of an I4ROW.
• I4ROW_SWAP swaps two rows of an I4ROW.
• I4VEC_HEAP_D reorders an array of integers into a descending heap.
• I4VEC_REVERSE reverses the elements of an integer vector.
• I4VEC_SORT_HEAP_A ascending sorts an array of integers using heap sort.
• LEVEL_SET generates the connected level structure rooted at a given node.
• PERM_CHECK checks that a vector represents a permutation.
• PERM_INVERSE3 produces the inverse of a given permutation.
• R8COL_PERMUTE permutes an R8COL in place.
• R8MAT_DATA_READ reads the data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8MAT_WRITE writes an R8MAT file with no header.
• RCM renumbers a connected component by the reverse Cuthill McKee algorithm.
• ROOT_FIND finds a pseudo-peripheral node.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• S_TO_I4 reads an I4 from a string.
• S_TO_I4VEC reads an I4VEC from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGULATION_ORDER3_ADJ_COUNT counts adjacencies in a triangulation.
• TRIANGULATION_ORDER3_ADJ_SET sets adjacencies in a triangulation.
• TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES determines triangle neighbors.
• TRIANGULATION_ORDER6_ADJ_COUNT counts adjacencies in a triangulation.
• TRIANGULATION_ORDER6_ADJ_SET sets adjacencies in a triangulation.
• TRIANGULATION_ORDER6_NEIGHBOR_TRIANGLES determines triangle neighbors.

You can go up one level to the C++ source codes.