TRIANGULATION_RCM
Reverse Cuthill-McKee Node Reordering

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

MESH_TO_XML, a FORTRAN90 program which reads information defining a 1D, 2D or 3D mesh, namely a file of node coordinates and a file of elements defined by node indices, and creates a corresponding XML file for input to DOLFIN or FENICS.

QUAD_MESH_RCM, a FORTRAN90 program which computes the reverse Cuthill-McKee (RCM) reordering for nodes in a mesh of 4-node quadrilaterals.

RCM, a FORTRAN90 library which carries out reverse Cuthill-McKee computations.

TABLE_DELAUNAY, a FORTRAN90 program which triangulates 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 FORTRAN90 library which carries out various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.

TRIANGULATION_BOUNDARY_NODES, a FORTRAN90 program which reads data defining a triangulation, determines which nodes lie on the boundary, and writes their coordinates to a file.

TRIANGULATION_CORNER, a FORTRAN90 program which patches triangulations so that no triangle has two sides on the boundary.

TRIANGULATION_DELAUNAY_DISCREPANCY, a FORTRAN90 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 FORTRAN90 program which computes histograms of data over a triangulation.

TRIANGULATION_L2Q, a FORTRAN90 program which reads data defining a 3-node triangulation and generates midside nodes and writes out the corresponding 6-node triangulation.

TRIANGULATION_MASK, a FORTRAN90 program which takes an existing triangulation and deletes triangles and their corresponding nodes as requested by the user.

TRIANGULATION_ORDER3, a data directory which contains a description and examples of order 3 triangulations.

TRIANGULATION_ORDER6, a data directory which contains a description and examples of order 6 triangulations.

TRIANGULATION_ORIENT, a FORTRAN90 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 FORTRAN90 program which reads data defining a triangulation and creates a PostScript image of the nodes and triangles.

TRIANGULATION_Q2L, a FORTRAN90 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 FORTRAN90 program which estimates the integral of a function over a triangulated region.

TRIANGULATION_QUALITY, a FORTRAN90 program which reads data defining a triangulation and computes a number of quality measures.

TRIANGULATION_REFINE, a FORTRAN90 program which can be used to refine a triangulation.

TRIANGULATION_TRIANGLE_NEIGHBORS, a FORTRAN90 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.f90, 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 a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DEGREE computes the degrees of the nodes in the connected component.
• DTABLE_DATA_READ reads data from a double precision table file.
• DTABLE_DATA_WRITE writes data to a double precision table file.
• DTABLE_HEADER_READ reads the header from a double precision table file.
• DTABLE_HEADER_WRITE writes the header to a double precision table file.
• DTABLE_WRITE writes a double precision table file.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_NAME_EXT_GET determines the "extension" of a file name.
• FILE_NAME_EXT_SWAP replaces the current "extension" of a file name.
• FILE_ROW_COUNT counts the number of row records in a file.
• GENRCM finds the reverse Cuthill-Mckee ordering for a general graph.
• GET_UNIT returns a free FORTRAN unit number.
• I4_SWAP swaps two integer values.
• 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 the rows of an I4ROW.
• I4ROW_SWAP swaps two rows of an I4ROW.
• I4VEC_REVERSE reverses the elements of an integer vector.
• ITABLE_DATA_READ reads data from an integer table file.
• ITABLE_DATA_WRITE writes data to an integer table file.
• ITABLE_HEADER_READ reads the header from an integer table file.
• ITABLE_HEADER_WRITE writes the header to an integer table file.
• ITABLE_WRITE writes an integer table file.
• LEVEL_SET generates the connected level structure rooted at a given node.
• PERM_CHECK checks that a vector represents a permutation.
• PERM_INVERSE produces the inverse of a given permutation.
• R8COL_PERMUTE permutes an R8COL in place.
• 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.
• S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
• S_INDEX_LAST_C finds the LAST occurrence of a given character.
• 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.
• TIMESTRING writes the current YMDHMS date into a string.
• 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 FORTRAN90 source codes.