mptopcom

# mptopcom

mptopcom is a software developed at TU Berlin and Hokkaido University for computing triangulations of point configurations in parallel. It is a combination of

• TOPCOM for triangulations,
• polymake for combinatorics, and
• mts for parallelising reverse search.

Our paper describes the algorithm and contains many details as well as examples. Please cite this reference in your papers if you are using mptopcom.

Please read this very carefully! mptopcom is highly optimized software dedicated to exceptionally large enumerations on suitable hardware. As a consequence it depends on a number of up-to-date-versions of other software, and the installation requires some diligence.

You need to have open-mpi or some other mpi implementation and polymake version at least 3.2 installed. In particular, you need the polymake callable library, which might not be installed by the package manager of your distribution. Furthermore, you need an installation of cdd, the bundled version coming with polymake does not build the cdd library.

mptopcom has been tested with several versions, at least 5.3.0, of gcc and clang, at least version 3.8.0. For MPI we have tested openmpi at least version 1.4.1 and Intel MPI 20150128.

Note that recent changes in polymake were breaking, so only the second version works with polymake 4.0 and newer.

Installation can be done with the following three commands:

./configure
ninja -C build/Opt -j #CPU's
ninja -C build/Opt install

There are several options to the ./configure command that can be viewed with

./configure --help

The most important options are

--with-polymake=/path/to/polymake/installation
--with-mpi=/path/to/mpi/installation
--prefix=/path/to/install/dir

The configure command will extract most of the information needed from polymake’s polymake-config command.

The binaries are in the build/Opt/bin folder after building and in the prefix/bin folder after installation. Besides the usual TOPCOM binaries there are four new binaries:

• mptopcom1: Runs reverse search single threaded.
• canonicalRepresentative: Compute the canonical representative from the orbit of a triangulation.
• randomBFS: Execute a random breadth first search on the flip-graph of triangulations, only reports the queue sizes.

You can test your build by running the testsuite:

perl testsuite.pl

Input files are formatted as in TOPCOM, please have a look at the files in the examples folder for samples. The points are given homogeneously, i.e. embedded at height one. After the points you can give the generators of a group acting on your point set. Here is the example of the input file for the two-dimensional 0-1-square:

[
[1,0,0],
[1,0,1],
[1,1,0],
[1,1,1]
]
[
[0,2,1,3],
[3,1,2,0],
[2,3,0,1]
]

You can then call mptopcom in the following way:

mpirun -np 5 mptopcom < examples/cube_4.dat

If you did not run make install you can call the binary in the build/Opt/bin folder:

mpirun -np 5 ./build/Opt/bin/mptopcom < examples/cube_4.dat

This will produce all triangulations of the four dimensional cube.

There are several options you can invoke to modify the behaviour of mptopcom:

1. --cyclic-flips : Use special combinatorial description of the reverse search graph for cyclic polytopes
2. --full : Only list full/fine/spanning triangulations
3. --flip-cache n : Sets size of flip cache to n (default: 2000)
4. --make-marked-tree : Draws a tree with classes having the same node color (mptopcom1 only)
5. --make-tree : Will produce polymake code to plot reverse search tree (mptopcom1 only)
6. --orbit-cache n : Size of symmetry cache set to n (default: 2000)
7. --regular : Only output regular triangulations
8. --regularity-cache n : Size of regularity cache set to n (default: 2000)
9. -v : verbose (every worker will tell when he found 1000 triangulations)
10. --central : Produce only triangulations with maximal gkz[0], depends on ordering of points
1. -maxnodes only affects the “bumps” at maxnodes and scale*maxnodes.
2. -scale sets the scaling factor when we have many jobs available (workers can work longer if we aren’t trying to split jobs)
3. -maxd sets the max depth to go to when we don’t have enough jobs available (2 is the default, which is aggressive. 0 disables)
4. -lmax sets the point where -scale is used (if |L| > lmax * numproc)
5. -lmin sets the point where -maxd is used (if |L| < lmin * numproc)

mts supports checkpointing and restarts; if you add options -checkp file1 -stop file2 then it will periodically check to see if file file2 exists, and if so then it will checkpoint to file1 when convenient (it waits for workers to finish their current jobs first). Then you can restart using option -restart file1 (and of course can use -checkp & -stop again, but you should give different filenames to the -restart and -checkp options). There is an example in the section below.

This section contains samples of how to call mptopcom1 and mptopcom using the above options. It is assumed that mptopcom was build, but not installed and that the current directory is the root of the source directory.

./build/Opt/bin/mptopcom1 < examples/cube_3.dat
./build/Opt/bin/mptopcom1 -v < examples/cube_3.dat
./build/Opt/bin/mptopcom1 -v < examples/moae.dat
./build/Opt/bin/mptopcom1 -v --regular < examples/moae.dat
./build/Opt/bin/mptopcom1 -v --regular --full < examples/moae.dat
./build/Opt/bin/mptopcom1 -v --flip-cache 2000 --orbit-cache 2000 < examples/moae.dat
./build/Opt/bin/mptopcom1 -v --make-tree < examples/moae.dat
mpirun -np 8 ./build/Opt/bin/mptopcom < examples/cube_3.dat
mpirun -np 8 ./build/Opt/bin/mptopcom -v < examples/cube_3.dat
mpirun -np 8 ./build/Opt/bin/mptopcom --regular < examples/cube_4.dat
mpirun -np 10 ./build/Opt/bin/mptopcom < examples/lattice_3_3.dat
mpirun -np 10 ./build/Opt/bin/mptopcom --full < examples/lattice_3_3.dat
mpirun -np 10 ./build/Opt/bin/mptopcom --full --flip-cache 2000 --orbit-cache 2000 < examples/lattice_3_3.dat
mpirun -np 10 ./build/Opt/bin/mptopcom < examples/lattice_3_3.dat 1>output.txt 2>error.txt

The following is an example for checkpointing and restarting:

mpirun -np 8 ./build/Opt/bin/mptopcom -v -stop stop_file -checkp checkp_file < examples/cube_4.dat
touch stop_file

Now the state will be written to the file checkp_file and after a while the computation stops. Restart it with

mpirun -np 8 ./build/Opt/bin/mptopcom -v -restart checkp_file < examples/cube_4.dat

mptopcom1 can output polymake code that one can paste into polymake to obtain the reverse search tree in the edge graph of the secondary polytope. This will not work for large examples, so handle with care. There are three possibilities:

1. Give mptopcom1 an example without symmetry group:
./build/Opt/bin/mptopcom1 --make-tree < mp_examples/nosym/moae.dat

This will just draw all nodes in the same color and the edges between them that the reverse search used. 2. Give mptopcom1 an example with symmetry group:

./build/Opt/bin/mptopcom1 --make-tree < examples/moae.dat

Now just the canonical representatives are drawn and the edges between them come from the reverse search, but they do not have to correspond to edges of the secondary polytope, since there can be a flip between classes of triangulations, while there is no flip between the canonical representatives. 3. Give mptopcom1 an example with symmetry group and the parameter –make-marked-tree:

./build/Opt/bin/mptopcom1 --make-marked-tree  < examples/moae.dat

This call will make mptopcom1 ignore the symmetry group. So the node number and the edges are the same as in 1. However, the tree generating procedure will use the symmetry group to color nodes according to their class membership.

mptopcom is joint work of the following three authors:

• Skip Jordan, Laboratory for Algorithmics, Hokkaido University, Japan
• Michael Joswig, Department of Mathematics, TU Berlin, Germany
• Lars Kastner, Department of Mathematics, TU Berlin, Germany

mptopcom is based on TOPCOM which is developed by

• Jörg Rambau, Department of Mathematics, Universität Bayreuth, Germany

See the files COPYING and LICENSE in the source for further details.