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505 lines
19 KiB
505 lines
19 KiB
2 years ago
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<!-------- @HEADER
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!
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! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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!
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! Zoltan Toolkit for Load-balancing, Partitioning, Ordering and Coloring
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! Copyright 2012 Sandia Corporation
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!
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! Under the terms of Contract DE-AC04-94AL85000 with Sandia Corporation,
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! the U.S. Government retains certain rights in this software.
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!
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! Redistribution and use in source and binary forms, with or without
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! modification, are permitted provided that the following conditions are
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! met:
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!
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! 1. Redistributions of source code must retain the above copyright
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! notice, this list of conditions and the following disclaimer.
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!
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! 2. Redistributions in binary form must reproduce the above copyright
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! notice, this list of conditions and the following disclaimer in the
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! documentation and/or other materials provided with the distribution.
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!
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! 3. Neither the name of the Corporation nor the names of the
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! contributors may be used to endorse or promote products derived from
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! this software without specific prior written permission.
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!
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! THIS SOFTWARE IS PROVIDED BY SANDIA CORPORATION "AS IS" AND ANY
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! EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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! IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
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! PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL SANDIA CORPORATION OR THE
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! CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
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! EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
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! PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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! PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
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! LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
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! NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
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! SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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!
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! Questions? Contact Karen Devine kddevin@sandia.gov
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! Erik Boman egboman@sandia.gov
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!
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! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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!
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! @HEADER
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------->
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<!DOCTYPE html PUBLIC "-//w3c//dtd html 4.0 transitional//en">
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<html>
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<head>
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<meta http-equiv="Content-Type"
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content="text/html; charset=iso-8859-1">
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<meta name="GENERATOR"
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content="Mozilla/4.76 [en] (X11; U; Linux 2.4.2-2smp i686) [Netscape]">
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<meta name="sandia.approved" content="SAND99-1377">
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<meta name="author" content="karen devine, kddevin@sandia.gov">
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<title>Zoltan Developer's Guide: Hypergraph Partitioning</title>
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</head>
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<body bgcolor="#ffffff">
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<div align="right"><b><i><a href="dev.html">Zoltan Developer's Guide</a>
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| <a href="dev_reftree.html">Next</a> | <a
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href="dev_parmetis.html">Previous</a></i></b></div>
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<h2>
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<a name="Hypergraph Partitioning"></a>Appendix: Hypergraph Partitioning</h2>
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Hypergraph partitioning is a useful partitioning and
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load balancing method when connectivity data is available. It can be
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viewed as a more sophisticated alternative to
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the traditional graph partitioning.
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<p>A hypergraph consists of vertices and hyperedges. A hyperedge
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connects
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one or more vertices. A graph is a special case of a hypergraph where
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each edge has size two (two vertices). The hypergraph model is well
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suited to parallel computing, where vertices correspond to data objects
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and hyperedges represent the communication requirements. The basic
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partitioning problem is to partition the vertices into <i>k</i>
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approximately equal sets such that the number of cut hyperedges is
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minimized. Most partitioners (including Zoltan-PHG) allows a more
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general
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model where both vertices and hyperedges can be assigned weights.
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It has been
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shown that the hypergraph model gives a more accurate representation
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of communication cost (volume) than the graph model. In particular,
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for sparse matrix-vector multiplication, the hypergraph model
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<strong>exactly</strong> represents communication volume. Sparse
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matrices can be partitioned either along rows or columns;
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in the row-net model the columns are vertices and each row corresponds
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to an hyperedge, while in the column-net model the roles of vertices
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and hyperedges are reversed. </p>
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<p>Zoltan contains a native parallel hypergraph partitioner, called PHG
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(Parallel HyperGraph partitioner). In addition, Zoltan provides
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access to <a href="https://bmi.osu.edu/%7Eumit/software.htm">PaToH</a>,
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a serial hypergraph partitioner.
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Note that PaToH is not part of Zoltan and should be obtained
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separately from the <a href="https://bmi.osu.edu/%7Eumit/software.htm">
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PaToH web site</a>.
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Zoltan-PHG is a fully parallel multilevel hypergraph partitioner. For
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further technical description, see <a
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href="ug_refs.html#hypergraph-ipdps06">[Devine et al, 2006]</a>.<br>
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</p>
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<h4>Algorithm:</h4>
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The algorithm used is multilevel hypergraph partitioning. For
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coarsening, several versions of inner product (heavy connectivity)
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matching are available.
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The refinement is based on Fiduccia-Mattheysis (FM) but in parallel it
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is only an approximation.
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<h4>Parallel implementation:</h4>
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A novel feature of our parallel implementation is that we use a 2D
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distribution of the hypergraph. That is, each processor owns partial
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data about some vertices and some hyperedges. The processors are
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logically organized in a 2D grid as well. Most communication is limited
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to either a processor row or column. This design should allow for
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good scalability on large number of processors.<br>
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<h4>Data structures:</h4>
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The hypergraph is the most important data structure. This is stored as
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a compressed sparse matrix. Note that in parallel, each processor owns
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a local part of the global hypergraph
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(a submatrix of the whole matrix).
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The hypergraph data type is <i>struct HGraph</i>, and contains
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information like number of vertices, hyperedges, pins, compressed
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storage of all pins, optional vertex and edge weights, pointers
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to relevant communicators, and more. One cryptic notation needs an
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explanation: The arrays <i>hindex, hvertex</i> are used to
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look up vertex info given a hyperedge, and <i>vindex, vedge</i> are
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used to look up hyperedge info given a vertex. Essentially,
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we store the hypergraph as a sparse matrix in both CSR and CSC formats.
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This doubles the memory cost but gives better performance.
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The data on each processor is stored using local indexing, starting at zero.
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In order to get the global vertex or edge number, use the macros
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<i>VTX_LNO_TO_GNO</i> and <i>EDGE_LNO_TO_GNO</i>. These macros will
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look up the correct offsets (using the dist_x and dist_y arrays).
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Note that <i>phg->nVtx</i> is always the local number of vertices,
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which may be zero on some processors.
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<h4>Parameters:</h4>
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In the User's Guide, only the most essential parameters have been
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documented. There are several other parameters, intended for developers
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and perhaps expert "power" users. We give a more complete list of all
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parameters below. Note that these parameters <span
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style="font-style: italic;">may change in future versions!<br>
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</span>
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For a precise list of parameters in a particular version of Zoltan, look at the source code (phg.c).
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<table nosave="" width="100%">
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<tbody>
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<tr>
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<td valign="top"><b>Method String:</b></td>
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<td><b>HYPERGRAPH</b></td>
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</tr>
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<tr>
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<td><b>Parameters:</b></td>
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<td><br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <span
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style="font-style: italic;">HYPERGRAPH_PACKAGE</span><br>
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</td>
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<td style="vertical-align: top;">PHG (parallel) or PaToH (serial)<br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <span
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style="font-style: italic;">CHECK_HYPERGRAPH</span><br>
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</td>
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<td style="vertical-align: top;">Check if input data is valid.
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(Slows performance;intended for debugging.)<br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"><span style="font-style: italic;">
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PHG_OUTPUT_LEVEL</span><br>
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</td>
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<td style="vertical-align: top;">Level of verbosity; 0 is silent.<br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <span
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style="font-style: italic;">PHG_FINAL_OUTPUT</span><br>
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</td>
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<td style="vertical-align: top;">Print stats about final
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partition? (0/1)<br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <span
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style="font-style: italic;">PHG_NPROC_VERTEX</span><br>
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</td>
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<td style="vertical-align: top;">Desired number of processes in
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the vertex direction (for 2D internal layout) </td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <span
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style="font-style: italic;">PHG_NPROC_HEDGE</span><br>
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</td>
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<td style="vertical-align: top;">Desired number of processes in
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the hyperedge direction (for 2D internal layout) </td>
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</tr>
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<tr>
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<td valign="top"><i> PHG_COARSENING_METHOD</i></td>
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<td>The method to use in matching/coarsening; currently these are
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available. <br>
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<span style="font-style: italic;">agg</span> - agglomerative inner product
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matching (a.k.a. heavy connectivity matching) <br>
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<span style="font-style: italic;">ipm</span> - inner product
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matching (a.k.a. heavy connectivity matching) <br>
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<span style="font-style: italic;">c-ipm</span> - column
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ipm; faster method based on ipm within processor columns <br>
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<span style="font-style: italic;">a-ipm </span>- alternate
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between fast method (l-ipm ) and ipm <br>
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<span style="font-style: italic;">l-ipm </span>- local ipm
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on each processor. Fastest option but often gives poor quality. <br>
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<i>h-ipm - </i>hybrid ipm that uses partial c-ipm followed
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by ipm on each level <br>
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<i><br>
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</i></td>
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</tr>
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<tr>
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<td> <span style="font-style: italic;">PHG_COARSENING_LIMIT</span><br>
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</td>
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<td>Number of vertices at which to stop coarsening.<br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <span
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style="font-style: italic;">PHG_VERTEX_VISIT_ORDER</span><br>
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</td>
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<td style="vertical-align: top;">Ordering of vertices in greedy
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matching scheme:<br>
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0 - random<br>
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1 - natural order (as given by the query functions)<br>
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2 - increasing vertex weights<br>
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3 - increasing vertex degree<br>
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4 - increasing vertex degree, weighted by pins<br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <span
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style="font-style: italic;">PHG_EDGE_SCALING</span><br>
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</td>
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<td style="vertical-align: top;">Scale edge weights by some
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function of size of the hyperedges:<br>
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0 - no scaling<br>
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1 - scale by 1/(size-1) [absorption scaling]<br>
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2 - scale by 2/((size*size-1)) [clique scaling]<br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <span
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style="font-style: italic;">PHG_VERTEX_SCALING</span><br>
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</td>
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<td style="vertical-align: top;">Variations in "inner product"
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similarity metric (for matching):<br>
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0 - Euclidean inner product: <x,y><br>
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1 - cosine similarity: <x,y>/(|x|*|y|)<br>
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2 - <x,y>/(|x|^2 * |y|^2)<br>
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3 - scale by sqrt of vertex weights<br>
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4 - scale by vertex weights<br>
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</td>
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</tr>
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<tr>
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<td valign="top"> <i>PHG_COARSEPARTITION_METHOD</i></td>
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<td>Method to partition the coarsest (smallest) hypergraph;
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typically done in serial:<br>
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<span style="font-style: italic;">random</span> - random<br>
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<span style="font-style: italic;">linear</span> - linear
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(natural) order<br>
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<span style="font-style: italic;">greedy </span>- greedy method
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based on minimizing cuts<br>
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<span style="font-style: italic;">auto </span>- automatically
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select from the above methods (in parallel, the processes will do
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different methods)<br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <span
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style="font-style: italic;">PHG_REFINEMENT_METHOD</span><br>
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</td>
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<td style="vertical-align: top;">Refinement algorithm:<br>
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<span style="font-style: italic;">fm </span>- two-way
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approximate FM<br>
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<span style="font-style: italic;">none</span> - no refinement<br>
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</td>
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</tr>
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<tr>
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<td> <i>PHG_REFINEMENT_LOOP_LIMIT</i></td>
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<td>Loop limit in FM refinement. Higher number means more
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refinement. <br>
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</td>
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</tr>
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<tr nosave="" valign="top">
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<td> <span style="font-style: italic;">PHG_REFINEMENT_MAX_NEG_MOVE</span><br>
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</td>
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<td nosave="">Maximum number of negative moves allowed in FM.<br>
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</td>
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</tr>
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<tr nosave="" valign="top">
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<td> <span style="font-style: italic;">PHG_BAL_TOL_ADJUSTMENT</span><br>
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</td>
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<td nosave="">Controls how the balance tolerance is adjusted at
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each level of bisection.<br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <span
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style="font-style: italic;">PHG_RANDOMIZE_INPUT</span><br>
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</td>
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<td style="vertical-align: top;">Randomize layout of vertices and
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hyperedges in internal parallel 2D layout? (0/1)<br>
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</td>
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</tr>
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<tr>
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<td style="vertical-align: top;"> <a
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name="PHG_EDGE_WEIGHT_OPERATION"></a><span style="font-style: italic;">PHG_EDGE_WEIGHT_OPERATION</span>
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</td>
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<td style="vertical-align: top;">Operation to be applied to edge
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weights supplied by different processes for the same hyperedge:<br>
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<i>add</i> - the hyperedge weight will be the sum of the supplied
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weights<br>
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<i>max</i> - the hyperedge weight will be the maximum of the
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supplied weights<br>
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<i>error</i> - if the hyperedge weights are not equal, Zoltan
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will flag an error, otherwise the hyperedge weight will be the value
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returned by the processes<br>
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</td>
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</tr>
|
||
|
|
<tr nosave="" valign="top">
|
||
|
|
<td> <span style="font-style: italic;">EDGE_SIZE_THRESHOLD</span><br>
|
||
|
|
</td>
|
||
|
|
<td nosave="">Ignore hyperedges greater than this fraction times
|
||
|
|
number of vertices.<br>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"> <span
|
||
|
|
style="font-style: italic;">PATOH_ALLOC_POOL0</span><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;">Memory allocation for PaToH; see
|
||
|
|
the PaToH manual for details.<br>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"> <span
|
||
|
|
style="font-style: italic;">PATOH_ALLOC_POOL1</span><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;">Memory allocation for PaToH; see
|
||
|
|
the PaToH manual for details.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td valign="top"><b>Default values:</b></td>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><i>HYPERGRAPH_PACKAGE = PHG<br>
|
||
|
|
</i></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;"><span style="font-style: italic;">CHECK_HYPERGRAPH</span>
|
||
|
|
= 0<br>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><span style="font-style: italic;">PHG_OUTPUT_LEVEL=0</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><span style="font-style: italic;">PHG_FINAL_OUTPUT=0</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><i>PHG_REDUCTION_METHOD=ipm</i></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><span style="font-style: italic;">PHG_REDUCTION_LIMIT=100</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><span style="font-style: italic;">PHG_VERTEX_VISIT_ORDER=0</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><span style="font-style: italic;">PHG_EDGE_SCALING=0</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><span style="font-style: italic;">PHG_VERTEX_SCALING=0</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;"><i>PHG_COARSEPARTITION_METHOD=greedy</i></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;"><span style="font-style: italic;">PHG_REFINEMENT_METHOD=fm</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;"><i>PHG_REFINEMENT_LOOP_LIMIT=10</i></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;"><span style="font-style: italic;">PHG_REFINEMENT_MAX_NEG_MOVE=100</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;"><span style="font-style: italic;">PHG_BAL_TOL_ADJUSTMENT=0.7</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;"><span style="font-style: italic;">PHG_RANDOMIZE_INPUT=0</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><span style="font-style: italic;">PHG_EDGE_WEIGHT_OPERATION=max</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;"><span style="font-style: italic;">EDGE_SIZE_THRESHOLD=0.25</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;"><span style="font-style: italic;">PATOH_ALLOC_POOL0=0</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td style="vertical-align: top;"><br>
|
||
|
|
</td>
|
||
|
|
<td style="vertical-align: top;"><span style="font-style: italic;">PATOH_ALLOC_POOL1=0</span></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td valign="top"><b>Required Query Functions:</b></td>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><b><a href="../ug_html/ug_query_lb.html#ZOLTAN_NUM_OBJ_FN">ZOLTAN_NUM_OBJ_FN</a></b></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><b><a href="../ug_html/ug_query_lb.html#ZOLTAN_OBJ_LIST_FN">ZOLTAN_OBJ_LIST_FN</a></b>
|
||
|
|
or <b><a href="../ug_html/ug_query_lb.html#ZOLTAN_FIRST_OBJ_FN">ZOLTAN_FIRST_OBJ_FN</a></b>/<b><a
|
||
|
|
href="../ug_html/ug_query_lb.html#ZOLTAN_NEXT_OBJ_FN">ZOLTAN_NEXT_OBJ_FN</a></b>
|
||
|
|
pair</td>
|
||
|
|
</tr>
|
||
|
|
<tr nosave="" valign="top">
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td nosave=""> <b><a href="../ug_html/ug_query_lb.html#ZOLTAN_HG_SIZE_CS_FN">ZOLTAN_HG_SIZE_CS_FN</a></b>
|
||
|
|
<br>
|
||
|
|
<b><a href="../ug_html/ug_query_lb.html#ZOLTAN_HG_CS_FN">ZOLTAN_HG_CS_FN</a></b>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td valign="top"><b>Optional Query Functions:</b></td>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><b><a href="../ug_html/ug_query_lb.html#ZOLTAN_HG_SIZE_EDGE_WTS_FN">ZOLTAN_HG_SIZE_EDGE_WTS_FN</a></b></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><br>
|
||
|
|
</td>
|
||
|
|
<td><b><a href="../ug_html/ug_query_lb.html#ZOLTAN_HG_EDGE_WTS_FN">ZOLTAN_HG_EDGE_WTS_FN</a></b></td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
<p>
|
||
|
|
It is possible to provide the graph query functions instead of the
|
||
|
|
hypergraph queries, though this is not recommended. If only graph query
|
||
|
|
functions are registered, Zoltan will automatically create a hypergraph
|
||
|
|
from the graph, but some information (specifically, edge weights) will
|
||
|
|
be lost. </p>
|
||
|
|
<hr width="100%">[<a href="ug.html">Table of Contents</a> | <a
|
||
|
|
href="dev_reftree.html">Next:
|
||
|
|
Refinement Tree Partitioning</a> | <a
|
||
|
|
href="dev_parmetis.html">Previous:
|
||
|
|
ParMetis</a> | <a href="https://www.sandia.gov/general/privacy-security/index.html">Privacy and Security</a>]
|
||
|
|
</body>
|
||
|
|
</html>
|