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<Title>Sparse Matrix Ordering Example</Title>
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<H1><A NAME="sec:sparse-matrix-ordering"></A>
Sparse Matrix Ordering
</H1>

<P>
Graph theory was identified as a powerful tool for sparse matrix
computation when Seymour Parter used undirected graphs to model
symmetric Gaussian elimination more than 30 years ago&nbsp;[<A HREF="bibliography.html#parter61:_gauss">28</A>].
Graphs can be used to model symmetric matrices, factorizations and
algorithms on non-symmetric matrices, such as fill paths in Gaussian
elimination, strongly connected components in matrix irreducibility,
bipartite matching, and alternating paths in linear dependence and
structural singularity. Not only do graphs make it easier to
understand and analyze sparse matrix algorithms, but they broaden the
area of manipulating sparse matrices using existing graph algorithms
and techniques&nbsp;[<A
 HREF="bibliography.html#george93:graphtheory">13</A>].  In this section, we are
going to illustrate how to use BGL in sparse matrix computation such
as ordering algorithms. Before we go further into the sparse matrix
algorithms, let us take a step back and review a few things.

<P>

<H2>Graphs and Sparse Matrices</H2>

<P>
A graph is fundamentally a way to represent a binary relation between
objects. The nonzero pattern of a sparse matrix also describes a
binary relation between unknowns. Therefore the nonzero pattern of a
sparse matrix of a linear system can be modeled with a graph
<I>G(V,E)</I>, whose <I>n</I> vertices in <I>V</I> represent the
<I>n</I> unknowns, and where there is an edge from vertex <I>i</I> to
vertex <I>j</I> when <i>A<sub>ij</sub></i> is nonzero.  Thus, when a
matrix has a symmetric nonzero pattern, the corresponding graph is
undirected.

<P>

<H2>Sparse Matrix Ordering Algorithms</H2>

<P>
The process for solving a sparse symmetric positive definite linear
system, <i>Ax=b</i>, can be divided into four stages as follows:
<DL>
<DT><STRONG>Ordering:</STRONG></DT>
<DD>Find a permutation <i>P</i> of matrix <i>A</i>,
</DD>
<DT><STRONG>Symbolic factorization:</STRONG></DT>
<DD>Set up a data structure for the Cholesky
 factor <i>L</i> of <i>PAP<sup>T</sup></i>,
</DD>
<DT><STRONG>Numerical factorization:</STRONG></DT>
<DD>Decompose <i>PAP<sup>T</sup></i> into <i>LL<sup>T</sup></i>,
</DD>
<DT><STRONG>Triangular system solution:</STRONG></DT>
<DD>Solve <i>LL<sup>T</sup>Px=Pb</i> for <i>x</i>.
</DD>
</DL>
Because the choice of permutation <i>P</i> will directly determine the
number of fill-in elements (elements present in the non-zero structure
of <i>L</i> that are not present in the non-zero structure of
<i>A</i>), the ordering has a significant impact on the memory and
computational requirements for the latter stages.  However, finding
the optimal ordering for <i>A</i> (in the sense of minimizing fill-in)
has been proven to be NP-complete&nbsp;[<a
href="bibliography.html#garey79:computers-and-intractability">30</a>]
requiring that heuristics be used for all but simple (or specially
structured) cases.

<P>
A widely used but rather simple ordering algorithm is a variant of the
Cuthill-McKee orderings, the reverse Cuthill-McKee ordering algorithm.
This algorithm can be used as a preordering method to improve ordering
in more sophisticated methods such as minimum degree
algorithms&nbsp;[<a href="bibliography.html#LIU:MMD">21</a>].

<P>

<H3><A NAME="SECTION001032100000000000000">
Reverse Cuthill-McKee Ordering Algorithm</A>
</H3>
The original Cuthill-McKee ordering algorithm is primarily designed to
reduce the profile of a matrix&nbsp;[<A
 HREF="bibliography.html#george81:__sparse_pos_def">14</A>].
George discovered that the reverse ordering often turned out to be
superior to the original ordering in 1971. Here we describe the
Reverse Cuthill-McKee algorithm in terms of a graph:

<OL>
<LI><I>Finding a starting vertex</I>: Determine a starting vertex
 <I>r</I> and assign <i>r</i> to <I>x<sub>1</sub></I>.
</LI>
<LI><I>Main part</I>: For <I>i=1,...,N</I>, find all
 the unnumbered neighbors of the vertex <I>x<sub>i</sub></I> and number them
 in increasing order of degree.
</LI>
<LI><I>Reversing ordering</I>: The reverse Cuthill-McKee ordering
 is given by <I>y<sub>1</sub>,...,y<sub>N</sub></I> where
 <I>y<sub>i</sub></I> is <I>x<sub>N-i+1</sub></I> for <I>i = 1,...,N</I>.
</LI>
</OL>
In the first step a good starting vertex needs to be determined. The
study by George and Liu&nbsp;[<A
HREF="bibliography.html#george81:__sparse_pos_def">14</A>] showed that
a pair of vertices which are at maximum or near maximum ``distance''
apart are good ones. They also proposed an algorithm to find such a
starting vertex in &nbsp;[<A
HREF="bibliography.html#george81:__sparse_pos_def">14</A>], which we
show in <A
HREF="#fig:ggcl_find_starting_vertex">Figure
1</A> implemented using the BGL interface.

<P>
<P></P>
<DIV ALIGN="CENTER"><A NAME="fig:ggcl_find_starting_vertex"></A>
<table border><tr><td>
<pre>
namespace boost {
  template &lt;class Graph, class Vertex, class Color, class Degree&gt;
  Vertex 
  pseudo_peripheral_pair(Graph& G, const Vertex& u, int& ecc,
                         Color color, Degree degree)
  {
    typename property_traits&lt;Color&gt;::value_type c = get(color, u);

    rcm_queue&lt;Vertex, Degree&gt; Q(degree);
    
    typename boost::graph_traits&lt;Graph&gt;::vertex_iterator ui, ui_end;
    for (tie(ui, ui_end) = vertices(G); ui != ui_end; ++ui)
      put(color, *ui, white(c));
    breadth_first_search(G, u, Q, bfs_visitor&lt;&gt;(), color);

    ecc = Q.eccentricity(); 
    return Q.spouse();
  }
  
  template &lt;class Graph, class Vertex, class Color, class Degree&gt; 
  Vertex find_starting_node(Graph& G, Vertex r, Color c, Degree d) {
    int eccen_r, eccen_x;
    Vertex x = pseudo_peripheral_pair(G, r, eccen_r, c, d);
    Vertex y = pseudo_peripheral_pair(G, x, eccen_x, c, d);

    while (eccen_x &gt; eccen_r) {
      r = x;
      eccen_r = eccen_x;
      x = y;
      y = pseudo_peripheral_pair(G, x, eccen_x, c, d);
    }
    return x;
  }
} // namespace boost
</pre>
</td></tr></table>
<CAPTION ALIGN="BOTTOM">
<table><tr><td>
<STRONG>Figure 1:</STRONG>
The BGL implementation of <TT>find_starting_node</TT>. The key
part <TT>pseudo_peripheral_pair</TT> is BFS with a custom queue
 type virtually.
</td></tr></table></CAPTION>
</DIV>
<P></P>

The key part of step one is a custom queue type with BFS as shown in
<A
HREF="#fig:ggcl_find_starting_vertex">Figure
1</A>.  The main Cuthill-McKee algorithm shown in Figure&nbsp;<A
HREF="#fig:cuthill-mckee">Figure 2</A>
is a fenced priority queue with a generalized BFS. If
<TT>inverse_permutation</TT> is a normal iterator, the result stored
is the inverse permutation of original Cuthill-McKee ordering. On the
other hand, if <TT>inverse_permutation</TT> is a reverse iterator, the
result stored is the inverse permutation of reverse Cuthill-McKee
ordering.  Our implementation is quite concise because many components
from BGL can be reused.


<P></P>
<DIV ALIGN="CENTER"><A NAME="fig:cuthill-mckee"></A><A NAME="6261"></A>
<table border><tr><td>
<pre>
  template &lt; class Graph, class Vertex, class OutputIterator, 
             class Color, class Degree &gt;
  inline void 
  cuthill_mckee_ordering(Graph& G, 
                         Vertex s,
                         OutputIterator inverse_permutation, 
                         Color color, Degree degree)
  {
    typedef typename property_traits&lt;Degree&gt;::value_type DS;
    typename property_traits&lt;Color&gt;::value_type c = get(color, s);
    typedef indirect_cmp&lt;Degree, std::greater&lt;DS&gt; &gt; Compare;
    Compare comp(degree);
    fenced_priority_queue&lt;Vertex, Compare &gt; Q(comp);

    typedef cuthill_mckee_visitor&lt;OutputIterator&gt;  CMVisitor;
    CMVisitor cm_visitor(inverse_permutation);
    
    typename boost::graph_traits&lt;Graph&gt;::vertex_iterator ui, ui_end;
    for (tie(ui, ui_end) = vertices(G); ui != ui_end; ++ui)
      put(color, *ui, white(c));
    breadth_first_search(G, s, Q, cm_visitor, color);
  }  
</pre>
</td></tr></table>
<CAPTION ALIGN="BOTTOM">
<TABLE>
<TR><TD> <STRONG>Figure 2:</STRONG> The BGL implementation of
Cuthill-McKee algoithm.  </TD></TR> </TABLE></CAPTION> </DIV><P></P>
<P>


<P>

<H3>Minimum Degree Ordering Algorithm</H3>

<P>
The pattern of another category of high-quality ordering algorithms in
wide use is based on a greedy approach such that the ordering is
chosen to minimize some quantity at each step of a simulated -step
symmetric Gaussian elimination process.  The algorithms using such an
approach are typically distinguished by their greedy minimization
criteria&nbsp;[<a href="bibliography.html#ng-raghavan">34</a>].

<P>
In graph terms, the basic ordering process used by most greedy
algorithms is as follows:

<OL>
<LI><EM>Start:</EM> Construct undirected graph <i>G<sup>0</sup></i>
 corresponding to matrix <i>A</i>
 
</LI>
<LI><EM>Iterate:</EM> For <i>k = 1, 2, ... </i> until 
  <i>G<sup>k</sup> = { }</i> do:
    
<UL>
<LI>Choose a vertex <i>v<sup>k</sup></i> from <i>G<sup>k</sup></i>
      according to some criterion
</LI>
<LI>Eliminate <i>v<sup>k</sup></i> from <i>G<sup>k</sup></i> to form 
  <i>G<sup>k+1</sup></i>
 
</LI>
</UL>
</LI>
</OL>
The resulting ordering is the sequence of vertices <i>{v<sup>0</sup>,
v<sup>1</sup>,...}</i> selected by the algorithm.

<P>
One of the most important examples of such an algorithm is the
<I>Minimum Degree</I> algorithm.  At each step the minimum degree
algorithm chooses the vertex with minimum degree in the corresponding
graph as <i>v<sup>k</sup></i>.  A number of enhancements to the basic
minimum degree algorithm have been developed, such as the use of a
quotient graph representation, mass elimination, incomplete degree
update, multiple elimination, and external degree.  See&nbsp;[<A
href="bibliography.html#George:evolution">35</a>] for a historical
survey of the minimum degree algorithm.

<P>
The BGL implementation of the Minimum Degree algorithm closely follows
the algorithmic descriptions of the one in &nbsp;[<A
HREF="bibliography.html#LIU:MMD">21</A>]. The implementation presently
includes the enhancements for mass elimination, incomplete degree
update, multiple elimination, and external degree.

<P>
In particular, we create a graph representation to improve the
performance of the algorithm. It is based on a templated ``vector of
vectors.''  The vector container used is an adaptor class built on top
the STL <TT>std::vector</TT> class.  Particular characteristics of this
adaptor class include the following:

<UL>
<LI>Erasing elements does not shrink the associated memory.
 Adding new elements after erasing will not need to allocate
  additional memory.
</LI>
<LI>Additional memory is allocated efficiently on demand when new
  elements are added (doubling the capacity every time it is
  increased).  This property comes from STL vector.
</LI>
</UL>

<P>
Note that this representation is similar to that used in Liu's
implementation, with some important differences due to dynamic memory
allocation.  With the dynamic memory allocation we do not need to
over-write portions of the graph that have been eliminated,
allowing for a more efficient graph traversal.  More importantly,
information about the elimination graph is preserved allowing for
trivial symbolic factorization.  Since symbolic factorization can be
an expensive part of the entire solution process, improving its
performance can result in significant computational savings.

<P>
The overhead of dynamic memory allocation could conceivably compromise
performance in some cases. However, in practice, memory allocation
overhead does not contribute significantly to run-time for our
implementation as shown in &nbsp;[] because it is not
done very often and the cost gets amortized.

<P>

<H2>Proper Self-Avoiding Walk</H2>

<P>
The finite element method (FEM) is a flexible and attractive numerical
approach for solving partial differential equations since it is
straightforward to handle geometrically complicated domains. However,
unstructured meshes generated by FEM does not provide an obvious
labeling (numbering) of the unknowns while it is vital to have it for
matrix-vector notation of the underlying algebraic equations. Special
numbering techniques have been developed to optimize memory usage and
locality of such algorithms. One novel technique is the self-avoiding
walk&nbsp;[].

<P>
If we think a mesh is a graph, a self-avoiding walk(SAW) over an
arbitrary unstructured two-dimensional mesh is an enumeration of all
the triangles of the mesh such that two successive triangles shares an
edge or a vertex. A proper SAW is a SAW where jumping twice over the
same vertex is forbidden for three consecutive triangles in the walk.
it can be used to improve parallel efficiency of several irregular
algorithms, in particular issues related to reducing runtime memory
access (improving locality) and interprocess communications. The
reference&nbsp;[] has proved the existence
of A PSAW for any arbitrary triangular mesh by extending an existing
PSAW over a small piece of mesh to a larger part. The proof
effectively provides a set of rules to construct a PSAW.

<P>
The algorithm family of constructing a PSAW on a mesh is to start from
any random triangle of the mesh, choose new triangles sharing an edge
with the current sub-mesh and extend the existing partial PSAW over
the new triangles.

<P>
Let us define a dual graph of a mesh.  Let a triangle in the mesh be a
vertex and two triangles sharing an edge in the mesh means there is an
edge between two vertices in the dual graph. By using a dual graph of
a mesh, one way to implement the algorithm family of constructing a
PSAW is to reuse  BGL algorithms such as BFS and DFS with a customized
visitor to provide operations during
traversal. The customized visitor has the function <TT>tree_edge()</TT>
which is to extend partial PSAW in case by case and function
<TT>start_vertex()</TT> which is to set up the PSAW for the starting vertex.

<br>
<HR>
<TABLE>
<TR valign=top>
<TD nowrap>Copyright &copy; 2000-2001</TD><TD>
<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
</TD></TR></TABLE>

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