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\begin{document}

\title{An Implementation of Isomorphism Testing}
\author{Jeremy G. Siek}

\maketitle

\section{Introduction}

This paper documents the implementation of the \code{isomorphism()}
function of the Boost Graph Library.  The implementation was by Jeremy
Siek with algorithmic improvements and test code from Douglas Gregor.
The \code{isomorphism()} function answers the question, ``are these
two graphs equal?''  By \emph{equal}, we mean the two graphs have the
same structure---the vertices and edges are connected in the same
way. The mathematical name for this kind of equality is
\emph{isomorphic}.

An \emph{isomorphism} is a one-to-one mapping of the vertices in one
graph to the vertices of another graph such that adjacency is
preserved. Another words, given graphs $G_{1} = (V_{1},E_{1})$ and
$G_{2} = (V_{2},E_{2})$, an isomorphism is a function $f$ such that
for all pairs of vertices $a,b$ in $V_{1}$, edge $(a,b)$ is in $E_{1}$
if and only if edge $(f(a),f(b))$ is in $E_{2}$.

Both graphs must be the same size, so let $N = |V_1| = |V_2|$. The
graph $G_1$ is \emph{isomorphic} to $G_2$ if an isomorphism exists
between the two graphs, which we denote by $G_1 \isomorphic G_2$.

In the following discussion we will need to use several notions from
graph theory. The graph $G_s=(V_s,E_s)$ is a \emph{subgraph} of graph
$G=(V,E)$ if $V_s \subseteq V$ and $E_s \subseteq E$.  An
\emph{induced subgraph}, denoted by $G[V_s]$, of a graph $G=(V,E)$
consists of the vertices in $V_s$, which is a subset of $V$, and every
edge $(u,v)$ in $E$ such that both $u$ and $v$ are in $V_s$.  We use
the notation $E[V_s]$ to mean the edges in $G[V_s]$.

In some places we express a function as a set of pairs, so the set $f
= \{ \pair{a_1}{b_1}, \ldots, \pair{a_n}{b_n} \}$
means $f(a_i) = b_i$ for $i=1,\ldots,n$.

\section{Exhaustive Backtracking Search}

The algorithm used by the \code{isomorphism()} function is, at
first approximation, an exhaustive search implemented via
backtracking.  The backtracking algorithm is a recursive function. At
each stage we will try to extend the match that we have found so far.
So suppose that we have already determined that some subgraph of $G_1$
is isomorphic to a subgraph of $G_2$.  We then try to add a vertex to
each subgraph such that the new subgraphs are still isomorphic to one
another. At some point we may hit a dead end---there are no vertices
that can be added to extend the isomorphic subgraphs. We then
backtrack to previous smaller matching subgraphs, and try extending
with a different vertex choice. The process ends by either finding a
complete mapping between $G_1$ and $G_2$ and return true, or by
exhausting all possibilities and returning false.

We are going to consider the vertices of $G_1$ in a specific order
(more about this later), so assume that the vertices of $G_1$ are
labeled $1,\ldots,N$ according to the order that we plan to add them
to the subgraph.  Let $G_1[k]$ denote the subgraph of $G_1$ induced by
the first $k$ vertices, with $G_1[0]$ being an empty graph. At each
stage of the recursion we start with an isomorphism $f_{k-1}$ between
$G_1[k-1]$ and a subgraph of $G_2$, which we denote by $G_2[S]$, so
$G_1[k-1] \isomorphic G_2[S]$. The vertex set $S$ is the subset of
$V_2$ that corresponds via $f_{k-1}$ to the first $k-1$ vertices in
$G_1$. We try to extend the isomorphism by finding a vertex $v \in V_2
- S$ that matches with vertex $k$. If a matching vertex is found, we
have a new isomorphism $f_k$ with $G_1[k] \isomorphic G_2[S \union \{
v \}]$.

\begin{tabbing}
IS\=O\=M\=O\=RPH($k$, $S$, $f_{k-1}$) $\equiv$ \\
\>\textbf{if} ($k = |V_1|+1$) \\
\>\>\textbf{return} true \\
\>\textbf{for} each vertex $v \in V_2 - S$ \\
\>\>\textbf{if} (MATCH($k$, $v$)) \\
\>\>\>$f_k = f_{k-1} \union \pair{k}{v}$ \\
\>\>\>ISOMORPH($k+1$, $S \union \{ v \}$, $f_k$)\\
\>\>\textbf{else}\\
\>\>\>\textbf{return} false \\
\\
ISOMORPH($0$, $G_1$, $\emptyset$, $G_2$)
\end{tabbing}

The basic idea of the match operation is to check whether $G_1[k]$ is
isomorphic to $G_2[S \union \{ v \}]$. We already know that $G_1[k-1]
\isomorphic G_2[S]$ with the mapping $f_{k-1}$, so all we need to do
is verify that the edges in $E_1[k] - E_1[k-1]$ connect vertices that
correspond to the vertices connected by the edges in $E_2[S \union \{
v \}] - E_2[S]$. The edges in $E_1[k] - E_1[k-1]$ are all the
out-edges $(k,j)$ and in-edges $(j,k)$ of $k$ where $j$ is less than
or equal to $k$ according to the ordering.  The edges in $E_2[S \union
\{ v \}] - E_2[S]$ consists of all the out-edges $(v,u)$ and
in-edges $(u,v)$ of $v$ where $u \in S$.

\begin{tabbing}
M\=ATCH($k$, $v$) $\equiv$ \\
\>$out \leftarrow \forall (k,j) \in E_1[k] - E_1[k-1] \Big( (v,f(j)) \in E_2[S \union \{ v \}] - E_2[S] \Big)$ \\
\>$in \leftarrow \forall (j,k) \in E_1[k] - E_1[k-1] \Big( (f(j),v) \in E_2[S \union \{ v \}] - E_2[S] \Big)$ \\
\>\textbf{return} $out \Land in$ 
\end{tabbing}

The problem with the exhaustive backtracking algorithm is that there
are $N!$ possible vertex mappings, and $N!$ gets very large as $N$
increases, so we need to prune the search space. We use the pruning
techniques described in
\cite{deo77:_new_algo_digraph_isomorph,fortin96:_isomorph,reingold77:_combin_algo}
that originated in
\cite{sussenguth65:_isomorphism,unger64:_isomorphism}.

\section{Vertex Invariants}
\label{sec:vertex-invariants}

One way to reduce the search space is through the use of \emph{vertex
invariants}. The idea is to compute a number for each vertex $i(v)$
such that $i(v) = i(v')$ if there exists some isomorphism $f$ where
$f(v) = v'$. Then when we look for a match to some vertex $v$, we only
need to consider those vertices that have the same vertex invariant
number. The number of vertices in a graph with the same vertex
invariant number $i$ is called the \emph{invariant multiplicity} for
$i$.  In this implementation, by default we use the out-degree of the
vertex as the vertex invariant, though the user can also supply there
own invariant function. The ability of the invariant function to prune
the search space varies widely with the type of graph.

As a first check to rule out graphs that have no possibility of
matching, one can create a list of computed vertex invariant numbers
for the vertices in each graph, sort the two lists, and then compare
them.  If the two lists are different then the two graphs are not
isomorphic.  If the two lists are the same then the two graphs may be
isomorphic.

Also, we extend the MATCH operation to use the vertex invariants to
help rule out vertices.

\begin{tabbing}
M\=A\=T\=C\=H-INVAR($k$, $v$) $\equiv$ \\
\>$out \leftarrow \forall (k,j) \in E_1[k] - E_1[k-1] \Big( (v,f(j)) \in E_2[S \union \{ v \}] - E_2[S] \Land i(v) = i(k) \Big)$ \\
\>$in \leftarrow \forall (j,k) \in E_1[k] - E_1[k-1] \Big( (f(j),v) \in E_2[S \union \{ v \}] - E_2[S] \Land i(v) = i(k) \Big)$ \\
\>\textbf{return} $out \Land in$ 
\end{tabbing}

\section{Vertex Order}

A good choice of the labeling for the vertices (which determines the
order in which the subgraph $G_1[k]$ is grown) can also reduce the
search space. In the following we discuss two labeling heuristics.

\subsection{Most Constrained First}

Consider the most constrained vertices first.  That is, examine
lower-degree vertices before higher-degree vertices. This reduces the
search space because it chops off a trunk before the trunk has a
chance to blossom out. We can generalize this to use vertex
invariants. We examine vertices with low invariant multiplicity
before examining vertices with high invariant multiplicity.

\subsection{Adjacent First}

The MATCH operation only considers edges when the other vertex already
has a mapping defined. This means that the MATCH operation can only
weed out vertices that are adjacent to vertices that have already been
matched. Therefore, when choosing the next vertex to examine, it is
desirable to choose one that is adjacent a vertex already in $S_1$.

\subsection{DFS Order, Starting with Lowest Multiplicity}

For this implementation, we combine the above two heuristics in the
following way. To implement the ``adjacent first'' heuristic we apply
DFS to the graph, and use the DFS discovery order as our vertex
order. To comply with the ``most constrained first'' heuristic we
order the roots of our DFS trees by invariant multiplicity.


\section{Implementation}

The following is the public interface for the \code{isomorphism}
function. The input to the function is the two graphs $G_1$ and $G_2$,
mappings from the vertices in the graphs to integers (in the range
$[0,|V|)$), and a vertex invariant function object. The output of the
function is an isomorphism $f$ if there is one. The \code{isomorphism}
function returns true if the graphs are isomorphic and false
otherwise. The requirements on type template parameters are described
below in the section ``Concept checking''.

@d Isomorphism Function Interface
@{
template <typename Graph1, typename Graph2, 
          typename IndexMapping, 
          typename VertexInvariant1, typename VertexInvariant2,
          typename IndexMap1, typename IndexMap2>
bool isomorphism(const Graph1& g1, const Graph2& g2, 
                 IndexMapping f, 
                 VertexInvariant1 invariant1, VertexInvariant2 invariant2,
                 IndexMap1 index_map1, IndexMap2 index_map2)
@}

The main outline of the \code{isomorphism} function is as
follows. Most of the steps in this function are for setting up the
vertex ordering, first ordering the vertices by invariant multiplicity
and then by DFS order. The last step is the call to the
\code{isomorph} function which starts the backtracking search.

@d Isomorphism Function Body
@{
{
  @<Some type definitions and iterator declarations@>
  @<Concept checking@>
  @<Quick return with false if $|V_1| \neq |V_2|$@>
  @<Compute vertex invariants@>
  @<Quick return if the graph's invariants do not match@>
  @<Compute invariant multiplicity@>
  @<Sort vertices by invariant multiplicity@>
  @<Order the vertices by DFS discover time@>
  @<Order the edges by DFS discover time@>
  @<Invoke recursive \code{isomorph} function@>
}
@}

There are some types that will be used throughout the function, which
we create shortened names for here. We will also need vertex
iterators for \code{g1} and \code{g2} in several places, so we define
them here.

@d Some type definitions and iterator declarations
@{
typedef typename graph_traits<Graph1>::vertex_descriptor vertex1_t;
typedef typename graph_traits<Graph2>::vertex_descriptor vertex2_t;
typedef typename graph_traits<Graph1>::vertices_size_type size_type;
typename graph_traits<Graph1>::vertex_iterator i1, i1_end;
typename graph_traits<Graph2>::vertex_iterator i2, i2_end;
@}

We use the Boost Concept Checking Library to make sure that the type
arguments to the function fulfill there requirements. The
\code{Graph1} type must be a \bglconcept{VertexListGraph} and a
\bglconcept{EdgeListGraph}. The \code{Graph2} type must be a
\bglconcept{VertexListGraph} and a
\bglconcept{BidirectionalGraph}. The \code{IndexMapping} type that
represents the isomorphism $f$ must be a
\pmconcept{ReadWritePropertyMap} that maps from vertices in $G_1$ to
vertices in $G_2$. The two other index maps are
\pmconcept{ReadablePropertyMap}s from vertices in $G_1$ and $G_2$ to
unsigned integers.

@d Concept checking
@{
// Graph requirements
function_requires< VertexListGraphConcept<Graph1> >();
function_requires< EdgeListGraphConcept<Graph1> >();
function_requires< VertexListGraphConcept<Graph2> >();
function_requires< BidirectionalGraphConcept<Graph2> >();

// Property map requirements
function_requires< ReadWritePropertyMapConcept<IndexMapping, vertex1_t> >();
typedef typename property_traits<IndexMapping>::value_type IndexMappingValue;
BOOST_STATIC_ASSERT((is_same<IndexMappingValue, vertex2_t>::value));

function_requires< ReadablePropertyMapConcept<IndexMap1, vertex1_t> >();
typedef typename property_traits<IndexMap1>::value_type IndexMap1Value;
BOOST_STATIC_ASSERT((is_convertible<IndexMap1Value, size_type>::value));

function_requires< ReadablePropertyMapConcept<IndexMap2, vertex2_t> >();
typedef typename property_traits<IndexMap2>::value_type IndexMap2Value;
BOOST_STATIC_ASSERT((is_convertible<IndexMap2Value, size_type>::value));
@}


\noindent If there are no vertices in either graph, then they are trivially
isomorphic.

@d Quick return with false if $|V_1| \neq |V_2|$
@{
if (num_vertices(g1) != num_vertices(g2))
  return false;
@}


\subsection{Ordering by Vertex Invariant Multiplicity}

The user can supply the vertex invariant functions as a
\stlconcept{AdaptableUnaryFunction} (with the addition of the
\code{max} function) in the \code{invariant1} and \code{invariant2}
parameters. We also define a default which uses the out-degree and
in-degree of a vertex. The following is the definition of the function
object for the default vertex invariant. User-defined vertex invariant
function objects should follow the same pattern.

@d Degree vertex invariant
@{
template <typename InDegreeMap, typename Graph>
class degree_vertex_invariant
{
public:
  typedef typename graph_traits<Graph>::vertex_descriptor argument_type;
  typedef typename graph_traits<Graph>::degree_size_type result_type;

  degree_vertex_invariant(const InDegreeMap& in_degree_map, const Graph& g)
    : m_in_degree_map(in_degree_map), m_g(g) { }

  result_type operator()(argument_type v) const {
    return (num_vertices(m_g) + 1) * out_degree(v, m_g)
      + get(m_in_degree_map, v);
  }
  // The largest possible vertex invariant number
  result_type max() const { 
    return num_vertices(m_g) * num_vertices(m_g) + num_vertices(m_g);
  }
private:
  InDegreeMap m_in_degree_map;
  const Graph& m_g;
};
@}

Since the invariant function may be expensive to compute, we
pre-compute the invariant numbers for every vertex in the two
graphs. The variables \code{invar1} and \code{invar2} are property
maps for accessing the stored invariants, which are described next.

@d Compute vertex invariants
@{
@<Setup storage for vertex invariants@>
for (tie(i1, i1_end) = vertices(g1); i1 != i1_end; ++i1)
  invar1[*i1] = invariant1(*i1);
for (tie(i2, i2_end) = vertices(g2); i2 != i2_end; ++i2)
  invar2[*i2] = invariant2(*i2);
@}

\noindent We store the invariants in two vectors, indexed by the vertex indices
of the two graphs. We then create property maps for accessing these
two vectors in a more convenient fashion (they go directly from vertex
to invariant, instead of vertex to index to invariant).

@d Setup storage for vertex invariants
@{
typedef typename VertexInvariant1::result_type InvarValue1;
typedef typename VertexInvariant2::result_type InvarValue2;
typedef std::vector<InvarValue1> invar_vec1_t;
typedef std::vector<InvarValue2> invar_vec2_t;
invar_vec1_t invar1_vec(num_vertices(g1));
invar_vec2_t invar2_vec(num_vertices(g2));
typedef typename invar_vec1_t::iterator vec1_iter;
typedef typename invar_vec2_t::iterator vec2_iter;
iterator_property_map<vec1_iter, IndexMap1, InvarValue1, InvarValue1&>
  invar1(invar1_vec.begin(), index_map1);
iterator_property_map<vec2_iter, IndexMap2, InvarValue2, InvarValue2&>
  invar2(invar2_vec.begin(), index_map2);
@}

As discussed in \S\ref{sec:vertex-invariants}, we can quickly rule out
the possibility of any isomorphism between two graphs by checking to
see if the vertex invariants can match up. We sort both vectors of vertex
invariants, and then check to see if they are equal.

@d Quick return if the graph's invariants do not match
@{
{ // check if the graph's invariants do not match
  invar_vec1_t invar1_tmp(invar1_vec);
  invar_vec2_t invar2_tmp(invar2_vec);
  std::sort(invar1_tmp.begin(), invar1_tmp.end());
  std::sort(invar2_tmp.begin(), invar2_tmp.end());
  if (! std::equal(invar1_tmp.begin(), invar1_tmp.end(), 
                   invar2_tmp.begin()))
    return false;
}
@}

Next we compute the invariant multiplicity, the number of vertices
with the same invariant number. The \code{invar\_mult} vector is
indexed by invariant number. We loop through all the vertices in the
graph to record the multiplicity.

@d Compute invariant multiplicity
@{
std::vector<std::size_t> invar_mult(invariant1.max(), 0);
for (tie(i1, i1_end) = vertices(g1); i1 != i1_end; ++i1)      
  ++invar_mult[invar1[*i1]];
@}

\noindent We then order the vertices by their invariant multiplicity.
This will allow us to search the more constrained vertices first.
Since we will need to know the permutation from the original order to
the new order, we do not sort the vertices directly. Instead we sort
the vertex indices, creating the \code{perm} array.  Once sorted, this
array provides a mapping from the new index to the old index.
We then use the \code{permute} function to sort the vertices of
the graph, which we store in the \code{g1\_vertices} vector.

@d Sort vertices by invariant multiplicity
@{
std::vector<size_type> perm;
integer_range<size_type> range(0, num_vertices(g1));
std::copy(range.begin(), range.end(), std::back_inserter(perm));
std::sort(perm.begin(), perm.end(),
          detail::compare_invariant_multiplicity(invar1_vec.begin(),
                                                 invar_mult.begin()));

std::vector<vertex1_t> g1_vertices;
for (tie(i1, i1_end) = vertices(g1); i1 != i1_end; ++i1)
  g1_vertices.push_back(*i1);
permute(g1_vertices.begin(), g1_vertices.end(), perm.begin());
@}

\noindent The definition of the \code{compare\_multiplicity} predicate
is shown below. This predicate provides the glue that binds
\code{std::sort} to our current purpose.

@d Compare multiplicity predicate
@{
namespace detail {
  template <typename InvarMap, typename MultMap>
  struct compare_invariant_multiplicity_predicate
  {
    compare_invariant_multiplicity_predicate(InvarMap i, MultMap m)
      : m_invar(i), m_mult(m) { }

    template <typename Vertex>
    bool operator()(const Vertex& x, const Vertex& y) const
      { return m_mult[m_invar[x]] < m_mult[m_invar[y]]; }

    InvarMap m_invar;
    MultMap m_mult;
  };
  template <typename InvarMap, typename MultMap>
  compare_invariant_multiplicity_predicate<InvarMap, MultMap>
  compare_invariant_multiplicity(InvarMap i, MultMap m) {
    return compare_invariant_multiplicity_predicate<InvarMap, MultMap>(i,m);
  }
} // namespace detail
@}


\subsection{Ordering by DFS Discover Time}

To implement the ``visit adjacent vertices first'' heuristic, we order
the vertices according to DFS discover time.  We replace the ordering
in \code{perm} with the new DFS ordering. Again, we use \code{permute}
to sort the vertices of graph \code{g1}.

@d Order the vertices by DFS discover time
@{
{
  perm.clear();
  @<Compute DFS discover times@>
  g1_vertices.clear();
  for (tie(i1, i1_end) = vertices(g1); i1 != i1_end; ++i1)
    g1_vertices.push_back(*i1);
  permute(g1_vertices.begin(), g1_vertices.end(), perm.begin());
}
@}

We implement the outer-loop of the DFS here, instead of calling the
\code{depth\_first\_search} function, because we want the roots of the
DFS tree's to be ordered by invariant multiplicity. We call
\code{depth\_\-first\_\-visit} to implement the recursive portion of
the DFS. The \code{record\_dfs\_order} adapts the DFS to record
the order in which DFS discovers the vertices.

@d Compute DFS discover times
@{
std::vector<default_color_type> color_vec(num_vertices(g1));
for (typename std::vector<vertex1_t>::iterator ui = g1_vertices.begin();
     ui != g1_vertices.end(); ++ui) {
  if (color_vec[get(index_map1, *ui)] 
      == color_traits<default_color_type>::white()) {
    depth_first_visit
      (g1, *ui, detail::record_dfs_order<Graph1, IndexMap1>(perm, 
                                                       index_map1), 
       make_iterator_property_map(&color_vec[0], index_map1, 
                                  color_vec[0]));
  }
}
@}

\noindent The definition of the \code{record\_dfs\_order} visitor
class is as follows. The index of each vertex is recorded in the
\code{dfs\_order} vector (which is the \code{perm} vector) in the
\code{discover\_vertex} event point.

@d Record DFS ordering visitor
@{
namespace detail {
  template <typename Graph1, typename IndexMap1>
  struct record_dfs_order : public default_dfs_visitor {
    typedef typename graph_traits<Graph1>::vertices_size_type size_type;
    typedef typename graph_traits<Graph1>::vertex_descriptor vertex;

    record_dfs_order(std::vector<size_type>& dfs_order, IndexMap1 index) 
      : dfs_order(dfs_order), index(index) { }

    void discover_vertex(vertex v, const Graph1& g) const {
      dfs_order.push_back(get(index, v));
    }
    std::vector<size_type>& dfs_order; 
    IndexMap1 index;
  };
} // namespace detail
@}


In the MATCH operation, we need to examine all the edges in the set
$E_1[k] - E_1[k-1]$. That is, we need to loop through all the edges of
the form $(k,j)$ or $(j,k)$ where $j \leq k$. To do this efficiently,
we create an array of all the edges in $G_1$ that has been sorted so
that $E_1[k] - E_1[k-1]$ forms a contiguous range.  To each edge
$e=(u,v)$ we assign the number $\max(u,v)$, and then sort the edges by
this number. All the edges $(u,v) \in E_1[k] - E_1[k-1]$ can then be
identified because $\max(u,v) = k$. The following code creates an
array of edges and then sorts them. The \code{edge\_\-ordering\_\-fun}
function object is described next.

@d Order the edges by DFS discover time
@{
typedef typename graph_traits<Graph1>::edge_descriptor edge1_t;
std::vector<edge1_t> edge_set;
std::copy(edges(g1).first, edges(g1).second, std::back_inserter(edge_set));

std::sort(edge_set.begin(), edge_set.end(), 
          detail::edge_ordering
          (make_iterator_property_map(perm.begin(), index_map1, perm[0]), g1));
@}

\noindent The \code{edge\_order} function computes the ordering number
for an edge, which for edge $e=(u,v)$ is $\max(u,v)$. The
\code{edge\_\-ordering\_\-fun} function object simply returns
comparison of two edge's ordering numbers.

@d Isomorph edge ordering predicate
@{
namespace detail {

  template <typename VertexIndexMap, typename Graph>
  std::size_t edge_order(const typename graph_traits<Graph>::edge_descriptor e,
                         VertexIndexMap index_map, const Graph& g) {
    return std::max(get(index_map, source(e, g)), get(index_map, target(e, g)));    
  }

  template <typename VertexIndexMap, typename Graph>
  class edge_ordering_fun {
  public:
    edge_ordering_fun(VertexIndexMap vip, const Graph& g)
      : m_index_map(vip), m_g(g) { }
    template <typename Edge>
    bool operator()(const Edge& e1, const Edge& e2) const {
      return edge_order(e1, m_index_map, m_g) < edge_order(e2, m_index_map, m_g);
    }
    VertexIndexMap m_index_map;
    const Graph& m_g;
  };
  template <class VertexIndexMap, class G>
  inline edge_ordering_fun<VertexIndexMap,G>
  edge_ordering(VertexIndexMap vip, const G& g)
  {
    return edge_ordering_fun<VertexIndexMap,G>(vip, g);
  }
} // namespace detail
@}


We are now ready to enter the main part of the algorithm, the
backtracking search implemented by the \code{isomorph} function (which
corresponds to the ISOMORPH algorithm).  The set $S$ is not
represented directly; instead we represent $V_2 - S$.  Initially $S =
\emptyset$ so $V_2 - S = V_2$.  We use the permuted indices for the
vertices of graph \code{g1}. We represent $V_2 - S$ with a bitset.  We
use \code{std::vector} instead of \code{boost::dyn\_bitset} for speed
instead of space.

@d Invoke recursive \code{isomorph} function
@{
std::vector<char> not_in_S_vec(num_vertices(g2), true);
iterator_property_map<char*, IndexMap2, char, char&>
  not_in_S(&not_in_S_vec[0], index_map2);

return detail::isomorph(g1_vertices.begin(), g1_vertices.end(), 
      edge_set.begin(), edge_set.end(), g1, g2,
      make_iterator_property_map(perm.begin(), index_map1, perm[0]),
      index_map2, f, invar1, invar2, not_in_S);
@}


\subsection{Implementation of ISOMORPH}

The ISOMORPH algorithm is implemented with the \code{isomorph}
function. The vertices of $G_1$ are searched in the order specified by
the iterator range \code{[k\_iter,last)}. The function returns true if
a isomorphism is found between the vertices of $G_1$ in
\code{[k\_iter,last)} and the vertices of $G_2$ in \code{not\_in\_S}.
The mapping is recorded in the parameter \code{f}.

@d Signature for the recursive isomorph function
@{
template <class VertexIter, class EdgeIter, class Graph1, class Graph2,
  class IndexMap1, class IndexMap2, class IndexMapping, 
  class Invar1, class Invar2, class Set>
bool isomorph(VertexIter k_iter, VertexIter last,
              EdgeIter edge_iter, EdgeIter edge_iter_end,
              const Graph1& g1, const Graph2& g2,
              IndexMap1 index_map1,
              IndexMap2 index_map2,
              IndexMapping f, Invar1 invar1, Invar2 invar2,
              const Set& not_in_S)
@}

\noindent The steps for this function are as follows.

@d Body of the isomorph function
@{
{
  @<Some typedefs and variable declarations@>
  @<Return true if matching is complete@>
  @<Create a copy of $f_{k-1}$ which will become $f_k$@>
  @<Compute $M$, the potential matches for $k$@>
  @<Invoke isomorph for each vertex in $M$@>
}
@}

\noindent Here we create short names for some often-used types
and declare some variables.

@d Some typedefs and variable declarations
@{
typedef typename graph_traits<Graph1>::vertex_descriptor vertex1_t;
typedef typename graph_traits<Graph2>::vertex_descriptor vertex2_t;
typedef typename graph_traits<Graph1>::vertices_size_type size_type;

vertex1_t k = *k_iter;
@}

\noindent We have completed creating an isomorphism if \code{k\_iter == last}.

@d Return true if matching is complete
@{
if (k_iter == last) 
  return true;
@}


In the pseudo-code for ISOMORPH, we iterate through each vertex in $v
\in V_2 - S$ and check if $k$ and $v$ can match.  A more efficient
approach is to directly iterate through the potential matches for $k$,
for this often is many fewer vertices than $V_2 - S$. Let $M$ be the
set of potential matches for $k$. $M$ consists of all the vertices $v
\in V_2 - S$ such that if $(k,j)$ or $(j,k) \in E_1[k] - E_1[k-1]$
then $(v,f(j)$ or $(f(j),v) \in E_2$ with $i(v) = i(k)$. Note that
this means if there are no edges in $E_1[k] - E_1[k-1]$ then $M = V_2
- S$. In the case where there are edges in $E_1[k] - E_1[k-1]$ we
break the computation of $M$ into two parts, computing $out$ sets
which are vertices that can match according to an out-edge of $k$, and
computing $in$ sets which are vertices that can match according to an
in-edge of $k$.

The implementation consists of a loop through the edges of $E_1[k] -
E_1[k-1]$. The straightforward implementation would initialize $M
\leftarrow V_2 - S$, and then intersect $M$ with the $out$ or $in$ set
for each edge. However, to reduce the cost of the intersection
operation, we start with $M \leftarrow \emptyset$, and on the first
iteration of the loop we do $M \leftarrow out$ or $M \leftarrow in$
instead of an intersection operation.

@d Compute $M$, the potential matches for $k$
@{
std::vector<vertex2_t> potential_matches;
bool some_edges = false;

for (; edge_iter != edge_iter_end; ++edge_iter) {
  if (get(index_map1, k) != edge_order(*edge_iter, index_map1, g1))
    break;      
  if (k == source(*edge_iter, g1)) { // (k,j)
    @<Compute the $out$ set@>
    if (some_edges == false) {
      @<Perform $M \leftarrow out$@>
    } else {
      @<Perform $M \leftarrow M \intersect out$@>
    }
    some_edges = true;
  } else { // (j,k)
    @<Compute the $in$ set@>
    if (some_edges == false) {
      @<Perform $M \leftarrow in$@>
    } else {
      @<Perform $M \leftarrow M \intersect in$@>
    }
    some_edges = true;
  }
  if (potential_matches.empty())
    break;
} // for edge_iter
if (some_edges == false) {
  @<Perform $M \leftarrow V_2 - S$@>
}
@}

To compute the $out$ set, we iterate through the out-edges $(k,j)$ of
$k$, and for each $j$ we iterate through the in-edges $(v,f(j))$ of
$f(j)$, putting all of the $v$'s in $out$ that have the same vertex
invariant as $k$, and which are in $V_2 - S$. Figure~\ref{fig:out}
depicts the computation of the $out$ set. The implementation is as
follows.

@d Compute the $out$ set
@{
vertex1_t j = target(*edge_iter, g1);
std::vector<vertex2_t> out;
typename graph_traits<Graph2>::in_edge_iterator ei, ei_end;
for (tie(ei, ei_end) = in_edges(get(f, j), g2); ei != ei_end; ++ei) {
  vertex2_t v = source(*ei, g2); // (v,f[j])
  if (invar1[k] == invar2[v] && not_in_S[v])
    out.push_back(v);
}
@}

\noindent Here initialize $M$ with the $out$ set. Since we are
representing sets with sorted vectors, we sort \code{out} before
copying to \code{potential\_matches}.

@d Perform $M \leftarrow out$
@{
indirect_cmp<IndexMap2,std::less<std::size_t> > cmp(index_map2);
std::sort(out.begin(), out.end(), cmp);
std::copy(out.begin(), out.end(), std::back_inserter(potential_matches));
@}

\noindent We use \code{std::set\_intersection} to implement $M
\leftarrow M \intersect out$. Since there is no version of
\code{std::set\_intersection} that works in-place, we create a
temporary for the result and then swap.

@d Perform $M \leftarrow M \intersect out$
@{
indirect_cmp<IndexMap2,std::less<std::size_t> > cmp(index_map2);
std::sort(out.begin(), out.end(), cmp);
std::vector<vertex2_t> tmp_matches;
std::set_intersection(out.begin(), out.end(),
                      potential_matches.begin(), potential_matches.end(),
                      std::back_inserter(tmp_matches), cmp);
std::swap(potential_matches, tmp_matches);
@}

% Shoot, there is some problem with f(j). Could have to do with the
% change from the edge set to just using out_edges and in_edges.
% Yes, have to visit edges in correct order to we don't hit
% part of f that is not yet defined.

\vizfig{out}{Computing the $out$ set.}

@c out.dot
@{
digraph G {
  node[shape=circle]
  size="4,2"
  ratio="fill"

  subgraph cluster0 { label="G_1"
    k -> j_1 
    k -> j_2 
    k -> j_3
  }

  subgraph cluster1 { label="G_2"

    subgraph cluster2 { label="out" v_1 v_2 v_3 v_4 v_5 v_6 }

    v_1 -> fj_1
    v_2 -> fj_1
    v_3 -> fj_1

    v_4 -> fj_2

    v_5 -> fj_3
    v_6 -> fj_3

    fj_1[label="f(j_1)"]
    fj_2[label="f(j_2)"]
    fj_3[label="f(j_3)"]
  }

  j_1 -> fj_1[style=dotted]
  j_2 -> fj_2[style=dotted]
  j_3 -> fj_3[style=dotted]
}
@}

The $in$ set is is constructed by iterating through the in-edges
$(j,k)$ of $k$, and for each $j$ we iterate through the out-edges
$(f(j),v)$ of $f(j)$. We put all of the $v$'s in $in$ that have the
same vertex invariant as $k$, and which are in $V_2 -
S$. Figure~\ref{fig:in} depicts the computation of the $in$ set.  The
following code computes the $in$ set.

@d Compute the $in$ set
@{
vertex1_t j = source(*edge_iter, g1);
std::vector<vertex2_t> in;
typename graph_traits<Graph2>::out_edge_iterator ei, ei_end;
for (tie(ei, ei_end) = out_edges(get(f, j), g2); ei != ei_end; ++ei) {
  vertex2_t v = target(*ei, g2); // (f[j],v)
  if (invar1[k] == invar2[v] && not_in_S[v])
    in.push_back(v);
}
@}

\noindent Here initialize $M$ with the $in$ set. Since we are
representing sets with sorted vectors, we sort \code{in} before
copying to \code{potential\_matches}.

@d Perform $M \leftarrow in$
@{
indirect_cmp<IndexMap2,std::less<std::size_t> > cmp(index_map2);
std::sort(in.begin(), in.end(), cmp);
std::copy(in.begin(), in.end(), std::back_inserter(potential_matches));
@}

\noindent Again we use \code{std::set\_intersection} on
sorted vectors to implement $M \leftarrow M \intersect in$.

@d Perform $M \leftarrow M \intersect in$
@{
indirect_cmp<IndexMap2, std::less<std::size_t> > cmp(index_map2);
std::sort(in.begin(), in.end(), cmp);
std::vector<vertex2_t> tmp_matches;
std::set_intersection(in.begin(), in.end(),
                      potential_matches.begin(), potential_matches.end(),
                      std::back_inserter(tmp_matches), cmp);
std::swap(potential_matches, tmp_matches);
@}

\vizfig{in}{Computing the $in$ set.}

@c in.dot
@{
digraph G {
  node[shape=circle]
  size="3,2"
  ratio="fill"
  subgraph cluster0 { label="G1"
    j_1 -> k
    j_2 -> k
  }

  subgraph cluster1 { label="G2"

    subgraph cluster2 { label="in" v_1 v_2 v_3 }

    v_1 -> fj_1
    v_2 -> fj_1

    v_3 -> fj_2

    fj_1[label="f(j_1)"]
    fj_2[label="f(j_2)"]
  }

  j_1 -> fj_1[style=dotted]
  j_2 -> fj_2[style=dotted]

}
@}

In the case where there were no edges in $E_1[k] - E_1[k-1]$, then $M
= V_2 - S$, so here we insert all the vertices from $V_2$ that are not
in $S$.

@d Perform $M \leftarrow V_2 - S$
@{
typename graph_traits<Graph2>::vertex_iterator vi, vi_end;
for (tie(vi, vi_end) = vertices(g2); vi != vi_end; ++vi)
  if (not_in_S[*vi])
    potential_matches.push_back(*vi);
@}

For each vertex $v$ in the potential matches $M$, we will create an
extended isomorphism $f_k = f_{k-1} \union \pair{k}{v}$. First
we create a local copy of $f_{k-1}$.

@d Create a copy of $f_{k-1}$ which will become $f_k$
@{
std::vector<vertex2_t> my_f_vec(num_vertices(g1));
typedef typename std::vector<vertex2_t>::iterator vec_iter;
iterator_property_map<vec_iter,  IndexMap1, vertex2_t, vertex2_t&>
  my_f(my_f_vec.begin(), index_map1);

typename graph_traits<Graph1>::vertex_iterator i1, i1_end;
for (tie(i1, i1_end) = vertices(g1); i1 != i1_end; ++i1)
  my_f[*i1] = get(f, *i1);
@}

Next we enter the loop through every vertex $v$ in $M$, and extend the
isomorphism with $\pair{k}{v}$. We then update the set $S$ (by
removing $v$ from $V_2 - S$) and make the recursive call to
\code{isomorph}. If \code{isomorph} returns successfully, we have
found an isomorphism for the complete graph, so we copy our local
mapping into the mapping from the previous calling function.

@d Invoke isomorph for each vertex in $M$
@{
for (std::size_t j = 0; j < potential_matches.size(); ++j) {
  my_f[k] = potential_matches[j];
  @<Perform $S' = S - \{ v \}$@>
  if (isomorph(boost::next(k_iter), last, edge_iter, edge_iter_end, g1, g2, 
               index_map1, index_map2, 
               my_f, invar1, invar2, my_not_in_S)) {
    for (tie(i1, i1_end) = vertices(g1); i1 != i1_end; ++i1)
      put(f, *i1, my_f[*i1]);
    return true;
  }
}
return false;
@}

We need to create the new set $S' = S - \{ v \}$, which will be the
$S$ for the next invocation to \code{isomorph}. As before, we
represent $V_2 - S'$ instead of $S'$ and use a bitset.

@d Perform $S' = S - \{ v \}$
@{
std::vector<char> my_not_in_S_vec(num_vertices(g2));
iterator_property_map<char*, IndexMap2, char, char&>
  my_not_in_S(&my_not_in_S_vec[0], index_map2);
typename graph_traits<Graph2>::vertex_iterator vi, vi_end;
for (tie(vi, vi_end) = vertices(g2); vi != vi_end; ++vi)
  my_not_in_S[*vi] = not_in_S[*vi];;
my_not_in_S[potential_matches[j]] = false;
@}


\section{Appendix}

Here we output the header file \code{isomorphism.hpp}. We add a
copyright statement, include some files, and then pull the top-level
code parts into namespace \code{boost}.

@o isomorphism.hpp -d
@{

// (C) Copyright Jeremy Siek 2001. Permission to copy, use, modify,
// sell and distribute this software is granted provided this
// copyright notice appears in all copies. This software is provided
// "as is" without express or implied warranty, and with no claim as
// to its suitability for any purpose.

// See http://www.boost.org/libs/graph/doc/isomorphism-impl.pdf 
// for a description of the implementation of the isomorphism function
// defined in this header file.

#ifndef BOOST_GRAPH_ISOMORPHISM_HPP
#define BOOST_GRAPH_ISOMORPHISM_HPP

#include <algorithm>
#include <boost/graph/detail/set_adaptor.hpp>
#include <boost/pending/indirect_cmp.hpp>
#include <boost/graph/detail/permutation.hpp>
#include <boost/graph/named_function_params.hpp>
#include <boost/graph/graph_concepts.hpp>
#include <boost/property_map/property_map.hpp>
#include <boost/pending/integer_range.hpp>
#include <boost/limits.hpp>
#include <boost/static_assert.hpp>
#include <boost/graph/depth_first_search.hpp>

namespace boost {

  @<Degree vertex invariant@>

  namespace detail {
    @<Signature for the recursive isomorph function@>
    @<Body of the isomorph function@>
  } // namespace detail

  @<Record DFS ordering visitor@>
  @<Compare multiplicity predicate@>
  @<Isomorph edge ordering predicate@>

  @<Isomorphism Function Interface@>
  @<Isomorphism Function Body@>

  namespace detail {
    // Should move this, make is public
    template <typename Graph, typename InDegreeMap, typename Cat>
    void compute_in_degree(const Graph& g, const InDegreeMap& in_degree_map,
                           Cat)
    {
      typename graph_traits<Graph>::vertex_iterator vi, vi_end;
      typename graph_traits<Graph>::out_edge_iterator ei, ei_end;
      for (tie(vi, vi_end) = vertices(g); vi != vi_end; ++vi)
        for (tie(ei, ei_end) = out_edges(*vi, g); ei != ei_end; ++ei) {
          typename graph_traits<Graph>::vertex_descriptor v = target(*ei, g);
          put(in_degree_map, v, get(in_degree_map, v) + 1);
        }
    }
    template <typename Graph, typename InDegreeMap>
    void compute_in_degree(const Graph& g, const InDegreeMap& in_degree_map,
                           edge_list_graph_tag)
    {
      typename graph_traits<Graph>::edge_iterator ei, ei_end;
      for (tie(ei, ei_end) = edges(g); ei != ei_end; ++ei) {
        typename graph_traits<Graph>::vertex_descriptor v = target(*ei, g);
        put(in_degree_map, v, get(in_degree_map, v) + 1);
      }
    }
    template <typename Graph, typename InDegreeMap>
    void compute_in_degree(const Graph& g, const InDegreeMap& in_degree_map)
    {
      typename graph_traits<Graph>::traversal_category cat;
      compute_in_degree(g, in_degree_map, cat);
    }


    template <typename Graph1, typename Graph2, 
              typename IndexMapping, typename IndexMap1, typename IndexMap2,
              typename P, typename T, typename R>
    bool isomorphism_impl(const Graph1& g1, const Graph2& g2, 
                          IndexMapping f, 
                          IndexMap1 index_map1, IndexMap2 index_map2,
                          const bgl_named_params<P,T,R>& params)
    {
      typedef typename graph_traits<Graph1>::vertices_size_type size_type;

      // Compute the in-degrees
      std::vector<size_type> in_degree_vec1(num_vertices(g1), 0);
      typedef iterator_property_map<size_type*, IndexMap1, 
         size_type, size_type&> InDegreeMap1;
      InDegreeMap1 in_degree_map1(&in_degree_vec1[0], index_map1);
      detail::compute_in_degree(g1, in_degree_map1);
      degree_vertex_invariant<InDegreeMap1, Graph1> 
        default_invar1(in_degree_map1, g1);

      std::vector<size_type> in_degree_vec2(num_vertices(g2), 0);
      typedef iterator_property_map<size_type*, IndexMap2, 
         size_type, size_type&> InDegreeMap2;
      InDegreeMap2 in_degree_map2(&in_degree_vec2[0], index_map2);
      detail::compute_in_degree(g2, in_degree_map2);
      degree_vertex_invariant<InDegreeMap2, Graph2>
         default_invar2(in_degree_map2, g2);

      return isomorphism(g1, g2, f, 
        choose_param(get_param(params, vertex_invariant_t()), default_invar1),
        choose_param(get_param(params, vertex_invariant_t()), default_invar2),
        index_map1, index_map2);
    }

  } // namespace detail

  // Named parameter interface
  template <typename Graph1, typename Graph2, class P, class T, class R>
  bool isomorphism(const Graph1& g1,
                   const Graph2& g2,
                   const bgl_named_params<P,T,R>& params)
  {
    typedef typename graph_traits<Graph2>::vertex_descriptor vertex2_t;
    typename std::vector<vertex2_t>::size_type
      n = is_default_param(get_param(params, vertex_isomorphism_t()))
        ? num_vertices(g1) : 1;
    std::vector<vertex2_t> f(n);
    vertex2_t x;
    return detail::isomorphism_impl
      (g1, g2, 
       choose_param(get_param(params, vertex_isomorphism_t()),
          make_iterator_property_map(f.begin(), 
            choose_const_pmap(get_param(params, vertex_index1),
                        g1, vertex_index), x)),
       choose_const_pmap(get_param(params, vertex_index1),
                     g1, vertex_index),
       choose_const_pmap(get_param(params, vertex_index2),
                     g2, vertex_index),
       params);
  }

  // All defaults interface
  template <typename Graph1, typename Graph2>
  bool isomorphism(const Graph1& g1, const Graph2& g2)
  {
    typedef typename graph_traits<Graph1>::vertices_size_type size_type;
    typedef typename graph_traits<Graph2>::vertex_descriptor vertex2_t;
    std::vector<vertex2_t> f(num_vertices(g1));

    // Compute the in-degrees
    std::vector<size_type> in_degree_vec1(num_vertices(g1), 0);
    typedef typename property_map<Graph1,vertex_index_t>::const_type IndexMap1;
    typedef iterator_property_map<size_type*, IndexMap1, 
       size_type, size_type&> InDegreeMap1;
    InDegreeMap1 in_degree_map1(&in_degree_vec1[0], get(vertex_index, g1));
    detail::compute_in_degree(g1, in_degree_map1);
    degree_vertex_invariant<InDegreeMap1, Graph1>
      invariant1(in_degree_map, g1);

    std::vector<size_type> in_degree_vec2(num_vertices(g2), 0);
    typedef typename property_map<Graph2,vertex_index_t>::const_type IndexMap2;
    typedef iterator_property_map<size_type*, IndexMap2, 
       size_type, size_type&> InDegreeMap2;
    InDegreeMap2 in_degree_map2(&in_degree_vec2[0], get(vertex_index, g2));
    detail::compute_in_degree(g2, in_degree_map2);
    degree_vertex_invariant<InDegreeMap2, Graph2> 
      invariant2(in_degree_map, g2);

    return isomorphism
      (g1, g2, make_iterator_property_map(f.begin(), get(vertex_index, g1), vertex2_t()),
       invariant1, invariant2, get(vertex_index, g1), get(vertex_index, g2));
  }

  // Verify that the given mapping iso_map from the vertices of g1 to the
  // vertices of g2 describes an isomorphism.
  // Note: this could be made much faster by specializing based on the graph
  // concepts modeled, but since we're verifying an O(n^(lg n)) algorithm,
  // O(n^4) won't hurt us.
  template<typename Graph1, typename Graph2, typename IsoMap>
  inline bool verify_isomorphism(const Graph1& g1, const Graph2& g2, 
				 IsoMap iso_map)
  {
    if (num_vertices(g1) != num_vertices(g2) || num_edges(g1) != num_edges(g2))
      return false;

    for (typename graph_traits<Graph1>::edge_iterator e1 = edges(g1).first;
	 e1 != edges(g1).second; ++e1) {
      bool found_edge = false;
      for (typename graph_traits<Graph2>::edge_iterator e2 = edges(g2).first;
	   e2 != edges(g2).second && !found_edge; ++e2) {
	if (source(*e2, g2) == get(iso_map, source(*e1, g1)) &&
	    target(*e2, g2) == get(iso_map, target(*e1, g1))) {
	  found_edge = true;
	}
      }

      if (!found_edge)
	return false;
    }

    return true;
  }

} // namespace boost

#endif // BOOST_GRAPH_ISOMORPHISM_HPP
@}

\bibliographystyle{abbrv}
\bibliography{ggcl}

\end{document}
% LocalWords:  Isomorphism Siek isomorphism adjacency subgraph subgraphs OM DFS
% LocalWords:  ISOMORPH Invariants invariants typename IndexMapping bool const
% LocalWords:  VertexInvariant VertexIndexMap iterator typedef VertexG Idx num
% LocalWords:  InvarValue struct invar vec iter tmp_matches mult inserter permute ui
% LocalWords:  dfs cmp isomorph VertexIter EdgeIter IndexMap desc RPH ATCH pre

% LocalWords:  iterators VertexListGraph EdgeListGraph BidirectionalGraph tmp
% LocalWords:  ReadWritePropertyMap VertexListGraphConcept EdgeListGraphConcept
% LocalWords:  BidirectionalGraphConcept ReadWritePropertyMapConcept indices ei
% LocalWords:  IndexMappingValue ReadablePropertyMapConcept namespace InvarMap
% LocalWords:  MultMap vip inline bitset typedefs fj hpp ifndef adaptor params
% LocalWords:  bgl param pmap endif