McGregor Common Subgraphs

Finds all common subgraphs between two graphs using McGregor’s backtracking algorithm.

Complexity: O(V1 * V2)
Defined in: <boost/graph/mcgregor_common_subgraphs.hpp>

Algorithmic semantics

These functions compute the induced common subgraph variant: for every pair of mapped vertices, an edge must be present in both graphs, or absent in both. Non-edges are part of the structural match, not just edges.

This is stricter than the non-induced (partial) common subgraph problem usually associated with McGregor’s 1982 paper, where only present edges need to agree and additional edges in either graph are ignored. BGL does not currently provide the non-induced variant.

The historical name is preserved because the underlying backtracking technique is McGregor’s, even though the problem solved here is the induced variant that became standard in later literature.

Example

#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/mcgregor_common_subgraphs.hpp>
#include <iostream>

using namespace boost;
using Graph = adjacency_list<vecS, vecS, undirectedS>;
using Size = graph_traits<Graph>::vertices_size_type;

struct max_callback {
    Size largest = 0;
    template <typename Map1, typename Map2>
    bool operator()(Map1, Map2, Size subgraph_size) {
        if (subgraph_size > largest) largest = subgraph_size;
        return true;
    }
};

int main() {
    Graph g1(4);
    add_edge(0, 1, g1); add_edge(1, 2, g1);
    add_edge(2, 3, g1); add_edge(3, 0, g1);

    Graph g2(5);
    add_edge(0, 1, g2); add_edge(1, 2, g2);
    add_edge(2, 3, g2); add_edge(3, 4, g2); add_edge(4, 0, g2);

    max_callback cb;
    mcgregor_common_subgraphs(g1, g2, true, std::ref(cb));
    std::cout << "Largest common subgraph size: " << cb.largest << "\n";
}

Largest common subgraph size: 3

mcgregor_common_subgraphs

// named parameter version
template <typename GraphFirst,
          typename GraphSecond,
          typename SubGraphCallback,
          typename Param,
          typename Tag,
          typename Rest>
void mcgregor_common_subgraphs(
    const GraphFirst& graph1,
    const GraphSecond& graph2,
    SubGraphCallback user_callback,
    bool only_connected_subgraphs,
    const bgl_named_params<Param, Tag, Rest>& params);

// non-named parameter version
template <typename GraphFirst,
          typename GraphSecond,
          typename VertexIndexMapFirst,
          typename VertexIndexMapSecond,
          typename EdgeEquivalencePredicate,
          typename VertexEquivalencePredicate,
          typename SubGraphCallback>
void mcgregor_common_subgraphs(
    const GraphFirst& graph1,
    const GraphSecond& graph2,
    const VertexIndexMapFirst vindex_map1,
    const VertexIndexMapSecond vindex_map2,
    EdgeEquivalencePredicate edges_equivalent,
    VertexEquivalencePredicate vertices_equivalent,
    bool only_connected_subgraphs,
    SubGraphCallback user_callback);

Direction Parameter Description

Direction	Parameter	Description
IN	`const GraphFirst& graph1`	The first graph of the pair to be searched. The type `GraphFirst` must be a model of Vertex List Graph and Incidence Graph. All parameters with a “1” (i.e. `vindex_map1`, `correspondence_map_1_to_2`) use this graph’s vertices as keys.
IN	`const GraphSecond& graph2`	The second graph of the pair to be searched. The type `GraphSecond` must be a model of Vertex List Graph and Incidence Graph. All parameters with a “2” (i.e. `vindex_map2`, `correspondence_map_2_to_1`) use this graph’s vertices as keys.
IN	`bool only_connected_subgraphs`	If `true`, subgraphs are expanded only when matching vertices have at least one edge connecting them to the existing subgraph.
OUT	`SubGraphCallback user_callback`	A function object that will be invoked when a subgraph has been discovered. See Callback Requirements for details.

const GraphFirst& graph1

The first graph of the pair to be searched. The type GraphFirst must be a model of Vertex List Graph and Incidence Graph. All parameters with a “1” (i.e. vindex_map1, correspondence_map_1_to_2) use this graph’s vertices as keys.

const GraphSecond& graph2

The second graph of the pair to be searched. The type GraphSecond must be a model of Vertex List Graph and Incidence Graph. All parameters with a “2” (i.e. vindex_map2, correspondence_map_2_to_1) use this graph’s vertices as keys.

bool only_connected_subgraphs

If true, subgraphs are expanded only when matching vertices have at least one edge connecting them to the existing subgraph.

OUT

SubGraphCallback user_callback

A function object that will be invoked when a subgraph has been discovered. See Callback Requirements for details.

Named parameters (named parameter version only):

Direction Named Parameter Description / Default

Direction	Named Parameter	Description / Default
IN	`vertex_index1(VertexIndexMapFirst vindex_map1)`	Maps each vertex to an integer in the range `[0, num_vertices(graph1)]`. This is necessary for efficient storage of the subgraphs. The type `VertexIndexMapFirst` must be a model of Readable Property Map. Default: `get(vertex_index, graph1)`
IN	`vertex_index2(VertexIndexMapSecond vindex_map2)`	Maps each vertex to an integer in the range `[0, num_vertices(graph2)]`. This is necessary for efficient storage of the subgraphs. The type `VertexIndexMapSecond` must be a model of Readable Property Map. Default: `get(vertex_index, graph2)`
IN	`edges_equivalent(EdgeEquivalencePredicate edges_equivalent)`	Used to determine if edges between `graph1` and `graph2` are equivalent. The `EdgeEquivalencePredicate` type must be a model of Binary Predicate and have argument types of `graph_traits<GraphFirst>::edge_descriptor` and `graph_traits<GraphSecond>::edge_descriptor`. A return value of `true` indicates that the edges are equivalent. Default: `always_equivalent`
IN	`vertices_equivalent(VertexEquivalencePredicate vertices_equivalent)`	Used to determine if vertices between `graph1` and `graph2` are equivalent. The `VertexEquivalencePredicate` type must be a model of Binary Predicate and have argument types of `graph_traits<GraphFirst>::vertex_descriptor` and `graph_traits<GraphSecond>::vertex_descriptor`. A return value of `true` indicates that the vertices are equivalent. Default: `always_equivalent`

vertex_index1(VertexIndexMapFirst vindex_map1)

Maps each vertex to an integer in the range [0, num_vertices(graph1)]. This is necessary for efficient storage of the subgraphs. The type VertexIndexMapFirst must be a model of Readable Property Map.
Default: get(vertex_index, graph1)

vertex_index2(VertexIndexMapSecond vindex_map2)

Maps each vertex to an integer in the range [0, num_vertices(graph2)]. This is necessary for efficient storage of the subgraphs. The type VertexIndexMapSecond must be a model of Readable Property Map.
Default: get(vertex_index, graph2)

edges_equivalent(EdgeEquivalencePredicate edges_equivalent)

Used to determine if edges between graph1 and graph2 are equivalent. The EdgeEquivalencePredicate type must be a model of Binary Predicate and have argument types of graph_traits<GraphFirst>::edge_descriptor and graph_traits<GraphSecond>::edge_descriptor. A return value of true indicates that the edges are equivalent.
Default: always_equivalent

vertices_equivalent(VertexEquivalencePredicate vertices_equivalent)

Used to determine if vertices between graph1 and graph2 are equivalent. The VertexEquivalencePredicate type must be a model of Binary Predicate and have argument types of graph_traits<GraphFirst>::vertex_descriptor and graph_traits<GraphSecond>::vertex_descriptor. A return value of true indicates that the vertices are equivalent.
Default: always_equivalent

mcgregor_common_subgraphs_unique

// Takes the same parameters as mcgregor_common_subgraphs
template <typename GraphFirst,
          typename GraphSecond,
          typename SubGraphCallback,
          typename Param,
          typename Tag,
          typename Rest>
void mcgregor_common_subgraphs_unique(
    const GraphFirst& graph1,
    const GraphSecond& graph2,
    SubGraphCallback user_callback,
    bool only_connected_subgraphs,
    const bgl_named_params<Param, Tag, Rest>& params);

Keeps an internal cache of all discovered subgraphs and only invokes the user_callback for unique subgraphs. Returning false from user_callback will terminate the search as expected.

Parameters are the same as mcgregor_common_subgraphs.

mcgregor_common_subgraphs_maximum

// Takes the same parameters as mcgregor_common_subgraphs
template <typename GraphFirst,
          typename GraphSecond,
          typename SubGraphCallback,
          typename Param,
          typename Tag,
          typename Rest>
void mcgregor_common_subgraphs_maximum(
    const GraphFirst& graph1,
    const GraphSecond& graph2,
    SubGraphCallback user_callback,
    bool only_connected_subgraphs,
    const bgl_named_params<Param, Tag, Rest>& params);

Explores the entire search space and invokes the user_callback afterward with each of the largest discovered subgraphs. Returning false from the user_callback will not terminate the search because it is invoked after the search has been completed.

Parameters are the same as mcgregor_common_subgraphs.

mcgregor_common_subgraphs_maximum_unique

// Takes the same parameters as mcgregor_common_subgraphs
template <typename GraphFirst,
          typename GraphSecond,
          typename SubGraphCallback,
          typename Param,
          typename Tag,
          typename Rest>
void mcgregor_common_subgraphs_maximum_unique(
    const GraphFirst& graph1,
    const GraphSecond& graph2,
    SubGraphCallback user_callback,
    bool only_connected_subgraphs,
    const bgl_named_params<Param, Tag, Rest>& params);

Explores the entire search space and invokes the user_callback afterward with each of the largest, unique discovered subgraphs. Returning false from the user_callback will not terminate the search because it is invoked after the search has been completed.

Parameters are the same as mcgregor_common_subgraphs.

Utility Functions and Structs

make_property_map_equivalent

template <typename PropertyMapFirst,
          typename PropertyMapSecond>
property_map_equivalent<PropertyMapFirst, PropertyMapSecond>
make_property_map_equivalent(
    const PropertyMapFirst property_map1,
    const PropertyMapSecond property_map2);

Returns a binary predicate function object (property_map_equivalent<PropertyMapFirst, PropertyMapSecond>) that compares vertices or edges between graphs using property maps. If, for example, vertex1 and vertex2 are the two parameters given when the function object is invoked, the operator() is effectively: return (get(m_property_map1, vertex1) == get(m_property_map2, vertex2));

always_equivalent

struct always_equivalent;

A binary function object that returns true for any pair of items.

fill_membership_map

template <typename GraphSecond,
          typename GraphFirst,
          typename CorrespondenceMapFirstToSecond,
          typename MembershipMapFirst>
void fill_membership_map(
    const GraphFirst& graph1,
    const CorrespondenceMapFirstToSecond correspondence_map_1_to_2,
    MembershipMapFirst membership_map1);

Takes a subgraph (represented as a correspondence map) and fills the vertex membership map (vertex → bool) (true means the vertex is present in the subgraph). MembershipMapFirst must model Writable Property Map.

make_membership_filtered_graph

template <typename Graph,
          typename MembershipMap>
typename membership_filtered_graph_traits<Graph, MembershipMap>::graph_type
make_membership_filtered_graph(
    const Graph& graph,
    MembershipMap& membership_map);

Creates a Filtered Graph from a subgraph, represented here as a vertex membership map (vertex → bool where a value of true means that the vertex is present in the subgraph). All edges between the included vertices are preserved. See the Writing a callback section for details.

Description

This algorithm finds all common subgraphs between graph1 and graph2 and outputs them to user_callback. The edges_equivalent and vertices_equivalent predicates are used to determine edges and vertex equivalence between the two graphs. To use property maps for equivalence, look at the make_property_map_equivalent function. By default, always_equivalent is used, which returns true for any pair of edges or vertices.

McGregor’s algorithm does a depth-first search on the space of possible common subgraphs. At each level, every unvisited pair of vertices from graph1 and graph2 are checked to see if they can extend the current subgraph. This is done in three steps (assume vertex1 is from graph1 and vertex2 is from graph2):

Verify that vertex1 and vertex2 are equivalent using the vertices_equivalent predicate.
For every vertex pair (existing_vertex1, existing_vertex2) in the current subgraph, ensure that any edges between vertex1 and existing_vertex1 in graph1 and between vertex2 and existing_vertex2 in graph2 match (i.e. either both exist or both don’t exist). If both edges exist, they are checked for equivalence using the edges_equivalent predicate.
Lastly (and optionally), make sure that the new subgraph vertex has at least one edge connecting it to the existing subgraph. When the only_connected_subgraphs parameter is false, this step is skipped.

All functions are defined in the boost namespace.

Callback Requirements

The user_callback function object’s operator() must have the following form:

template <typename CorrespondenceMapFirstToSecond,
          typename CorrespondenceMapSecondToFirst>
bool operator()(CorrespondenceMapFirstToSecond correspondence_map_1_to_2,
                CorrespondenceMapSecondToFirst correspondence_map_2_to_1,
                typename graph_traits<GraphFirst>::vertices_size_type subgraph_size);

Both the CorrespondenceMapFirstToSecond and CorrespondenceMapSecondToFirst types are models of Readable Property Map and map equivalent vertices across the two graphs given to mcgregor_common_subgraphs. For example, if vertex1 is from graph1, vertex2 is from graph2, and the vertices can be considered equivalent in the subgraph, then get(correspondence_map_1_to_2, vertex1) will be vertex2 and get(correspondence_map_2_to_1, vertex2) will be vertex1. If any vertex, say vertex1 in graph1, doesn’t match a vertex in the other graph (graph2 in this example), then get(correspondence_map_1_to_2, vertex1) will be graph_traits<GraphSecond>::null_vertex(). Likewise for any un-matched vertices from graph2, get(correspondence_map_2_to_1, vertex2) will be graph_traits<GraphFirst>::null_vertex().

The subgraph_size parameter is the number of vertices in the subgraph, or the number of matched vertex pairs contained in the correspondence maps. This can be used to quickly filter out subgraphs whose sizes do not fall within the desired range.

Returning false from the callback will abort the search immediately. Otherwise, the entire search space will be explored [1].

Writing a callback

Before calling any of the mcgregor_common_subgraphs algorithms, you must create a function object to act as a callback:

template <typename GraphFirst,
          typename GraphSecond>
struct print_callback {

  print_callback(const GraphFirst& graph1,
                 const GraphSecond& graph2) :
    m_graph1(graph1), m_graph2(graph2) { }

  template <typename CorrespondenceMapFirstToSecond,
            typename CorrespondenceMapSecondToFirst>
  bool operator()(CorrespondenceMapFirstToSecond correspondence_map_1_to_2,
                  CorrespondenceMapSecondToFirst correspondence_map_2_to_1,
                  typename graph_traits<GraphFirst>::vertices_size_type subgraph_size) {

    // Print out correspondences between vertices
    BGL_FORALL_VERTICES_T(vertex1, m_graph1, GraphFirst) {

      // Skip unmapped vertices
      if (get(correspondence_map_1_to_2, vertex1) != graph_traits<GraphSecond>::null_vertex()) {
        std::cout << vertex1 << " <-> " << get(correspondence_map_1_to_2, vertex1) << std::endl;
      }

    }

    std::cout << "---" << std::endl;

    return (true);
  }

  private:
    const GraphFirst& m_graph1;
    const GraphSecond& m_graph2;

};

// Assuming the graph types GraphFirst and GraphSecond have already been defined
GraphFirst graph1;
GraphSecond graph2;

print_callback<GraphFirst, GraphSecond> my_callback(graph1, graph2);

Example 1

If all the vertices and edges in your graph are identical, you can start enumerating subgraphs immediately:

// Print out all connected common subgraphs between graph1 and graph2.
// All vertices and edges are assumed to be equivalent and both graph1 and graph2
// have implicit vertex index properties.
mcgregor_common_subgraphs(graph1, graph2, true, my_callback);

Example 2

If the vertices and edges of your graphs can be differentiated with property maps, you can use the make_property_map_equivalent function to create a binary predicate for vertex or edge equivalence:

// Assume both graphs were defined with implicit vertex name,
// edge name, and vertex index properties
property_map<GraphFirst, vertex_name_t> vname_map1 = get(vertex_name, graph1);
property_map<GraphSecond, vertex_name_t> vname_map2 = get(vertex_name, graph2);

property_map<GraphFirst, edge_name_t> ename_map1 = get(edge_name, graph1);
property_map<GraphSecond, edge_name_t> ename_map2 = get(edge_name, graph2);

// Print out all connected common subgraphs between graph1 and graph2.
mcgregor_common_subgraphs(graph1, graph2, true, my_callback,
  edges_equivalent(make_property_map_equivalent(ename_map1, ename_map2)).
  vertices_equivalent(make_property_map_equivalent(vname_map1, vname_map2)));

Example 3

There are some helper functions that can be used to obtain a filtered graph from the correspondence maps given in your callback:

// Assuming we're inside the operator() of the callback with a member
// variable m_graph1 representing the first graph passed to
// mcgregor_common_subgraphs.
// ...

// Step 1: Transform a correspondence map into a membership map.
// Any vertex -> bool property map will work
typedef shared_array_property_map<bool, VertexIndexMap> MembershipMap;
MembershipMap membership_map1(num_vertices(m_graph1), get(vertex_index, m_graph1));

// Fill the membership map for m_graph1. GraphSecond is the type of the
// second graph given to mcgregor_common_subgraphs.
fill_membership_map<GraphSecond>(m_graph1, correspondence_map_1_to_2, membership_map1);

// Step 2: Use the membership map from Step 1 to obtain a filtered graph
typedef typename membership_filtered_graph_traits<GraphFirst, MembershipMap>::graph_type
  MembershipFilteredGraph;

MembershipFilteredGraph subgraph1 = make_membership_filtered_graph(m_graph1, membership_map1);

// The filtered graph can be used like a regular BGL graph...
BGL_FORALL_VERTICES_T(vertex1, subgraph1, MembershipFilteredGraph) {
  std::cout << vertex1 << " is present in the subgraph of graph1" << std::endl;
}

Notes

[1] For mcgregor_common_subgraphs_maximum and mcgregor_common_subgraphs_maximum_unique the entire search space is always explored, so the return value of the callback has no effect.

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