# Six Degrees of Kevin Bacon

A fun application of graph theory comes up in the popular game ``Six Degrees of Kevin Bacon''. The idea of the game is to connect an actor to Kevin Bacon through a trail of actors who appeared together in movies, and do so in less than six steps. For example, Theodore Hesburgh (President Emeritus of the University of Notre Dame) was in the movie ``Rudy'' with the actor Gerry Becker, who was in the movie ``Sleepers'' with Kevin Bacon. Why Kevin Bacon? For some reason the three students who invented the game, Mike Ginelli, Craig Fass, and Brian Turtle decided that Kevin Bacon is the center of the entertainment world. The Kevin Bacon game is quite similar to the Erdös Number which has ``been a part of the folklore of mathematicians throughout the world for many years''.

The ``Six Degrees of Kevin Bacon'' game is really a graph problem. If you assign each actor to a vertex, and add an edge between two actors if they have appeared together in a movie, then you have a graph that represents the data for this game. Then the problem of finding a trail of actors to Kevin Bacon becomes a traditional graph problem: that of finding a path between two vertices. Since we wish to find a path that is shorter than six steps, ideally we would find the shortest path between the vertices. Now it turns out that there are no algorithms for finding the shortest path between a pair of vertices that are asymptotically faster than the algorithms for finding the shortest path from one vertex to all other vertices in the graph. For this reason we do not have algorithms for finding a shortest path between a pair of vertices in BGL. There are two single-source shortest paths algorithms implemented in BGL: Dijkstra's and the Bellman-Ford algorithm. In the following example we will show how to use Dijkstra's algorithm to calculate shortest paths to Kevin Bacon.

## Input File and Graph Setup

For this example we will use a much smaller graph than all movies and all actors. The full source code for this example is in `examples/kevin_bacon.cpp`. We have supplied a file `kevin_bacon.dat` which contains a list of actor pairs who appeared in the same movie. Each line of the file contains an actor's name, a movie, and another actor that appeared in the movie. The ``|'' character is used as a separator. Here is an excerpt.

    Patrick Stewart|Prince of Egypt, The (1998)|Steve Martin

Our first task will be to read in the file and create a graph from it. We use a `std::ifstream` to input the file.

      std::ifstream datafile("./kevin_bacon.dat");
      if (!datafile) {
        cerr << "No ./kevin_bacon.dat file" << endl;
        return -1;
      }

We will want to attach the actor's names to the vertices in the graph, and the movie names to the edges. In addition, Dijkstra's algorithm will expect edges to have a weight property (we will just set the weights to one). We use the `property` class to specify the addition of these vertex and edge properties to the graph. We choose to use an undirected graph, because the relationship of actors appearing together in a movie is symmetric.

      using namespace boost;
      typedef adjacency_list<vecS, vecS, undirectedS,
        property<vertex_name_t, string>,
        property<edge_name_t, string, property<edge_weight_t, int> >
      > Graph;

To access the properties, we will need to obtain property map objects from the graph. The following code shows how this is done.

      boost::property_map<Graph, vertex_name_t>::type actor_name = get(vertex_name, g);
      boost::property_map<Graph, edge_name_t>::type connecting_movie = get(edge_name, g);
      boost::property_map<Graph, edge_weight_t>::type weight = get(edge_weight, g);

Next we can start to loop through the file, parsing each line into a list of tokens. `stringtok` is a C++ version of the old `strtok` standard C function.

      std::string line;
      while (std::getline(datafile,line)) {
        std::list<std::string> line_toks;
        boost::stringtok(line_toks, line, "|");
        ...
      }

Each line of the input corresponds to an edge in the graph, with the two actors as the vertices incident to the edge, and the movie name will be a property attached to the edge. One challenge in creating the graph from this format of input is that the edges are specified by a pair of actor names. As we add vertices to the graph, we'll need to create a map from actor names to their vertices so that later appearances of the same actor (on a different edge) can be linked with the correct vertex in the graph. The `std::map` can be used to implement this mapping.

      typedef graph_traits<Graph>::vertex_descriptor Vertex;
      typedef std::map<string, Vertex> NameVertexMap;
      NameVertexMap actors;

The first token will be an actor's name. If the actor is not already in our actor's map we will add a vertex to the graph, set the name property of the vertex, and record the vertex descriptor in the map. If the actor is already in the graph, we will retreive the vertex descriptor from the map's `pos` iterator.

      NameVertexMap::iterator pos;
      bool inserted;
      Vertex u, v;

      std::list<std::string>::iterator i = line_toks.begin();

      boost::tie(pos, inserted) = actors.insert(make_pair(*i, Vertex()));
      if (inserted) {
        u = add_vertex(g);
        actor_name[u] = *i;
        pos->second = u;
      } else
        u = pos->second;
      ++i;

The second token is the name of the movie, and the third token is the other actor. We use the same technique as above to insert the second actor into the graph.

      std::string movie_name = *i++;

      boost::tie(pos, inserted) = actors.insert(make_pair(*i, Vertex()));
      if (inserted) {
        v = add_vertex(g);
        actor_name[v] = *i;
        pos->second = v;
      } else
        v = pos->second;

The final step is to add an edge connecting the two actors, and record the name of the connecting movie. We also set the weight of the edge to one. Since we chose `setS` for the `EdgeList` type of the `adjacency_list`, any parallel edges in the input will not be inserted into the graph.

      Edge e;
      boost::tie(e, inserted) = add_edge(u, v, g);
      if (inserted) {
        connecting_movie[e] = movie_name;
        weight[e] = 1;
      }

## Applying Shortest Paths

The documentation for variant 4 of `dijkstra_shortest_paths()` algorithm shows that we will need to provide several more property maps for distance, color, and vertex ID (edge weights are already on the graph) . The distance will be the shortest distances from each actor to Kevin Bacon calculated by the algorithm. This distance has also been referred to as the ``Bacon Number'' in memory of the ``Erdos Number'' used to rank mathematicians according to how many publications separate them from Paul Erdos. We will use a vector to store the bacon numbers, and another vector to store the color property needed by Dijkstra's algorithm. The `predecessor` vector will be used to record the shortest path from each actor to Kevin. For each vertex, the `predecessor` vector will contain a pointer to the next vertex on the shortest path towards Kevin Bacon.

      std::vector<int> bacon_number(num_vertices(g));
      std::vector<int> color(num_vertices(g));
      std::vector<Vertex> predecessor(num_vertices(g));

Dijkstra's algorithm also requires a source vertex as input; of course this will be Kevin Bacon. We now call the BGL Dijkstra algorithm, passing in the graph, source vertex, and all the property maps. We use the `predecessor_visitor` to record the predecessors, since this is not part of the basic algorithm.

        Vertex src = actors["Kevin Bacon"];

        dijkstra_shortest_paths
          (g, src, bacon_number.begin(),
           get(edge_weight, g), color.begin(), get(vertex_index, g),
           make_ucs_visitor(record_predecessors(predecessor.begin(),
                                                on_edge_relaxed())));

We can output the Bacon number for each actor simply by looping through all the vertices in the graph and use them to index into the `bacon_number` vector.

        graph_traits<Graph>::vertex_iterator i, end;
        for (boost::tie(i, end) = vertices(g); i != end; ++i)
          std::cout << actor_name[*i] << "'s bacon number is "
                    << bacon_number[*i] << std::endl;

Note that vertex descriptor objects can not always be used as indices into vectors or arrays such as `bacon_number`. This is valid with the `adjacency_list` class with `VertexList=vecS`, but not with other variations of `adjacency_list`. A more generic way to index based on vertices is to use the ID property map (`vertex_index_t`) in coordination with the `random_access_iterator_property_map`.

Here are some excepts from the output of the program.

    William Shatner was in Loaded Weapon 1 (1993) with Denise Richards
    Denise Richards was in Wild Things (1998) with Kevin Bacon
    Patrick Stewart was in Prince of Egypt, The (1998) with Steve Martin
    ...
    William Shatner's bacon number is 2
    Denise Richards's bacon number is 1
    Kevin Bacon's bacon number is 0
    Patrick Stewart's bacon number is 2
    Steve Martin's bacon number is 1
    ...

Copyright © 2000 | Jeremy Siek, Univ.of Notre Dame (jsiek@lsc.nd.edu)