From：http://stackoverflow.com/questions/8950188/is-it-possible-to-apply-breadth-first-search-algorithm-of-boost-library-to-matri/8953750#8953750

我的任务是找出矩阵中一个点到另一个点的最短路径，且只能按照上下左右移动

0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 1 F 0
0 1 0 1 0 0 0 0
0 0 0 1 0 0 0 0
0 S 0 1 0 0 1 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 1 0

S – 起点

F – 终点

0 – 可以穿过的点

1 – 不能穿过的点

很明显广度优先搜索是解决该问题的最佳方法，我知道Boost库支持该算法，但是我对Boost库不是特别熟悉。

我如何使用Boost库的广度优先搜索算法解决我的问题？据我了解，Boost库的广度优先搜索算法是专门为图而设计的，我感觉把m*n的矩阵转换成一个包含m*n个点和m*(n-1) + (m-1)*n条边的图不是一个好主意

我可以直接将Boost库的广度优先搜索算法应用于矩阵（不是转换成图）吗，或者是能使用我自己定义的函数就更好了？

注：以上是对提问者的翻译

以下是对回答者的翻译：

 （首先很抱歉给的答案篇幅这么长。我使用Boost Graph Library已经有些日子了，写这些也可以很好的复习一下。全部代码在最底部）

         Boost图形库（和通常的泛型编程）的好处就在于你不需要在既定的算法中使用特定的数据结构。你上述提到的遍历矩阵的规则已经和图差不多了,你需要做的就是将这些规则归纳成可以被BGL（Boost图形库）算法使用的traits class。

         明确的说，就是我们要为你的graph定义一个特定的boost::graph_traits<T>结构。让我们假定你的矩阵是一个以行优先的int型的一维数组。但很不幸的是，只包含int[N]的graph_traits是不够的，因为它没有提供矩阵中维度的信息。所以我们按照以下形式定义你的graph：

namespace matrix
{
  typedef int cell;

  static const int FREE = 0;
  static const int WALL = 1;

  template< size_t ROWS, size_t COLS >
  struct graph
  {
    cell cells[ROWS*COLS];
  };
}

       我这里使用了一个数组来表示单元格数据，如果在外面管理这些数据的话你可以使用指针更简单一些。现在我们已经有了一个可以用于graph_traits的表示矩阵的类型了，但是首先让我们定义一些我们需要的类型和函数：

顶点类型和辅助函数：

namespace matrix
{
  typedef size_t vertex_descriptor;

  template< size_t ROWS, size_t COLS >
  size_t get_row(
    vertex_descriptor vertex,
    graph< ROWS, COLS > const & )
  {
    return vertex / COLS;
  }

  template< size_t ROWS, size_t COLS >
  size_t get_col(
    vertex_descriptor vertex,
    graph< ROWS, COLS > const & )
  {
    return vertex % COLS;
  }

  template< size_t ROWS, size_t COLS >
  vertex_descriptor make_vertex(
    size_t row,
    size_t col,
    graph< ROWS, COLS > const & )
  {
    return row * COLS + col;
  }
}

遍历顶点的类型和函数：

namespace matrix
{
  typedef const cell * vertex_iterator;

  template< size_t ROWS, size_t COLS >
  std::pair< vertex_iterator, vertex_iterator >
  vertices( graph< ROWS, COLS > const & g )
  {
    return std::make_pair( g.cells, g.cells + ROWS*COLS );
  }

  typedef size_t vertices_size_type;

  template< size_t ROWS, size_t COLS >
  vertices_size_type
  num_vertices( graph< ROWS, COLS > const & g )
  {
    return ROWS*COLS;
  }
}

边类型：

namespace matrix
{
  typedef std::pair< vertex_descriptor, vertex_descriptor > edge_descriptor;

  bool operator==(
    edge_descriptor const & lhs,
    edge_descriptor const & rhs )
  {
    return
      lhs.first == rhs.first && lhs.second == rhs.second ||
      lhs.first == rhs.second && lhs.second == rhs.first;
  }

  bool operator!=(
    edge_descriptor const & lhs,
    edge_descriptor const & rhs )
  {
    return !(lhs == rhs);
  }
}

最后，迭代器和函数来帮助我们遍历出顶点和边存在的关联关系：

namespace matrix
{
  template< size_t ROWS, size_t COLS >
  vertex_descriptor
  source(
    edge_descriptor const & edge,
    graph< ROWS, COLS > const & )
  {
    return edge.first;
  }

  template< size_t ROWS, size_t COLS >
  vertex_descriptor
  target(
    edge_descriptor const & edge,
    graph< ROWS, COLS > const & )
  {
    return edge.second;
  }

  typedef boost::shared_container_iterator< std::vector< edge_descriptor > > out_edge_iterator;

  template< size_t ROWS, size_t COLS >
  std::pair< out_edge_iterator, out_edge_iterator >
  out_edges(
    vertex_descriptor vertex,
    graph< ROWS, COLS > const & g )
  {
    boost::shared_ptr< std::vector< edge_descriptor > > edges( new std::vector< edge_descriptor >() );

    if( g.cells[vertex] == FREE )
    {
      size_t
        row = get_row( vertex, g ),
        col = get_col( vertex, g );

      if( row != 0 )
      {
        vertex_descriptor up = make_vertex( row - 1, col, g );

        if( g.cells[up] == FREE )
          edges->push_back( edge_descriptor( vertex, up ) );
      }

      if( row != ROWS-1 )
      {
        vertex_descriptor down = make_vertex( row + 1, col, g );

        if( g.cells[down] == FREE )
          edges->push_back( edge_descriptor( vertex, down ) );
      }

      if( col != 0 )
      {
        vertex_descriptor left = make_vertex( row, col - 1, g );

        if( g.cells[left] == FREE )
          edges->push_back( edge_descriptor( vertex, left ) );
      }

      if( col != COLS-1 )
      {
        vertex_descriptor right = make_vertex( row, col + 1, g );

        if( g.cells[right] == FREE )
          edges->push_back( edge_descriptor( vertex, right ) );
      }
    }

    return boost::make_shared_container_range( edges );
  }

  typedef size_t degree_size_type;

  template< size_t ROWS, size_t COLS >
  degree_size_type
  out_degree(
    vertex_descriptor vertex,
    graph< ROWS, COLS > const & g )
  {
    std::pair< out_edge_iterator, out_edge_iterator > edges = out_edges( vertex, g );
    return std::distance( edges.first, edges.second );
  }
}

到现在未知我们已经定义好了我们需要的boost::graph_traits结构

namespace boost
{
  template< size_t ROWS, size_t COLS >
  struct graph_traits< matrix::graph< ROWS, COLS > >
  {
    typedef matrix::vertex_descriptor vertex_descriptor;
    typedef matrix::edge_descriptor edge_descriptor;

    typedef matrix::out_edge_iterator out_edge_iterator;
    typedef matrix::vertex_iterator vertex_iterator;

    typedef boost::undirected_tag directed_category;
    typedef boost::disallow_parallel_edge_tag edge_parallel_category;
    struct traversal_category :
      virtual boost::vertex_list_graph_tag,
      virtual boost::incidence_graph_tag {};

    typedef matrix::vertices_size_type vertices_size_type;
    typedef matrix::degree_size_type degree_size_type;

    static vertex_descriptor null_vertex() { return ROWS*COLS; }
  };
}

接下来就是如何展示使用BFS算法查找最短路径：

int main()
{
  const size_t rows = 8, cols = 8;

  using namespace matrix;

  typedef graph< rows, cols > my_graph;

  my_graph g =
  {
    FREE, FREE, FREE, FREE, WALL, FREE, FREE, FREE,
    WALL, FREE, FREE, FREE, FREE, FREE, FREE, FREE,
    FREE, FREE, FREE, WALL, FREE, WALL, FREE, FREE,
    FREE, WALL, FREE, WALL, FREE, FREE, FREE, FREE,
    FREE, FREE, FREE, WALL, FREE, FREE, FREE, FREE,
    FREE, FREE, FREE, WALL, FREE, FREE, WALL, FREE,
    FREE, FREE, FREE, FREE, FREE, FREE, WALL, FREE,
    FREE, FREE, FREE, FREE, FREE, FREE, WALL, FREE,
  };

  const vertex_descriptor
    start_vertex = make_vertex( 5, 1, g ),
    finish_vertex = make_vertex( 2, 6, g );

  vertex_descriptor predecessors[rows*cols] = { 0 };

  using namespace boost;

  breadth_first_search(
    g,
    start_vertex,
    visitor( make_bfs_visitor( record_predecessors( predecessors, on_tree_edge() ) ) ).
    vertex_index_map( identity_property_map() ) );

  typedef std::list< vertex_descriptor > path;

  path p;

  for( vertex_descriptor vertex = finish_vertex; vertex != start_vertex; vertex = predecessors[vertex] )
    p.push_front( vertex );

  p.push_front( start_vertex );

  for( path::const_iterator cell = p.begin(); cell != p.end(); ++cell )
    std::cout << "[" << get_row( *cell, g ) << ", " << get_col( *cell, g ) << "]\n" ;

  return 0;
}

以下是输出的从起点到终点的最短路径：

[5, 1]
[4, 1]
[4, 2]
[3, 2]
[2, 2]
[1, 2]
[1, 3]
[1, 4]
[1, 5]
[1, 6]
[2, 6]

源码如下：

#include <boost/graph/graph_traits.hpp>
#include <boost/graph/breadth_first_search.hpp>
#include <boost/graph/visitors.hpp>
#include <boost/shared_container_iterator.hpp>
#include <boost/shared_ptr.hpp>
#include <vector>
#include <boost/iterator/counting_iterator.hpp>
#include <list>

namespace matrix
{
  typedef int cell;

  static const int FREE = 0;
  static const int WALL = 1;

  template< size_t ROWS, size_t COLS >
  struct graph
  {
    cell cells[ROWS*COLS];
  };

  typedef size_t vertex_descriptor;

  template< size_t ROWS, size_t COLS >
  size_t get_row(
    vertex_descriptor vertex,
    graph< ROWS, COLS > const & )
  {
    return vertex / COLS;
  }

  template< size_t ROWS, size_t COLS >
  size_t get_col(
    vertex_descriptor vertex,
    graph< ROWS, COLS > const & )
  {
    return vertex % COLS;
  }

  template< size_t ROWS, size_t COLS >
  vertex_descriptor make_vertex(
    size_t row,
    size_t col,
    graph< ROWS, COLS > const & )
  {
    return row * COLS + col;
  }

  typedef const cell * vertex_iterator;

  template< size_t ROWS, size_t COLS >
  std::pair< vertex_iterator, vertex_iterator >
  vertices( graph< ROWS, COLS > const & g )
  {
    return std::make_pair( g.cells, g.cells + ROWS*COLS );
  }

  typedef size_t vertices_size_type;

  template< size_t ROWS, size_t COLS >
  vertices_size_type
  num_vertices( graph< ROWS, COLS > const & g )
  {
    return ROWS*COLS;
  }
  
  typedef std::pair< vertex_descriptor, vertex_descriptor > edge_descriptor;

  bool operator==(
    edge_descriptor const & lhs,
    edge_descriptor const & rhs )
  {
    return
      lhs.first == rhs.first && lhs.second == rhs.second ||
      lhs.first == rhs.second && lhs.second == rhs.first;
  }

  bool operator!=(
    edge_descriptor const & lhs,
    edge_descriptor const & rhs )
  {
    return !(lhs == rhs);
  }

  template< size_t ROWS, size_t COLS >
  vertex_descriptor
  source(
    edge_descriptor const & edge,
    graph< ROWS, COLS > const & )
  {
    return edge.first;
  }

  template< size_t ROWS, size_t COLS >
  vertex_descriptor
  target(
    edge_descriptor const & edge,
    graph< ROWS, COLS > const & )
  {
    return edge.second;
  }

  typedef boost::shared_container_iterator< std::vector< edge_descriptor > > out_edge_iterator;

  template< size_t ROWS, size_t COLS >
  std::pair< out_edge_iterator, out_edge_iterator >
  out_edges(
    vertex_descriptor vertex,
    graph< ROWS, COLS > const & g )
  {
    boost::shared_ptr< std::vector< edge_descriptor > > edges( new std::vector< edge_descriptor >() );

    if( g.cells[vertex] == FREE )
    {
      size_t
        row = get_row( vertex, g ),
        col = get_col( vertex, g );

      if( row != 0 )
      {
        vertex_descriptor up = make_vertex( row - 1, col, g );

        if( g.cells[up] == FREE )
          edges->push_back( edge_descriptor( vertex, up ) );
      }

      if( row != ROWS-1 )
      {
        vertex_descriptor down = make_vertex( row + 1, col, g );

        if( g.cells[down] == FREE )
          edges->push_back( edge_descriptor( vertex, down ) );
      }

      if( col != 0 )
      {
        vertex_descriptor left = make_vertex( row, col - 1, g );

        if( g.cells[left] == FREE )
          edges->push_back( edge_descriptor( vertex, left ) );
      }

      if( col != COLS-1 )
      {
        vertex_descriptor right = make_vertex( row, col + 1, g );

        if( g.cells[right] == FREE )
          edges->push_back( edge_descriptor( vertex, right ) );
      }
    }

    return boost::make_shared_container_range( edges );
  }

  typedef size_t degree_size_type;

  template< size_t ROWS, size_t COLS >
  degree_size_type
  out_degree(
    vertex_descriptor vertex,
    graph< ROWS, COLS > const & g )
  {
    std::pair< out_edge_iterator, out_edge_iterator > edges = out_edges( vertex, g );
    return std::distance( edges.first, edges.second );
  }
}

namespace boost
{
  template< size_t ROWS, size_t COLS >
  struct graph_traits< matrix::graph< ROWS, COLS > >
  {
    typedef matrix::vertex_descriptor vertex_descriptor;
    typedef matrix::edge_descriptor edge_descriptor;

    typedef matrix::out_edge_iterator out_edge_iterator;
    typedef matrix::vertex_iterator vertex_iterator;

    typedef boost::undirected_tag directed_category;
    typedef boost::disallow_parallel_edge_tag edge_parallel_category;
    struct traversal_category :
      virtual boost::vertex_list_graph_tag,
      virtual boost::incidence_graph_tag {};

    typedef matrix::vertices_size_type vertices_size_type;
    typedef matrix::degree_size_type degree_size_type;

    static vertex_descriptor null_vertex() { return ROWS*COLS; }
  };
}

int main()
{
  const size_t rows = 8, cols = 8;

  using namespace matrix;

  typedef matrix::graph< rows, cols > my_graph;
  
  my_graph g =
  {
    FREE, FREE, FREE, FREE, WALL, FREE, FREE, FREE,
    WALL, FREE, FREE, FREE, FREE, FREE, FREE, FREE,
    FREE, FREE, FREE, WALL, FREE, WALL, FREE, FREE,
    FREE, WALL, FREE, WALL, FREE, FREE, FREE, FREE,
    FREE, FREE, FREE, WALL, FREE, FREE, FREE, FREE,
    FREE, FREE, FREE, WALL, FREE, FREE, WALL, FREE,
    FREE, FREE, FREE, FREE, FREE, FREE, WALL, FREE,
    FREE, FREE, FREE, FREE, FREE, FREE, WALL, FREE,
  };

  const vertex_descriptor
    start_vertex = make_vertex( 5, 1, g ),
    finish_vertex = make_vertex( 2, 6, g );

  vertex_descriptor predecessors[rows*cols] = { 0 };

  using namespace boost;

  breadth_first_search(
    g,
    start_vertex,
    visitor( make_bfs_visitor( record_predecessors( predecessors, on_tree_edge() ) ) ).
    vertex_index_map( boost::identity_property_map() ) );

  typedef std::list< vertex_descriptor > path;
  
  path p;

  for( vertex_descriptor vertex = finish_vertex; vertex != start_vertex; vertex = predecessors[vertex] )
    p.push_front( vertex );

  p.push_front( start_vertex );

  for( path::const_iterator cell = p.begin(); cell != p.end(); ++cell )
    std::cout << "[" << get_row( *cell, g ) << ", " << get_col( *cell, g ) << "]\n" ;

  return 0;
}

输出：

    [5, 1]
    [4, 1]
    [4, 2]
    [3, 2]
    [2, 2]
    [1, 2]
    [1, 3]
    [1, 4]
    [1, 5]
    [1, 6]
    [2, 6]