Здраствуйте. Позарез нужно реализовать алгоритм Слоана. Этот алгоритм работает с неориентированным графом. Он перенумеровывает вершины графа , тем самым уменьшая профиль матрицы, связанной с этим графом.
Сам этот алгоритм и видимо его программная реализация на языке Фортран приведена в статье S.W.Sloan. A Fortran program for profile and wavefront reduction. Int. J. for Numer. Meth. in Eng., vol. 28, 2651-2679 (1989).
Этой статьи в интернете я найти не смог. Да и про сам алгоритм в интернете практически ничего нет. Я нашел только его реализацию в библиотеке boost.Но его код там довольно сложен и запутан и разобраться в нем я не смог.
Хотелось бы понять сам алгоритм. В виде словесного описания по шагам или лучше всего на небольшом примере. Если кто - нибудь знает как работает или кто -нибудь его сам программировал помогите мне! Если у кого -нибудь есть эта статья или ссылка на неё выложите плиз.
//
//=======================================================================
// Copyright 2002 Marc Wintermantel (wintermantel@even-ag.ch)
// ETH Zurich, Center of Structure Technologies (www.imes.ethz.ch/st)
//
// Distributed under the Boost Software License, Version 1.0. (See
// accompanying file LICENSE_1_0.txt or copy at
// http://www.boost.org/LICENSE_1_0.txt)
//=======================================================================
//
#ifndef BOOST_GRAPH_SLOAN_HPP
#define BOOST_GRAPH_SLOAN_HPP
#define WEIGHT1 1 //default weight for the distance in the Sloan algorithm
#define WEIGHT2 2 //default weight for the degree in the Sloan algorithm
#define MAXINT 2147483647 //Maximum value for a 32bit integer
#include <boost/config.hpp>
#include <vector>
#include <queue>
#include <boost/pending/queue.hpp>
#include <boost/graph/graph_traits.hpp>
#include <boost/graph/breadth_first_search.hpp>
#include <boost/graph/properties.hpp>
#include <boost/pending/indirect_cmp.hpp>
#include <boost/property_map.hpp>
#include <algorithm>
#include <utility>
#include <boost/graph/visitors.hpp>
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/cuthill_mckee_ordering.hpp>
////////////////////////////////////////////////////////////
//
//Sloan-Algorithm for graph reordering
//(optimzes profile and wavefront, not primiraly bandwidth
//
////////////////////////////////////////////////////////////
namespace boost {
/////////////////////////////////////////////////////////////////////////
// Function that returns the maximum depth of
// a rooted level strucutre (RLS)
//
/////////////////////////////////////////////////////////////////////////
template<class Distance>
unsigned RLS_depth(Distance& d)
{
unsigned h_s = 0;
typename Distance::iterator iter;
for (iter = d.begin(); iter != d.end(); ++iter)
{
if(*iter > h_s)
{
h_s = *iter;
}
}
return h_s;
}
/////////////////////////////////////////////////////////////////////////
// Function that returns the width of the largest level of
// a rooted level strucutre (RLS)
//
/////////////////////////////////////////////////////////////////////////
template<class Distance, class my_int>
unsigned RLS_max_width(Distance& d, my_int depth)
{
//Searching for the maximum width of a level
std::vector<unsigned> dummy_width(depth+1, 0);
std::vector<unsigned>::iterator my_it;
typename Distance::iterator iter;
unsigned w_max = 0;
for (iter = d.begin(); iter != d.end(); ++iter)
{
dummy_width[*iter]++;
}
for(my_it = dummy_width.begin(); my_it != dummy_width.end(); ++my_it)
{
if(*my_it > w_max) w_max = *my_it;
}
return w_max;
}
/////////////////////////////////////////////////////////////////////////
// Function for finding a good starting node for Sloan algorithm
//
// This is to find a good starting node. "good" is in the sense
// of the ordering generated.
/////////////////////////////////////////////////////////////////////////
template <class Graph, class ColorMap, class DegreeMap>
typename graph_traits<Graph>::vertex_descriptor
sloan_start_end_vertices(Graph& G,
typename graph_traits<Graph>::vertex_descriptor &s,
ColorMap color,
DegreeMap degree)
{
typedef typename property_traits<DegreeMap>::value_type Degree;
typedef typename graph_traits<Graph>::vertex_descriptor Vertex;
typedef typename std::vector< typename graph_traits<Graph>::vertices_size_type>::iterator vec_iter;
typedef typename graph_traits<Graph>::vertices_size_type size_type;
typedef typename property_map<Graph, vertex_index_t>::const_type VertexID;
s = *(vertices(G).first);
Vertex e = s;
Vertex i;
unsigned my_degree = get(degree, s );
unsigned dummy, h_i, h_s, w_i, w_e;
bool new_start = true;
unsigned maximum_degree = 0;
//Creating a std-vector for storing the distance from the start vertex in dist
std::vector<typename graph_traits<Graph>::vertices_size_type> dist(num_vertices(G), 0);
//Wrap a property_map_iterator around the std::iterator
boost::iterator_property_map<vec_iter, VertexID, size_type, size_type&> dist_pmap(dist.begin(), get(vertex_index, G));
//Creating a property_map for the indices of a vertex
typename property_map<Graph, vertex_index_t>::type index_map = get(vertex_index, G);
//Creating a priority queue
typedef indirect_cmp<DegreeMap, std::greater<Degree> > Compare;
Compare comp(degree);
std::priority_queue<Vertex, std::vector<Vertex>, Compare> degree_queue(comp);
//step 1
//Scan for the vertex with the smallest degree and the maximum degree
typename graph_traits<Graph>::vertex_iterator ui, ui_end;
for (tie(ui, ui_end) = vertices(G); ui != ui_end; ++ui)
{
dummy = get(degree, *ui);
if(dummy < my_degree)
{
my_degree = dummy;
s = *ui;
}
if(dummy > maximum_degree)
{
maximum_degree = dummy;
}
}
//end 1
do{
new_start = false; //Setting the loop repetition status to false
//step 2
//initialize the the disance std-vector with 0
for(typename std::vector<typename graph_traits<Graph>::vertices_size_type>::iterator iter = dist.begin(); iter != dist.end(); ++iter) *iter = 0;
//generating the RLS (rooted level structure)
breadth_first_search
(G, s, visitor
(
make_bfs_visitor(record_distances(dist_pmap, on_tree_edge() ) )
)
);
//end 2
//step 3
//calculating the depth of the RLS
h_s = RLS_depth(dist);
//step 4
//pushing one node of each degree in an ascending manner into degree_queue
std::vector<bool> shrink_trace(maximum_degree, false);
for (tie(ui, ui_end) = vertices(G); ui != ui_end; ++ui)
{
dummy = get(degree, *ui);
if( (dist[index_map[*ui]] == h_s ) && ( !shrink_trace[ dummy ] ) )
{
degree_queue.push(*ui);
shrink_trace[ dummy ] = true;
}
}
//end 3 & 4
//step 5
//Initializing w
w_e = MAXINT;
//end 5
//step 6
//Testing for termination
while( !degree_queue.empty() )
{
i = degree_queue.top(); //getting the node with the lowest degree from the degree queue
degree_queue.pop(); //ereasing the node with the lowest degree from the degree queue
//generating a RLS
for(typename std::vector<typename graph_traits<Graph>::vertices_size_type>::iterator iter = dist.begin(); iter != dist.end(); ++iter) *iter = 0;
breadth_first_search
(G, i, boost::visitor
(
make_bfs_visitor(record_distances(dist_pmap, on_tree_edge() ) )
)
);
//Calculating depth and width of the rooted level
h_i = RLS_depth(dist);
w_i = RLS_max_width(dist, h_i);
//Testing for termination
if( (h_i > h_s) && (w_i < w_e) )
{
h_s = h_i;
s = i;
while(!degree_queue.empty()) degree_queue.pop();
new_start = true;
}
else if(w_i < w_e)
{
w_e = w_i;
e = i;
}
}
//end 6
}while(new_start);
return e;
}
//////////////////////////////////////////////////////////////////////////
// Sloan algorithm with a given starting Vertex.
//
// This algorithm requires user to provide a starting vertex to
// compute Sloan ordering.
//////////////////////////////////////////////////////////////////////////
template <class Graph, class OutputIterator,
class ColorMap, class DegreeMap,
class PriorityMap, class Weight>
OutputIterator
sloan_ordering(Graph& g,
typename graph_traits<Graph>::vertex_descriptor s,
typename graph_traits<Graph>::vertex_descriptor e,
OutputIterator permutation,
ColorMap color,
DegreeMap degree,
PriorityMap priority,
Weight W1,
Weight W2)
{
//typedef typename property_traits<DegreeMap>::value_type Degree;
typedef typename property_traits<PriorityMap>::value_type Degree;
typedef typename property_traits<ColorMap>::value_type ColorValue;
typedef color_traits<ColorValue> Color;
typedef typename graph_traits<Graph>::vertex_descriptor Vertex;
typedef typename std::vector<typename graph_traits<Graph>::vertices_size_type>::iterator vec_iter;
typedef typename graph_traits<Graph>::vertices_size_type size_type;
typedef typename property_map<Graph, vertex_index_t>::const_type VertexID;
//Creating a std-vector for storing the distance from the end vertex in it
typename std::vector<typename graph_traits<Graph>::vertices_size_type> dist(num_vertices(g), 0);
//Wrap a property_map_iterator around the std::iterator
boost::iterator_property_map<vec_iter, VertexID, size_type, size_type&> dist_pmap(dist.begin(), get(vertex_index, g));
breadth_first_search
(g, e, visitor
(
make_bfs_visitor(record_distances(dist_pmap, on_tree_edge() ) )
)
);
//Creating a property_map for the indices of a vertex
typename property_map<Graph, vertex_index_t>::type index_map = get(vertex_index, g);
//Sets the color and priority to their initial status
unsigned cdeg;
typename graph_traits<Graph>::vertex_iterator ui, ui_end;
for (tie(ui, ui_end) = vertices(g); ui != ui_end; ++ui)
{
put(color, *ui, Color::white());
cdeg=get(degree, *ui)+1;
put(priority, *ui, W1*dist[index_map[*ui]]-W2*cdeg );
}
//Priority list
typedef indirect_cmp<PriorityMap, std::greater<Degree> > Compare;
Compare comp(priority);
std::list<Vertex> priority_list;
//Some more declarations
typename graph_traits<Graph>::out_edge_iterator ei, ei_end, ei2, ei2_end;
Vertex u, v, w;
put(color, s, Color::green()); //Sets the color of the starting vertex to gray
priority_list.push_front(s); //Puts s into the priority_list
while ( !priority_list.empty() )
{
priority_list.sort(comp); //Orders the elements in the priority list in an ascending manner
u = priority_list.front(); //Accesses the last element in the priority list
priority_list.pop_front(); //Removes the last element in the priority list
if(get(color, u) == Color::green() )
{
//for-loop over all out-edges of vertex u
for (tie(ei, ei_end) = out_edges(u, g); ei != ei_end; ++ei)
{
v = target(*ei, g);
put( priority, v, get(priority, v) + W2 ); //updates the priority
if (get(color, v) == Color::white() ) //test if the vertex is inactive
{
put(color, v, Color::green() ); //giving the vertex a preactive status
priority_list.push_front(v); //writing the vertex in the priority_queue
}
}
}
//Here starts step 8
*permutation++ = u; //Puts u to the first position in the permutation-vector
put(color, u, Color::black() ); //Gives u an inactive status
//for loop over all the adjacent vertices of u
for (tie(ei, ei_end) = out_edges(u, g); ei != ei_end; ++ei) {
v = target(*ei, g);
if (get(color, v) == Color::green() ) { //tests if the vertex is inactive
put(color, v, Color::red() ); //giving the vertex an active status
put(priority, v, get(priority, v)+W2); //updates the priority
//for loop over alll adjacent vertices of v
for (tie(ei2, ei2_end) = out_edges(v, g); ei2 != ei2_end; ++ei2) {
w = target(*ei2, g);
if(get(color, w) != Color::black() ) { //tests if vertex is postactive
put(priority, w, get(priority, w)+W2); //updates the priority
if(get(color, w) == Color::white() ){
put(color, w, Color::green() ); // gives the vertex a preactive status
priority_list.push_front(w); // puts the vertex into the priority queue
} //end if
} //end if
} //end for
} //end if
} //end for
} //end while
return permutation;
}
/////////////////////////////////////////////////////////////////////////////////////////
// Same algorithm as before, but without the weights given (taking default weights
template <class Graph, class OutputIterator,
class ColorMap, class DegreeMap,
class PriorityMap>
OutputIterator
sloan_ordering(Graph& g,
typename graph_traits<Graph>::vertex_descriptor s,
typename graph_traits<Graph>::vertex_descriptor e,
OutputIterator permutation,
ColorMap color,
DegreeMap degree,
PriorityMap priority)
{
return sloan_ordering(g, s, e, permutation, color, degree, priority, WEIGHT1, WEIGHT2);
}
//////////////////////////////////////////////////////////////////////////
// Sloan algorithm without a given starting Vertex.
//
// This algorithm finds a good starting vertex itself to
// compute Sloan-ordering.
//////////////////////////////////////////////////////////////////////////
template < class Graph, class OutputIterator,
class Color, class Degree,
class Priority, class Weight>
inline OutputIterator
sloan_ordering(Graph& G,
OutputIterator permutation,
Color color,
Degree degree,
Priority priority,
Weight W1,
Weight W2 )
{
typedef typename boost::graph_traits<Graph>::vertex_descriptor Vertex;
Vertex s, e;
e = sloan_start_end_vertices(G, s, color, degree);
return sloan_ordering(G, s, e, permutation, color, degree, priority, W1, W2);
}
/////////////////////////////////////////////////////////////////////////////////////////
// Same as before, but without given weights (default weights are taken instead)
template < class Graph, class OutputIterator,
class Color, class Degree,
class Priority >
inline OutputIterator
sloan_ordering(Graph& G,
OutputIterator permutation,
Color color,
Degree degree,
Priority priority)
{
return sloan_ordering(G, permutation, color, degree, priority, WEIGHT1, WEIGHT2);
}
} // namespace boost
#endif // BOOST_GRAPH_SLOAN_HPP
Если кто -нибудь сможет в нем разобраться буду только рад...
