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// $Id: graph_coloring.h 31932 2013-12-08 02:15:54Z heister $
//
// Copyright (C) 2013 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef __deal2__graph_coloring_h
#define __deal2__graph_coloring_h
#include <deal.II/base/config.h>
#include <deal.II/base/thread_management.h>
#include <deal.II/base/std_cxx1x/function.h>
#include <deal.II/dofs/dof_handler.h>
#include <boost/unordered_map.hpp>
#include <boost/unordered_set.hpp>
#include <set>
#include <vector>
DEAL_II_NAMESPACE_OPEN
/**
* A namespace containing functions that can color graphs.
*/
namespace GraphColoring
{
namespace internal
{
/**
* Given two sets of indices that are assumed to be sorted, determine
* whether they will have a nonempty intersection. The actual intersection is
* not computed.
* @param indices1 A set of indices, assumed sorted.
* @param indices2 A set of indices, assumed sorted.
* @return Whether the two sets of indices do have a nonempty intersection.
*/
inline
bool
have_nonempty_intersection (const std::vector<types::global_dof_index> &indices1,
const std::vector<types::global_dof_index> &indices2)
{
// we assume that both arrays are sorted, so we can walk
// them in lockstep and see if we encounter an index that's
// in both arrays. once we reach the end of either array,
// we know that there is no intersection
std::vector<types::global_dof_index>::const_iterator
p = indices1.begin(),
q = indices2.begin();
while ((p != indices1.end()) && (q != indices2.end()))
{
if (*p < *q)
++p;
else if (*p > *q)
++q;
else
// conflict found!
return true;
}
// no conflict found!
return false;
}
/**
* Create a partitioning of the given range of iterators using a simplified
* version of the Cuthill-McKee algorithm (Breadth First Search algorithm).
* The function creates partitions that contain "zones" of iterators
* where the first partition contains the first iterator, the second
* zone contains all those iterators that have conflicts with the single
* element in the first zone, the third zone contains those iterators that
* have conflicts with the iterators of the second zone and have not previously
* been assigned to a zone, etc. If the iterators represent cells, then this
* generates partitions that are like onion shells around the very first
* cell. Note that elements in each zone may conflict with other elements in
* the same zone.
*
* The question whether two iterators conflict is determined by a user-provided
* function. The meaning of this function is discussed in the documentation of
* the GraphColoring::make_graph_coloring() function.
*
* @param[in] begin The first element of a range of iterators for which a
* partitioning is sought.
* @param[in] end The element past the end of the range of iterators.
* @param[in] get_conflict_indices A user defined function object returning
* a set of indicators that are descriptive of what represents a
* conflict. See above for a more thorough discussion.
* @return A set of sets of iterators (where sets are represented by
* std::vector for efficiency). Each element of the outermost set
* corresponds to the iterators pointing to objects that are in the
* same partition (i.e., the same zone).
*
* @author Martin Kronbichler, Bruno Turcksin
*/
template <typename Iterator>
std::vector<std::vector<Iterator> >
create_partitioning(const Iterator &begin,
const typename identity<Iterator>::type &end,
const std_cxx1x::function<std::vector<types::global_dof_index> (const Iterator &)> &get_conflict_indices)
{
// Number of iterators.
unsigned int n_iterators = 0;
// Create a map from conflict indices to iterators
boost::unordered_map<types::global_dof_index,std::vector<Iterator> > indices_to_iterators;
for (Iterator it=begin; it!=end; ++it)
{
const std::vector<types::global_dof_index> conflict_indices = get_conflict_indices(it);
const unsigned int n_conflict_indices = conflict_indices.size();
for (unsigned int i=0; i<n_conflict_indices; ++i)
indices_to_iterators[conflict_indices[i]].push_back(it);
++n_iterators;
}
// create the very first zone which contains only the first
// iterator. then create the other zones. keep track of all the
// iterators that have already been assigned to a zone
std::vector<std::vector<Iterator> > zones(1,std::vector<Iterator> (1,begin));
std::set<Iterator> used_it;
used_it.insert(begin);
while (used_it.size()!=n_iterators)
{
// loop over the elements of the previous zone. for each element of
// the previous zone, get the conflict indices and from there get
// those iterators that are conflicting with the current element
typename std::vector<Iterator>::iterator previous_zone_it(zones.back().begin());
typename std::vector<Iterator>::iterator previous_zone_end(zones.back().end());
std::vector<Iterator> new_zone;
for (; previous_zone_it!=previous_zone_end; ++previous_zone_it)
{
const std::vector<types::global_dof_index>
conflict_indices = get_conflict_indices(*previous_zone_it);
const unsigned int n_conflict_indices(conflict_indices.size());
for (unsigned int i=0; i<n_conflict_indices; ++i)
{
const std::vector<Iterator> &conflicting_elements
= indices_to_iterators[conflict_indices[i]];
for (unsigned int j=0; j<conflicting_elements.size(); ++j)
{
// check that the iterator conflicting with the current one is not
// associated to a zone yet and if so, assign it to the current
// zone. mark it as used
//
// we can shortcut this test if the conflicting iterator is the
// current iterator
if ((conflicting_elements[j] != *previous_zone_it)
&&
(used_it.count(conflicting_elements[j])==0))
{
new_zone.push_back(conflicting_elements[j]);
used_it.insert(conflicting_elements[j]);
}
}
}
}
// If there are iterators in the new zone, then the zone is added to the
// partition. Otherwise, the graph is disconnected and we need to find
// an iterator on the other part of the graph. start the whole process again
// with the first iterator that hasn't been assigned to a zone yet
if (new_zone.size()!=0)
zones.push_back(new_zone);
else
for (Iterator it=begin; it!=end; ++it)
if (used_it.count(it)==0)
{
zones.push_back(std::vector<Iterator> (1,it));
used_it.insert(it);
break;
}
}
return zones;
}
/**
* This function uses DSATUR (Degree SATURation) to color the elements of
* a set. DSATUR works as follows:
* -# Arrange the vertices by decreasing order of degrees.
* -# Color a vertex of maximal degree with color 1.
* -# Choose a vertex with a maximal saturation degree. If there is equality,
* choose any vertex of maximal degree in the uncolored subgraph.
* -# Color the chosen vertex with the least possible (lowest numbered) color.
* -# If all the vertices are colored, stop. Otherwise, return to 3.
*
* @param[in] partition The set of iterators that should be colored.
* @param[in] get_conflict_indices A user defined function object returning
* a set of indicators that are descriptive of what represents a
* conflict. See above for a more thorough discussion.
* @param[out] partition_coloring A set of sets of iterators (where sets are represented by
* std::vector for efficiency). Each element of the outermost set
* corresponds to the iterators pointing to objects that are in the
* same partition (have the same color) and consequently do not
* conflict. The elements of different sets may conflict.
*/
template <typename Iterator>
void
make_dsatur_coloring(std::vector<Iterator> &partition,
const std_cxx1x::function<std::vector<types::global_dof_index> (const Iterator &)> &get_conflict_indices,
std::vector<std::vector<Iterator> > &partition_coloring)
{
partition_coloring.clear ();
// Number of zones composing the partitioning.
const unsigned int partition_size(partition.size());
std::vector<unsigned int> sorted_vertices(partition_size);
std::vector<int> degrees(partition_size);
std::vector<std::vector<types::global_dof_index> > conflict_indices(partition_size);
std::vector<std::vector<unsigned int> > graph(partition_size);
// Get the conflict indices associated to each iterator. The conflict_indices have to
// be sorted so we can more easily find conflicts later on
for (unsigned int i=0; i<partition_size; ++i)
{
conflict_indices[i] = get_conflict_indices(partition[i]);
std::sort(conflict_indices[i].begin(), conflict_indices[i].end());
}
// Compute the degree of each vertex of the graph using the
// intersection of the conflict indices.
for (unsigned int i=0; i<partition_size; ++i)
for (unsigned int j=i+1; j<partition_size; ++j)
// If the two iterators share indices then we increase the degree of the
// vertices and create an ''edge'' in the graph.
if (have_nonempty_intersection (conflict_indices[i], conflict_indices[j]))
{
++degrees[i];
++degrees[j];
graph[i].push_back(j);
graph[j].push_back(i);
}
// Sort the vertices by decreasing degree.
std::vector<int>::iterator degrees_it;
for (unsigned int i=0; i<partition_size; ++i)
{
// Find the largest element.
degrees_it = std::max_element(degrees.begin(),degrees.end());
sorted_vertices[i] = degrees_it-degrees.begin();
// Put the largest element to -1 so it cannot be chosen again.
*degrees_it = -1;
}
// Color the graph.
std::vector<boost::unordered_set<unsigned int> > colors_used;
for (unsigned int i=0; i<partition_size; ++i)
{
const unsigned int current_vertex(sorted_vertices[i]);
bool new_color(true);
// Try to use an existing color, i.e., try to find a color which is not
// associated to one of the vertices linked to current_vertex.
// Loop over the color.
for (unsigned int j=0; j<partition_coloring.size(); ++j)
{
// Loop on the vertices linked to current_vertex. If one vertex linked
// to current_vertex is already using the color j, this color cannot
// be used anymore.
bool unused_color(true);
for (unsigned int k=0; k<graph[current_vertex].size(); ++k)
if (colors_used[j].count(graph[current_vertex][k])==1)
{
unused_color = false;
break;
}
if (unused_color)
{
partition_coloring[j].push_back(partition[current_vertex]);
colors_used[j].insert(current_vertex);
new_color = false;
break;
}
}
// Add a new color.
if (new_color)
{
partition_coloring.push_back(std::vector<Iterator> (1,
partition[current_vertex]));
boost::unordered_set<unsigned int> tmp;
tmp.insert(current_vertex);
colors_used.push_back(tmp);
}
}
}
/**
* Given a partition-coloring graph, i.e., a set of zones (partitions) each
* of which is colored, produce a combined coloring for the entire set of
* iterators. This is possible because any color on an
* even (resp. odd) zone does not conflict with any color of any other even
* (resp. odd) zone. Consequently, we can combine colors from all even and all
* odd zones. This function tries to create colors of similar number of elements.
*/
template <typename Iterator>
std::vector<std::vector<Iterator> >
gather_colors(const std::vector<std::vector<std::vector<Iterator> > > &partition_coloring)
{
std::vector<std::vector<Iterator> > coloring;
// Count the number of iterators in each color.
const unsigned int partition_size(partition_coloring.size());
std::vector<std::vector<unsigned int> > colors_counter(partition_size);
for (unsigned int i=0; i<partition_size; ++i)
{
const unsigned int n_colors(partition_coloring[i].size());
colors_counter[i].resize(n_colors);
for (unsigned int j=0; j<n_colors; ++j)
colors_counter[i][j] = partition_coloring[i][j].size();
}
// Find the partition with the largest number of colors for the even partition.
unsigned int i_color(0);
unsigned int max_even_n_colors(0);
const unsigned int colors_size(colors_counter.size());
for (unsigned int i=0; i<colors_size; i+=2)
{
if (max_even_n_colors<colors_counter[i].size())
{
max_even_n_colors = colors_counter[i].size();
i_color = i;
}
}
coloring.resize(max_even_n_colors);
for (unsigned int j=0; j<colors_counter[i_color].size(); ++j)
coloring[j] = partition_coloring[i_color][j];
for (unsigned int i=0; i<partition_size; i+=2)
{
if (i!=i_color)
{
boost::unordered_set<unsigned int> used_k;
for (unsigned int j=0; j<colors_counter[i].size(); ++j)
{
// Find the color in the current partition with the largest number of
// iterators.
std::vector<unsigned int>::iterator it;
it = std::max_element(colors_counter[i].begin(),colors_counter[i].end());
unsigned int min_iterators(-1);
unsigned int pos(0);
// Find the color of coloring with the least number of colors among
// the colors that have not been used yet.
for (unsigned int k=0; k<max_even_n_colors; ++k)
if (used_k.count(k)==0)
if (colors_counter[i_color][k]<min_iterators)
{
min_iterators = colors_counter[i_color][k];
pos = k;
}
colors_counter[i_color][pos] += *it;
// Concatenate the current color with the existing coloring.
coloring[pos].insert(coloring[pos].end(),
partition_coloring[i][it-colors_counter[i].begin()].begin(),
partition_coloring[i][it-colors_counter[i].begin()].end());
used_k.insert(pos);
// Put the number of iterators to the current color to zero.
*it = 0;
}
}
}
// If there is more than one partition, do the same thing that we did for the even partitions
// to the odd partitions
if (partition_size>1)
{
unsigned int max_odd_n_colors(0);
for (unsigned int i=1; i<partition_size; i+=2)
{
if (max_odd_n_colors<colors_counter[i].size())
{
max_odd_n_colors = colors_counter[i].size();
i_color = i;
}
}
coloring.resize(max_even_n_colors+max_odd_n_colors);
for (unsigned int j=0; j<colors_counter[i_color].size(); ++j)
coloring[max_even_n_colors+j] = partition_coloring[i_color][j];
for (unsigned int i=1; i<partition_size; i+=2)
{
if (i!=i_color)
{
boost::unordered_set<unsigned int> used_k;
for (unsigned int j=0; j<colors_counter[i].size(); ++j)
{
// Find the color in the current partition with the largest number of
// iterators.
std::vector<unsigned int>::iterator it;
it = std::max_element(colors_counter[i].begin(),colors_counter[i].end());
unsigned int min_iterators(-1);
unsigned int pos(0);
// Find the color of coloring with the least number of colors among
// the colors that have not been used yet.
for (unsigned int k=0; k<max_odd_n_colors; ++k)
if (used_k.count(k)==0)
if (colors_counter[i_color][k]<min_iterators)
{
min_iterators = colors_counter[i_color][k];
pos = k;
}
colors_counter[i_color][pos] += *it;
// Concatenate the current color with the existing coloring.
coloring[max_even_n_colors+pos].insert(coloring[max_even_n_colors+pos].end(),
partition_coloring[i][it-colors_counter[i].begin()].begin(),
partition_coloring[i][it-colors_counter[i].begin()].end());
used_k.insert(pos);
// Put the number of iterators to the current color to zero.
*it = 0;
}
}
}
}
return coloring;
}
}
/**
* Create a partitioning of the given range of iterators so that
* iterators that point to conflicting objects will be placed
* into different partitions, where the question whether two objects conflict
* is determined by a user-provided function.
*
* This function can also be considered as a graph coloring: each object
* pointed to by an iterator is considered to be a node and there is an
* edge between each two nodes that conflict. The graph coloring algorithm
* then assigns a color to each node in such a way that two nodes connected
* by an edge do not have the same color.
*
* A typical use case for this function is in assembling a matrix in parallel.
* There, one would like to assemble local contributions on different cells
* at the same time (an operation that is purely local and so requires
* no synchronization) but then we need to add these local contributions
* to the global matrix. In general, the contributions from different cells
* may be to the same matrix entries if the cells share degrees of freedom
* and, consequently, can not happen at the same time unless we want to
* risk a race condition (see http://en.wikipedia.org/wiki/Race_condition ).
* Thus, we call these two cells in conflict, and we can only allow operations
* in parallel from cells that do not conflict. In other words, two cells
* are in conflict if the set of matrix entries (for example characterized
* by the rows) have a nonempty intersection.
*
* In this generality, computing the graph of conflicts would require calling
* a function that determines whether two iterators (or the two objects they
* represent) conflict, and calling it for every pair of iterators, i.e.,
* $\frac 12 N (N-1)$ times. This is too expensive in general. A better
* approach is to require a user-defined function that returns for every
* iterator it is called for a set of indicators of some kind that characterize
* a conflict; two iterators are in conflict if their conflict indicator sets
* have a nonempty intersection. In the example of assembling a matrix,
* the conflict indicator set would contain the indices of all degrees of
* freedom on the cell pointed to (in the case of continuous Galerkin methods)
* or the union of indices of degree of freedom on the current cell and all
* cells adjacent to the faces of the current cell (in the case of
* discontinuous Galerkin methods, because there one computes face integrals
* coupling the degrees of freedom connected by a common face -- see step-12).
*
* @note The conflict set returned by the user defined function passed as
* third argument needs to accurately describe <i>all</i> degrees of
* freedom for which anything is written into the matrix or right hand side.
* In other words, if the writing happens through a function like
* ConstraintMatrix::copy_local_to_global(), then the set of conflict indices
* must actually contain not only the degrees of freedom on the current
* cell, but also those they are linked to by constraints such as hanging
* nodes.
*
* In other situations, the conflict indicator sets may represent
* something different altogether -- it is up to the caller of this function
* to describe what it means for two iterators to conflict. Given this,
* computing conflict graph edges can be done significantly more cheaply
* than with ${\cal O}(N^2)$ operations.
*
* In any case, the result of the function will be so that iterators whose
* conflict indicator sets have overlap will not be assigned to the same
* color.
*
* @note The algorithm used in this function is described in a paper by
* Turcksin, Kronbichler and Bangerth, see @ref workstream_paper .
*
* @param[in] begin The first element of a range of iterators for which a
* coloring is sought.
* @param[in] end The element past the end of the range of iterators.
* @param[in] get_conflict_indices A user defined function object returning
* a set of indicators that are descriptive of what represents a
* conflict. See above for a more thorough discussion.
* @return A set of sets of iterators (where sets are represented by
* std::vector for efficiency). Each element of the outermost set
* corresponds to the iterators pointing to objects that are in the
* same partition (have the same color) and consequently do not
* conflict. The elements of different sets may conflict.
*
* @author Martin Kronbichler, Bruno Turcksin
*/
template <typename Iterator>
std::vector<std::vector<Iterator> >
make_graph_coloring(const Iterator &begin,
const typename identity<Iterator>::type &end,
const std_cxx1x::function<std::vector<types::global_dof_index> (const typename identity<Iterator>::type &)> &get_conflict_indices)
{
Assert (begin != end, ExcMessage ("GraphColoring is not prepared to deal with empty ranges!"));
// Create the partitioning.
std::vector<std::vector<Iterator> >
partitioning = internal::create_partitioning (begin,
end,
get_conflict_indices);
// Color the iterators within each partition.
// Run the coloring algorithm on each zone in parallel
const unsigned int partitioning_size(partitioning.size());
std::vector<std::vector<std::vector<Iterator> > >
partition_coloring(partitioning_size);
Threads::TaskGroup<> tasks;
for (unsigned int i=0; i<partitioning_size; ++i)
tasks += Threads::new_task (&internal::make_dsatur_coloring<Iterator>,
partitioning[i],
get_conflict_indices,
partition_coloring[i]);
tasks.join_all();
// Gather the colors together.
return internal::gather_colors(partition_coloring);
}
} // End graph_coloring namespace
DEAL_II_NAMESPACE_CLOSE
//---------------------------- graph_coloring.h ---------------------------
// end of #ifndef __deal2__graph_coloring_h
#endif
//---------------------------- graph_coloring.h ---------------------------
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