index_set.h

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// ---------------------------------------------------------------------
//
// Copyright (C) 2009 - 2020 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.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------

#ifndef dealii_index_set_h
#define dealii_index_set_h

#include <deal.II/base/config.h>

#include <deal.II/base/exceptions.h>
#include <deal.II/base/thread_management.h>
#include <deal.II/base/utilities.h>

#include <boost/serialization/vector.hpp>

#include <algorithm>
#include <iterator>
#include <vector>


#ifdef DEAL_II_WITH_TRILINOS
#  include <Epetra_Map.h>
#  ifdef DEAL_II_TRILINOS_WITH_TPETRA
#    include <Tpetra_Map.hpp>
#  endif
#endif

#if defined(DEAL_II_WITH_MPI) || defined(DEAL_II_WITH_PETSC)
#  include <mpi.h>
#else
using MPI_Comm = int;
#  ifndef MPI_COMM_WORLD
#    define MPI_COMM_WORLD 0
#  endif
#endif

DEAL_II_NAMESPACE_OPEN

class IndexSet
{
public:
  // forward declarations:
  class ElementIterator;
  class IntervalIterator;

  using size_type = types::global_dof_index;

  using value_type = signed int;


  IndexSet();

  explicit IndexSet(const size_type size);

  IndexSet(const IndexSet &) = default;

  IndexSet &
  operator=(const IndexSet &) = default;

  IndexSet(IndexSet &&is) noexcept;

  IndexSet &
  operator=(IndexSet &&is) noexcept;

#ifdef DEAL_II_WITH_TRILINOS

  explicit IndexSet(const Epetra_BlockMap &map);
#endif

  void
  clear();

  void
  set_size(const size_type size);

  size_type
  size() const;

  void
  add_range(const size_type begin, const size_type end);

  void
  add_index(const size_type index);

  template <typename ForwardIterator>
  void
  add_indices(const ForwardIterator &begin, const ForwardIterator &end);

  void
  add_indices(const IndexSet &other, const size_type offset = 0);

  bool
  is_element(const size_type index) const;

  bool
  is_contiguous() const;

  bool
  is_empty() const;

  bool
  is_ascending_and_one_to_one(const MPI_Comm &communicator) const;

  size_type
  n_elements() const;

  size_type
  nth_index_in_set(const size_type local_index) const;

  size_type
  index_within_set(const size_type global_index) const;

  unsigned int
  n_intervals() const;

  size_type
  largest_range_starting_index() const;

  void
  compress() const;

  bool
  operator==(const IndexSet &is) const;

  bool
  operator!=(const IndexSet &is) const;

  IndexSet operator&(const IndexSet &is) const;

  IndexSet
  get_view(const size_type begin, const size_type end) const;

  std::vector<IndexSet>
  split_by_block(
    const std::vector<types::global_dof_index> &n_indices_per_block) const;

  void
  subtract_set(const IndexSet &other);

  IndexSet
  tensor_product(const IndexSet &other) const;

  size_type
  pop_back();

  size_type
  pop_front();

  void
  fill_index_vector(std::vector<size_type> &indices) const;

  template <typename VectorType>
  void
  fill_binary_vector(VectorType &vector) const;

  template <class StreamType>
  void
  print(StreamType &out) const;

  void
  write(std::ostream &out) const;

  void
  read(std::istream &in);

  void
  block_write(std::ostream &out) const;

  void
  block_read(std::istream &in);

#ifdef DEAL_II_WITH_TRILINOS

  Epetra_Map
  make_trilinos_map(const MPI_Comm &communicator = MPI_COMM_WORLD,
                    const bool      overlapping  = false) const;

#  ifdef DEAL_II_TRILINOS_WITH_TPETRA
  Tpetra::Map<int, types::global_dof_index>
  make_tpetra_map(const MPI_Comm &communicator = MPI_COMM_WORLD,
                  const bool      overlapping  = false) const;
#  endif
#endif


  std::size_t
  memory_consumption() const;

  DeclException1(ExcIndexNotPresent,
                 size_type,
                 << "The global index " << arg1
                 << " is not an element of this set.");

  template <class Archive>
  void
  serialize(Archive &ar, const unsigned int version);


  class IntervalAccessor
  {
  public:
    IntervalAccessor(const IndexSet *idxset, const size_type range_idx);

    explicit IntervalAccessor(const IndexSet *idxset);

    size_type
    n_elements() const;

    bool
    is_valid() const;

    ElementIterator
    begin() const;

    ElementIterator
    end() const;

    size_type
    last() const;

  private:
    IntervalAccessor(const IntervalAccessor &other);
    IntervalAccessor &
    operator=(const IntervalAccessor &other);

    bool
    operator==(const IntervalAccessor &other) const;
    bool
    operator<(const IntervalAccessor &other) const;
    void
    advance();
    const IndexSet *index_set;

    size_type range_idx;

    friend class IntervalIterator;
  };

  class IntervalIterator
  {
  public:
    IntervalIterator(const IndexSet *idxset, const size_type range_idx);

    explicit IntervalIterator(const IndexSet *idxset);

    IntervalIterator();

    IntervalIterator(const IntervalIterator &other) = default;

    IntervalIterator &
    operator=(const IntervalIterator &other) = default;

    IntervalIterator &
    operator++();

    IntervalIterator
    operator++(int);

    const IntervalAccessor &operator*() const;

    const IntervalAccessor *operator->() const;

    bool
    operator==(const IntervalIterator &) const;

    bool
    operator!=(const IntervalIterator &) const;

    bool
    operator<(const IntervalIterator &) const;

    int
    operator-(const IntervalIterator &p) const;

    using iterator_category = std::forward_iterator_tag;
    using value_type        = IntervalAccessor;
    using difference_type   = std::ptrdiff_t;
    using pointer           = IntervalAccessor *;
    using reference         = IntervalAccessor &;

  private:
    IntervalAccessor accessor;
  };

  class ElementIterator
  {
  public:
    ElementIterator(const IndexSet *idxset,
                    const size_type range_idx,
                    const size_type index);

    explicit ElementIterator(const IndexSet *idxset);

    size_type operator*() const;

    bool
    is_valid() const;

    ElementIterator &
    operator++();

    ElementIterator
    operator++(int);

    bool
    operator==(const ElementIterator &) const;

    bool
    operator!=(const ElementIterator &) const;

    bool
    operator<(const ElementIterator &) const;

    std::ptrdiff_t
    operator-(const ElementIterator &p) const;

    using iterator_category = std::forward_iterator_tag;
    using value_type        = size_type;
    using difference_type   = std::ptrdiff_t;
    using pointer           = size_type *;
    using reference         = size_type &;

  private:
    void
    advance();

    const IndexSet *index_set;
    size_type range_idx;
    size_type idx;
  };

  ElementIterator
  begin() const;

  ElementIterator
  at(const size_type global_index) const;

  ElementIterator
  end() const;

  IntervalIterator
  begin_intervals() const;

  IntervalIterator
  end_intervals() const;

private:
  struct Range
  {
    size_type begin;
    size_type end;

    size_type nth_index_in_set;

    Range();

    Range(const size_type i1, const size_type i2);

    friend inline bool
    operator<(const Range &range_1, const Range &range_2)
    {
      return (
        (range_1.begin < range_2.begin) ||
        ((range_1.begin == range_2.begin) && (range_1.end < range_2.end)));
    }

    static bool
    end_compare(const IndexSet::Range &x, const IndexSet::Range &y)
    {
      return x.end < y.end;
    }

    static bool
    nth_index_compare(const IndexSet::Range &x, const IndexSet::Range &y)
    {
      return (x.nth_index_in_set + (x.end - x.begin) <
              y.nth_index_in_set + (y.end - y.begin));
    }

    friend inline bool
    operator==(const Range &range_1, const Range &range_2)
    {
      return ((range_1.begin == range_2.begin) && (range_1.end == range_2.end));
    }

    static std::size_t
    memory_consumption()
    {
      return sizeof(Range);
    }

    template <class Archive>
    void
    serialize(Archive &ar, const unsigned int version);
  };

  mutable std::vector<Range> ranges;

  mutable bool is_compressed;

  size_type index_space_size;

  mutable size_type largest_range;

  mutable Threads::Mutex compress_mutex;

  void
  do_compress() const;
};


inline IndexSet
complete_index_set(const IndexSet::size_type N)
{
  IndexSet is(N);
  is.add_range(0, N);
  is.compress();
  return is;
}

/* ------------------ inline functions ------------------ */


/* IntervalAccessor */

inline IndexSet::IntervalAccessor::IntervalAccessor(
  const IndexSet *          idxset,
  const IndexSet::size_type range_idx)
  : index_set(idxset)
  , range_idx(range_idx)
{
  Assert(range_idx < idxset->n_intervals(),
         ExcInternalError("Invalid range index"));
}



inline IndexSet::IntervalAccessor::IntervalAccessor(const IndexSet *idxset)
  : index_set(idxset)
  , range_idx(numbers::invalid_dof_index)
{}



inline IndexSet::IntervalAccessor::IntervalAccessor(
  const IndexSet::IntervalAccessor &other)
  : index_set(other.index_set)
  , range_idx(other.range_idx)
{
  Assert(range_idx == numbers::invalid_dof_index || is_valid(),
         ExcMessage("invalid iterator"));
}



inline IndexSet::size_type
IndexSet::IntervalAccessor::n_elements() const
{
  Assert(is_valid(), ExcMessage("invalid iterator"));
  return index_set->ranges[range_idx].end - index_set->ranges[range_idx].begin;
}



inline bool
IndexSet::IntervalAccessor::is_valid() const
{
  return index_set != nullptr && range_idx < index_set->n_intervals();
}



inline IndexSet::ElementIterator
IndexSet::IntervalAccessor::begin() const
{
  Assert(is_valid(), ExcMessage("invalid iterator"));
  return {index_set, range_idx, index_set->ranges[range_idx].begin};
}



inline IndexSet::ElementIterator
IndexSet::IntervalAccessor::end() const
{
  Assert(is_valid(), ExcMessage("invalid iterator"));

  // point to first index in next interval unless we are the last interval.
  if (range_idx < index_set->ranges.size() - 1)
    return {index_set, range_idx + 1, index_set->ranges[range_idx + 1].begin};
  else
    return index_set->end();
}



inline IndexSet::size_type
IndexSet::IntervalAccessor::last() const
{
  Assert(is_valid(), ExcMessage("invalid iterator"));

  return index_set->ranges[range_idx].end - 1;
}



inline IndexSet::IntervalAccessor &
IndexSet::IntervalAccessor::operator=(const IndexSet::IntervalAccessor &other)
{
  index_set = other.index_set;
  range_idx = other.range_idx;
  Assert(range_idx == numbers::invalid_dof_index || is_valid(),
         ExcMessage("invalid iterator"));
  return *this;
}



inline bool
IndexSet::IntervalAccessor::
operator==(const IndexSet::IntervalAccessor &other) const
{
  Assert(index_set == other.index_set,
         ExcMessage(
           "Can not compare accessors pointing to different IndexSets"));
  return range_idx == other.range_idx;
}



inline bool
IndexSet::IntervalAccessor::
operator<(const IndexSet::IntervalAccessor &other) const
{
  Assert(index_set == other.index_set,
         ExcMessage(
           "Can not compare accessors pointing to different IndexSets"));
  return range_idx < other.range_idx;
}



inline void
IndexSet::IntervalAccessor::advance()
{
  Assert(
    is_valid(),
    ExcMessage(
      "Impossible to advance an IndexSet::IntervalIterator that is invalid"));
  ++range_idx;

  // set ourselves to invalid if we walk off the end
  if (range_idx >= index_set->ranges.size())
    range_idx = numbers::invalid_dof_index;
}


/* IntervalIterator */

inline IndexSet::IntervalIterator::IntervalIterator(
  const IndexSet *          idxset,
  const IndexSet::size_type range_idx)
  : accessor(idxset, range_idx)
{}



inline IndexSet::IntervalIterator::IntervalIterator()
  : accessor(nullptr)
{}



inline IndexSet::IntervalIterator::IntervalIterator(const IndexSet *idxset)
  : accessor(idxset)
{}



inline IndexSet::IntervalIterator &
IndexSet::IntervalIterator::operator++()
{
  accessor.advance();
  return *this;
}



inline IndexSet::IntervalIterator
IndexSet::IntervalIterator::operator++(int)
{
  const IndexSet::IntervalIterator iter = *this;
  accessor.advance();
  return iter;
}



inline const IndexSet::IntervalAccessor &IndexSet::IntervalIterator::
                                         operator*() const
{
  return accessor;
}



inline const IndexSet::IntervalAccessor *IndexSet::IntervalIterator::
                                         operator->() const
{
  return &accessor;
}



inline bool
IndexSet::IntervalIterator::
operator==(const IndexSet::IntervalIterator &other) const
{
  return accessor == other.accessor;
}



inline bool
IndexSet::IntervalIterator::
operator!=(const IndexSet::IntervalIterator &other) const
{
  return !(*this == other);
}



inline bool
IndexSet::IntervalIterator::
operator<(const IndexSet::IntervalIterator &other) const
{
  return accessor < other.accessor;
}



inline int
IndexSet::IntervalIterator::
operator-(const IndexSet::IntervalIterator &other) const
{
  Assert(accessor.index_set == other.accessor.index_set,
         ExcMessage(
           "Can not compare iterators belonging to different IndexSets"));

  const size_type lhs = (accessor.range_idx == numbers::invalid_dof_index) ?
                          accessor.index_set->ranges.size() :
                          accessor.range_idx;
  const size_type rhs =
    (other.accessor.range_idx == numbers::invalid_dof_index) ?
      accessor.index_set->ranges.size() :
      other.accessor.range_idx;

  if (lhs > rhs)
    return static_cast<int>(lhs - rhs);
  else
    return -static_cast<int>(rhs - lhs);
}



/* ElementIterator */

inline IndexSet::ElementIterator::ElementIterator(
  const IndexSet *          idxset,
  const IndexSet::size_type range_idx,
  const IndexSet::size_type index)
  : index_set(idxset)
  , range_idx(range_idx)
  , idx(index)
{
  Assert(range_idx < index_set->ranges.size(),
         ExcMessage(
           "Invalid range index for IndexSet::ElementIterator constructor."));
  Assert(
    idx >= index_set->ranges[range_idx].begin &&
      idx < index_set->ranges[range_idx].end,
    ExcInternalError(
      "Invalid index argument for IndexSet::ElementIterator constructor."));
}



inline IndexSet::ElementIterator::ElementIterator(const IndexSet *idxset)
  : index_set(idxset)
  , range_idx(numbers::invalid_dof_index)
  , idx(numbers::invalid_dof_index)
{}



inline bool
IndexSet::ElementIterator::is_valid() const
{
  Assert((range_idx == numbers::invalid_dof_index &&
          idx == numbers::invalid_dof_index) ||
           (range_idx < index_set->ranges.size() &&
            idx < index_set->ranges[range_idx].end),
         ExcInternalError("Invalid ElementIterator state."));

  return (range_idx < index_set->ranges.size() &&
          idx < index_set->ranges[range_idx].end);
}



inline IndexSet::size_type IndexSet::ElementIterator::operator*() const
{
  Assert(
    is_valid(),
    ExcMessage(
      "Impossible to dereference an IndexSet::ElementIterator that is invalid"));
  return idx;
}



inline bool
IndexSet::ElementIterator::
operator==(const IndexSet::ElementIterator &other) const
{
  Assert(index_set == other.index_set,
         ExcMessage(
           "Can not compare iterators belonging to different IndexSets"));
  return range_idx == other.range_idx && idx == other.idx;
}



inline void
IndexSet::ElementIterator::advance()
{
  Assert(
    is_valid(),
    ExcMessage(
      "Impossible to advance an IndexSet::ElementIterator that is invalid"));
  if (idx < index_set->ranges[range_idx].end)
    ++idx;
  // end of this range?
  if (idx == index_set->ranges[range_idx].end)
    {
      // point to first element in next interval if possible
      if (range_idx < index_set->ranges.size() - 1)
        {
          ++range_idx;
          idx = index_set->ranges[range_idx].begin;
        }
      else
        {
          // we just fell off the end, set to invalid:
          range_idx = numbers::invalid_dof_index;
          idx       = numbers::invalid_dof_index;
        }
    }
}



inline IndexSet::ElementIterator &
IndexSet::ElementIterator::operator++()
{
  advance();
  return *this;
}



inline IndexSet::ElementIterator
IndexSet::ElementIterator::operator++(int)
{
  const IndexSet::ElementIterator it = *this;
  advance();
  return it;
}



inline bool
IndexSet::ElementIterator::
operator!=(const IndexSet::ElementIterator &other) const
{
  return !(*this == other);
}



inline bool
IndexSet::ElementIterator::
operator<(const IndexSet::ElementIterator &other) const
{
  Assert(index_set == other.index_set,
         ExcMessage(
           "Can not compare iterators belonging to different IndexSets"));
  return range_idx < other.range_idx ||
         (range_idx == other.range_idx && idx < other.idx);
}



inline std::ptrdiff_t
IndexSet::ElementIterator::
operator-(const IndexSet::ElementIterator &other) const
{
  Assert(index_set == other.index_set,
         ExcMessage(
           "Can not compare iterators belonging to different IndexSets"));
  if (*this == other)
    return 0;
  if (!(*this < other))
    return -(other - *this);

  // only other can be equal to end() because of the checks above.
  Assert(is_valid(), ExcInternalError());

  // Note: we now compute how far advance *this in "*this < other" to get other,
  // so we need to return -c at the end.

  // first finish the current range:
  std::ptrdiff_t c = index_set->ranges[range_idx].end - idx;

  // now walk in steps of ranges (need to start one behind our current one):
  for (size_type range = range_idx + 1;
       range < index_set->ranges.size() && range <= other.range_idx;
       ++range)
    c += index_set->ranges[range].end - index_set->ranges[range].begin;

  Assert(
    other.range_idx < index_set->ranges.size() ||
      other.range_idx == numbers::invalid_dof_index,
    ExcMessage(
      "Inconsistent iterator state. Did you invalidate iterators by modifying the IndexSet?"));

  // We might have walked too far because we went until the end of
  // other.range_idx, so walk backwards to other.idx:
  if (other.range_idx != numbers::invalid_dof_index)
    c -= index_set->ranges[other.range_idx].end - other.idx;

  return -c;
}


/* Range */

inline IndexSet::Range::Range()
  : begin(numbers::invalid_dof_index)
  , end(numbers::invalid_dof_index)
  , nth_index_in_set(numbers::invalid_dof_index)
{}



inline IndexSet::Range::Range(const size_type i1, const size_type i2)
  : begin(i1)
  , end(i2)
  , nth_index_in_set(numbers::invalid_dof_index)
{}



/* IndexSet itself */

inline IndexSet::IndexSet()
  : is_compressed(true)
  , index_space_size(0)
  , largest_range(numbers::invalid_unsigned_int)
{}



inline IndexSet::IndexSet(const size_type size)
  : is_compressed(true)
  , index_space_size(size)
  , largest_range(numbers::invalid_unsigned_int)
{}



inline IndexSet::IndexSet(IndexSet &&is) noexcept
  : ranges(std::move(is.ranges))
  , is_compressed(is.is_compressed)
  , index_space_size(is.index_space_size)
  , largest_range(is.largest_range)
{
  is.ranges.clear();
  is.is_compressed    = true;
  is.index_space_size = 0;
  is.largest_range    = numbers::invalid_unsigned_int;

  compress();
}



inline IndexSet &
IndexSet::operator=(IndexSet &&is) noexcept
{
  ranges           = std::move(is.ranges);
  is_compressed    = is.is_compressed;
  index_space_size = is.index_space_size;
  largest_range    = is.largest_range;

  is.ranges.clear();
  is.is_compressed    = true;
  is.index_space_size = 0;
  is.largest_range    = numbers::invalid_unsigned_int;

  compress();

  return *this;
}



inline IndexSet::ElementIterator
IndexSet::begin() const
{
  compress();
  if (ranges.size() > 0)
    return {this, 0, ranges[0].begin};
  else
    return end();
}



inline IndexSet::ElementIterator
IndexSet::at(const size_type global_index) const
{
  compress();
  AssertIndexRange(global_index, size());

  if (ranges.empty())
    return end();

  std::vector<Range>::const_iterator main_range =
    ranges.begin() + largest_range;

  Range r(global_index, global_index + 1);
  // This optimization makes the bounds for lower_bound smaller by checking
  // the largest range first.
  std::vector<Range>::const_iterator range_begin, range_end;
  if (global_index < main_range->begin)
    {
      range_begin = ranges.begin();
      range_end   = main_range;
    }
  else
    {
      range_begin = main_range;
      range_end   = ranges.end();
    }

  // This will give us the first range p=[a,b[ with b>=global_index using
  // a binary search
  const std::vector<Range>::const_iterator p =
    Utilities::lower_bound(range_begin, range_end, r, Range::end_compare);

  // We couldn't find a range, which means we have no range that contains
  // global_index and also no range behind it, meaning we need to return end().
  if (p == ranges.end())
    return end();

  // Finally, we can have two cases: Either global_index is not in [a,b[,
  // which means we need to return an iterator to a because global_index, ...,
  // a-1 is not in the IndexSet (if branch). Alternatively, global_index is in
  // [a,b[ and we will return an iterator pointing directly at global_index
  // (else branch).
  if (global_index < p->begin)
    return {this, static_cast<size_type>(p - ranges.begin()), p->begin};
  else
    return {this, static_cast<size_type>(p - ranges.begin()), global_index};
}



inline IndexSet::ElementIterator
IndexSet::end() const
{
  compress();
  return IndexSet::ElementIterator(this);
}



inline IndexSet::IntervalIterator
IndexSet::begin_intervals() const
{
  compress();
  if (ranges.size() > 0)
    return IndexSet::IntervalIterator(this, 0);
  else
    return end_intervals();
}



inline IndexSet::IntervalIterator
IndexSet::end_intervals() const
{
  compress();
  return IndexSet::IntervalIterator(this);
}



inline void
IndexSet::clear()
{
  // reset so that there are no indices in the set any more; however,
  // as documented, the index set retains its size
  ranges.clear();
  is_compressed = true;
  largest_range = numbers::invalid_unsigned_int;
}



inline void
IndexSet::set_size(const size_type sz)
{
  Assert(ranges.empty(),
         ExcMessage("This function can only be called if the current "
                    "object does not yet contain any elements."));
  index_space_size = sz;
  is_compressed    = true;
}



inline IndexSet::size_type
IndexSet::size() const
{
  return index_space_size;
}



inline void
IndexSet::compress() const
{
  if (is_compressed == true)
    return;

  do_compress();
}



inline void
IndexSet::add_index(const size_type index)
{
  AssertIndexRange(index, index_space_size);

  const Range new_range(index, index + 1);
  if (ranges.size() == 0 || index > ranges.back().end)
    ranges.push_back(new_range);
  else if (index == ranges.back().end)
    ranges.back().end++;
  else
    ranges.insert(Utilities::lower_bound(ranges.begin(),
                                         ranges.end(),
                                         new_range),
                  new_range);
  is_compressed = false;
}



inline void
IndexSet::add_range(const size_type begin, const size_type end)
{
  Assert((begin < index_space_size) ||
           ((begin == index_space_size) && (end == index_space_size)),
         ExcIndexRangeType<size_type>(begin, 0, index_space_size));
  Assert(end <= index_space_size,
         ExcIndexRangeType<size_type>(end, 0, index_space_size + 1));
  AssertIndexRange(begin, end + 1);

  if (begin != end)
    {
      const Range new_range(begin, end);

      // the new index might be larger than the last index present in the
      // ranges. Then we can skip the binary search
      if (ranges.size() == 0 || begin > ranges.back().end)
        ranges.push_back(new_range);
      else
        ranges.insert(Utilities::lower_bound(ranges.begin(),
                                             ranges.end(),
                                             new_range),
                      new_range);
      is_compressed = false;
    }
}



template <typename ForwardIterator>
inline void
IndexSet::add_indices(const ForwardIterator &begin, const ForwardIterator &end)
{
  if (begin == end)
    return;

  // identify ranges in the given iterator range by checking whether some
  // indices happen to be consecutive. to avoid quadratic complexity when
  // calling add_range many times (as add_range() going into the middle of an
  // already existing range must shift entries around), we first collect a
  // vector of ranges.
  std::vector<std::pair<size_type, size_type>> tmp_ranges;
  bool                                         ranges_are_sorted = true;
  for (ForwardIterator p = begin; p != end;)
    {
      const size_type begin_index = *p;
      size_type       end_index   = begin_index + 1;
      ForwardIterator q           = p;
      ++q;
      while ((q != end) && (*q == end_index))
        {
          ++end_index;
          ++q;
        }

      tmp_ranges.emplace_back(begin_index, end_index);
      p = q;

      // if the starting index of the next go-around of the for loop is less
      // than the end index of the one just identified, then we will have at
      // least one pair of ranges that are not sorted, and consequently the
      // whole collection of ranges is not sorted.
      if (p != end && *p < end_index)
        ranges_are_sorted = false;
    }

  if (!ranges_are_sorted)
    std::sort(tmp_ranges.begin(), tmp_ranges.end());

  // if we have many ranges, we first construct a temporary index set (where
  // we add ranges in a consecutive way, so fast), otherwise, we work with
  // add_range(). the number 9 is chosen heuristically given the fact that
  // there are typically up to 8 independent ranges when adding the degrees of
  // freedom on a 3D cell or 9 when adding degrees of freedom of faces. if
  // doing cell-by-cell additions, we want to avoid repeated calls to
  // IndexSet::compress() which gets called upon merging two index sets, so we
  // want to be in the other branch then.
  if (tmp_ranges.size() > 9)
    {
      IndexSet tmp_set(size());
      tmp_set.ranges.reserve(tmp_ranges.size());
      for (const auto &i : tmp_ranges)
        tmp_set.add_range(i.first, i.second);
      this->add_indices(tmp_set);
    }
  else
    for (const auto &i : tmp_ranges)
      add_range(i.first, i.second);
}



inline bool
IndexSet::is_element(const size_type index) const
{
  if (ranges.empty() == false)
    {
      compress();

      // fast check whether the index is in the largest range
      Assert(largest_range < ranges.size(), ExcInternalError());
      if (index >= ranges[largest_range].begin &&
          index < ranges[largest_range].end)
        return true;

      // get the element after which we would have to insert a range that
      // consists of all elements from this element to the end of the index
      // range plus one. after this call we know that if p!=end() then
      // p->begin<=index unless there is no such range at all
      //
      // if the searched for element is an element of this range, then we're
      // done. otherwise, the element can't be in one of the following ranges
      // because otherwise p would be a different iterator
      //
      // since we already know the position relative to the largest range (we
      // called compress!), we can perform the binary search on ranges with
      // lower/higher number compared to the largest range
      std::vector<Range>::const_iterator p = std::upper_bound(
        ranges.begin() +
          (index < ranges[largest_range].begin ? 0 : largest_range + 1),
        index < ranges[largest_range].begin ? ranges.begin() + largest_range :
                                              ranges.end(),
        Range(index, size() + 1));

      if (p == ranges.begin())
        return ((index >= p->begin) && (index < p->end));

      Assert((p == ranges.end()) || (p->begin > index), ExcInternalError());

      // now move to that previous range
      --p;
      Assert(p->begin <= index, ExcInternalError());

      return (p->end > index);
    }

  // didn't find this index, so it's not in the set
  return false;
}



inline bool
IndexSet::is_contiguous() const
{
  compress();
  return (ranges.size() <= 1);
}



inline bool
IndexSet::is_empty() const
{
  return ranges.empty();
}



inline IndexSet::size_type
IndexSet::n_elements() const
{
  // make sure we have non-overlapping ranges
  compress();

  size_type v = 0;
  if (!ranges.empty())
    {
      Range &r = ranges.back();
      v        = r.nth_index_in_set + r.end - r.begin;
    }

#ifdef DEBUG
  size_type s = 0;
  for (const auto &range : ranges)
    s += (range.end - range.begin);
  Assert(s == v, ExcInternalError());
#endif

  return v;
}



inline unsigned int
IndexSet::n_intervals() const
{
  compress();
  return ranges.size();
}



inline IndexSet::size_type
IndexSet::largest_range_starting_index() const
{
  Assert(ranges.empty() == false, ExcMessage("IndexSet cannot be empty."));

  compress();
  const std::vector<Range>::const_iterator main_range =
    ranges.begin() + largest_range;

  return main_range->nth_index_in_set;
}



inline IndexSet::size_type
IndexSet::nth_index_in_set(const size_type n) const
{
  AssertIndexRange(n, n_elements());

  compress();

  // first check whether the index is in the largest range
  Assert(largest_range < ranges.size(), ExcInternalError());
  std::vector<Range>::const_iterator main_range =
    ranges.begin() + largest_range;
  if (n >= main_range->nth_index_in_set &&
      n < main_range->nth_index_in_set + (main_range->end - main_range->begin))
    return main_range->begin + (n - main_range->nth_index_in_set);

  // find out which chunk the local index n belongs to by using a binary
  // search. the comparator is based on the end of the ranges. Use the
  // position relative to main_range to subdivide the ranges
  Range r(n, n + 1);
  r.nth_index_in_set = n;
  std::vector<Range>::const_iterator range_begin, range_end;
  if (n < main_range->nth_index_in_set)
    {
      range_begin = ranges.begin();
      range_end   = main_range;
    }
  else
    {
      range_begin = main_range + 1;
      range_end   = ranges.end();
    }

  const std::vector<Range>::const_iterator p =
    Utilities::lower_bound(range_begin, range_end, r, Range::nth_index_compare);

  Assert(p != ranges.end(), ExcInternalError());
  return p->begin + (n - p->nth_index_in_set);
}



inline IndexSet::size_type
IndexSet::index_within_set(const size_type n) const
{
  // to make this call thread-safe, compress() must not be called through this
  // function
  Assert(is_compressed == true, ExcMessage("IndexSet must be compressed."));
  AssertIndexRange(n, size());

  // return immediately if the index set is empty
  if (is_empty())
    return numbers::invalid_dof_index;

  // check whether the index is in the largest range. use the result to
  // perform a one-sided binary search afterward
  Assert(largest_range < ranges.size(), ExcInternalError());
  std::vector<Range>::const_iterator main_range =
    ranges.begin() + largest_range;
  if (n >= main_range->begin && n < main_range->end)
    return (n - main_range->begin) + main_range->nth_index_in_set;

  Range                              r(n, n);
  std::vector<Range>::const_iterator range_begin, range_end;
  if (n < main_range->begin)
    {
      range_begin = ranges.begin();
      range_end   = main_range;
    }
  else
    {
      range_begin = main_range + 1;
      range_end   = ranges.end();
    }

  std::vector<Range>::const_iterator p =
    Utilities::lower_bound(range_begin, range_end, r, Range::end_compare);

  // if n is not in this set
  if (p == range_end || p->end == n || p->begin > n)
    return numbers::invalid_dof_index;

  Assert(p != ranges.end(), ExcInternalError());
  Assert(p->begin <= n, ExcInternalError());
  Assert(n < p->end, ExcInternalError());
  return (n - p->begin) + p->nth_index_in_set;
}



inline bool
IndexSet::operator==(const IndexSet &is) const
{
  Assert(size() == is.size(), ExcDimensionMismatch(size(), is.size()));

  compress();
  is.compress();

  return ranges == is.ranges;
}



inline bool
IndexSet::operator!=(const IndexSet &is) const
{
  Assert(size() == is.size(), ExcDimensionMismatch(size(), is.size()));

  compress();
  is.compress();

  return ranges != is.ranges;
}



template <typename Vector>
void
IndexSet::fill_binary_vector(Vector &vector) const
{
  Assert(vector.size() == size(), ExcDimensionMismatch(vector.size(), size()));

  compress();
  // first fill all elements of the vector with zeroes.
  std::fill(vector.begin(), vector.end(), 0);

  // then write ones into the elements whose indices are contained in the
  // index set
  for (const auto &range : ranges)
    for (size_type i = range.begin; i < range.end; ++i)
      vector[i] = 1;
}



template <class StreamType>
inline void
IndexSet::print(StreamType &out) const
{
  compress();
  out << "{";
  std::vector<Range>::const_iterator p;
  for (p = ranges.begin(); p != ranges.end(); ++p)
    {
      if (p->end - p->begin == 1)
        out << p->begin;
      else
        out << "[" << p->begin << "," << p->end - 1 << "]";

      if (p != --ranges.end())
        out << ", ";
    }
  out << "}" << std::endl;
}



template <class Archive>
inline void
IndexSet::Range::serialize(Archive &ar, const unsigned int)
{
  ar &begin &end &nth_index_in_set;
}



template <class Archive>
inline void
IndexSet::serialize(Archive &ar, const unsigned int)
{
  ar &ranges &is_compressed &index_space_size &largest_range;
}

DEAL_II_NAMESPACE_CLOSE

#endif