psimpl.hpp

#ifndef PSIMPL_HPP
#define PSIMPL_HPP

/* ***** BEGIN LICENSE BLOCK *****
 * Version: MPL 1.1
 *
 * The contents of this file are subject to the Mozilla Public License Version
 * 1.1 (the "License"); you may not use this file except in compliance with
 * the License. You may obtain a copy of the License at
 * http://www.mozilla.org/MPL/
 *
 * Software distributed under the License is distributed on an "AS IS" basis,
 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
 * for the specific language governing rights and limitations under the
 * License.
 *
 * The Original Code is
 * 'psimpl - generic n-dimensional polyline simplification'.
 *
 * The Initial Developer of the Original Code is
 * Elmar de Koning.
 * Portions created by the Initial Developer are Copyright (C) 2010-2011
 * the Initial Developer. All Rights Reserved.
 *
 * Contributor(s):
 *
 * ***** END LICENSE BLOCK ***** */

/*
    psimpl - generic n-dimensional polyline simplification
    Copyright (C) 2010-2011 Elmar de Koning, edekoning@gmail.com

    This file is part of psimpl, and is hosted at SourceForge:
    http://sourceforge.net/projects/psimpl/
*/

/*!
    \mainpage psimpl - generic n-dimensional polyline simplification

<pre>
    Author  - Elmar de Koning
    Support - edekoning@gmail.com
    Website - http://psimpl.sf.net
    Article - http://www.codeproject.com/KB/recipes/PolylineSimplification.aspx
    License - MPL 1.1
</pre><br>

    \section sec_psimpl psimpl
<pre>
    'psimpl' is a c++ polyline simplification library that is generic, easy to
use, and supports the following algorithms:

    Simplification
    + Nth point - A naive algorithm that keeps only each nth point
    + Distance between points - Removes successive points that are clustered
together
    + Perpendicular distance - Removes points based on their distance to the
line segment defined by their left and right neighbors
    + Reumann-Witkam - Shifts a strip along the polyline and removes points that
fall outside
    + Opheim - A constrained version of Reumann-Witkam
    + Lang - Similar to the Perpendicular distance routine, but instead of
looking only at direct neighbors, an entire search region is processed
    + Douglas-Peucker - A classic simplification algorithm that provides an
excellent approximation of the original line
    + A variation on the Douglas-Peucker algorithm - Slower, but yields better
results at lower resolutions

    Errors
    + positional error - Distance of each polyline point to its simplification

    All the algorithms have been implemented in a single standalone C++ header
using an STL-style interface that operates on input and output iterators.
Polylines can be of any dimension, and defined using floating point or signed
integer data types.
</pre><br>

































    \section sec_changelog changelog
<pre>
    28-09-2010 - Initial version
    23-10-2010 - Changed license from CPOL to MPL
    26-10-2010 - Clarified input (type) requirements, and changed the behavior
of the algorithms under invalid input 01-12-2010 - Added the nth point,
perpendicular distance and Reumann-Witkam routines; moved all functions related
to distance calculations to the math namespace 10-12-2010 - Fixed a bug in the
perpendicular distance routine 27-02-2011 - Added Opheim simplification, and
functions for computing positional errors due to simplification; renamed
simplify_douglas_peucker_alt to simplify_douglas_peucker_n 18-06-2011 - Added
Lang simplification; fixed divide by zero bug when using integers; fixed a bug
where incorrect output iterators were returned under invalid input; fixed a bug
                 in douglas_peucker_n where an incorrect number of points could
be returned; fixed a bug in compute_positional_errors2 that required the output
and input iterator types to be the same; fixed a bug in
compute_positional_error_statistics where invalid statistics could be returned
under questionable input; documented input iterator requirements for each
algorithm; miscellaneous refactoring of most algorithms.
</pre>
*/

#ifndef PSIMPL_GENERIC
#define PSIMPL_GENERIC

#include <algorithm>
#include <cmath>
#include <functional>
#include <iterator>
#include <numeric>
#include <queue>
#include <stack>

/*!
    \brief Root namespace of the polyline simplification library.
*/
namespace psimpl {
/*!
    \brief Contains utility functions and classes.
*/
namespace util {
/*!
    \brief A smart pointer for holding a dynamically allocated array.

    Similar to boost::scoped_array.
*/
template<typename T> class scoped_array {
public:
  scoped_array(unsigned n) { array = new T[n]; }

  ~scoped_array() { delete[] array; }

  T&
  operator[](int offset) {
    return array[offset];
  }

  const T&
  operator[](int offset) const {
    return array[offset];
  }

  T*
  get() const {
    return array;
  }

  void
  swap(scoped_array& b) {
    T* tmp = b.array;
    b.array = array;
    array = tmp;
  }

private:
  scoped_array(const scoped_array&);
  scoped_array& operator=(const scoped_array&);

private:
  T* array;
};

template<typename T>
inline void
swap(scoped_array<T>& a, scoped_array<T>& b) {
  a.swap(b);
}

} // namespace util

/*!
    \brief Contains functions for calculating statistics and distances between
   various geometric entities.
*/
namespace math {
/*!
    \brief POD structure for storing several statistical values
*/
struct Statistics {
  Statistics() : max(0), sum(0), mean(0), std(0) {}

  double max;
  double sum;
  double mean;
  double std; //! standard deviation
};

/*!
    \brief Determines if two points have the exact same coordinates.

    \param[in] p1       the first coordinate of the first point
    \param[in] p2       the first coordinate of the second point
    \return             true when the points are equal; false otherwise
*/
template<unsigned DIM, class InputIterator>
inline bool
equal(InputIterator p1, InputIterator p2) {
  for(unsigned d = 0; d < DIM; ++d) {
    if(*p1 != *p2) {
      return false;
    }
    ++p1;
    ++p2;
  }

  return true;
}

/*!
    \brief Creates a vector from two points.

    \param[in] p1       the first coordinate of the first point
    \param[in] p2       the first coordinate of the second point
    \param[in] result   the resulting vector (p2-p1)
    \return             one beyond the last coordinate of the resulting vector
*/
template<unsigned DIM, class InputIterator, class OutputIterator>
inline OutputIterator
make_vector(InputIterator p1, InputIterator p2, OutputIterator result) {
  for(unsigned d = 0; d < DIM; ++d) {
    *result = *p2 - *p1;
    ++result;
    ++p1;
    ++p2;
  }

  return result;
}

/*!
    \brief Computes the dot product of two vectors.

    \param[in] v1   the first coordinate of the first vector
    \param[in] v2   the first coordinate of the second vector
    \return         the dot product (v1 * v2)
*/
template<unsigned DIM, class InputIterator>
inline typename std::iterator_traits<InputIterator>::value_type
dot(InputIterator v1, InputIterator v2) {
  typename std::iterator_traits<InputIterator>::value_type result = 0;
  for(unsigned d = 0; d < DIM; ++d) {
    result += (*v1) * (*v2);
    ++v1;
    ++v2;
  }

  return result;
}

/*!
    \brief Peforms linear interpolation between two points.

    \param[in] p1           the first coordinate of the first point
    \param[in] p2           the first coordinate of the second point
    \param[in] fraction     the fraction used during interpolation
    \param[in] result       the interpolation result (p1 + fraction * (p2 - p1))
    \return                 one beyond the last coordinate of the interpolated
   point
*/
template<unsigned DIM, class InputIterator, class OutputIterator>
inline OutputIterator
interpolate(InputIterator p1, InputIterator p2, float fraction, OutputIterator result) {
  typedef typename std::iterator_traits<InputIterator>::value_type value_type;

  for(unsigned d = 0; d < DIM; ++d) {
    *result = *p1 + static_cast<value_type>(fraction * (*p2 - *p1));
    ++result;
    ++p1;
    ++p2;
  }

  return result;
}

/*!
    \brief Computes the squared distance of two points

    \param[in] p1   the first coordinate of the first point
    \param[in] p2   the first coordinate of the second point
    \return         the squared distance
*/
template<unsigned DIM, class InputIterator1, class InputIterator2>
inline typename std::iterator_traits<InputIterator1>::value_type
point_distance2(InputIterator1 p1, InputIterator2 p2) {
  typename std::iterator_traits<InputIterator1>::value_type result = 0;
  for(unsigned d = 0; d < DIM; ++d) {
    result += (*p1 - *p2) * (*p1 - *p2);
    ++p1;
    ++p2;
  }

  return result;
}

/*!
    \brief Computes the squared distance between an infinite line (l1, l2) and a
   point p

    \param[in] l1   the first coordinate of the first point on the line
    \param[in] l2   the first coordinate of the second point on the line
    \param[in] p    the first coordinate of the test point
    \return         the squared distance
*/
template<unsigned DIM, class InputIterator>
inline typename std::iterator_traits<InputIterator>::value_type
line_distance2(InputIterator l1, InputIterator l2, InputIterator p) {
  typedef typename std::iterator_traits<InputIterator>::value_type value_type;

  value_type v[DIM]; // vector l1 --> l2
  value_type w[DIM]; // vector l1 --> p

  make_vector<DIM>(l1, l2, v);
  make_vector<DIM>(l1, p, w);

  value_type cv = dot<DIM>(v, v); // squared length of v
  value_type cw = dot<DIM>(w, v); // project w onto v

  // avoid problems with divisions when value_type is an integer type
  float fraction = cv == 0 ? 0 : static_cast<float>(cw) / static_cast<float>(cv);

  value_type proj[DIM]; // p projected onto line (l1, l2)
  interpolate<DIM>(l1, l2, fraction, proj);

  return point_distance2<DIM>(p, proj);
}

/*!
    \brief Computes the squared distance between a line segment (s1, s2) and a
   point p

    \param[in] s1   the first coordinate of the start point of the segment
    \param[in] s2   the first coordinate of the end point of the segment
    \param[in] p    the first coordinate of the test point
    \return         the squared distance
*/
template<unsigned DIM, class InputIterator>
inline typename std::iterator_traits<InputIterator>::value_type
segment_distance2(InputIterator s1, InputIterator s2, InputIterator p) {
  typedef typename std::iterator_traits<InputIterator>::value_type value_type;

  value_type v[DIM]; // vector s1 --> s2
  value_type w[DIM]; // vector s1 --> p

  make_vector<DIM>(s1, s2, v);
  make_vector<DIM>(s1, p, w);

  value_type cw = dot<DIM>(w, v); // project w onto v
  if(cw <= 0) {
    // projection of w lies to the left of s1
    return point_distance2<DIM>(p, s1);
  }

  value_type cv = dot<DIM>(v, v); // squared length of v
  if(cv <= cw) {
    // projection of w lies to the right of s2
    return point_distance2<DIM>(p, s2);
  }

  // avoid problems with divisions when value_type is an integer type
  float fraction = cv == 0 ? 0 : static_cast<float>(cw) / static_cast<float>(cv);

  value_type proj[DIM]; // p projected onto segement (s1, s2)
  interpolate<DIM>(s1, s2, fraction, proj);

  return point_distance2<DIM>(p, proj);
}

/*!
    \brief Computes the squared distance between a ray (r1, r2) and a point p

    \param[in] r1   the first coordinate of the start point of the ray
    \param[in] r2   the first coordinate of a point on the ray
    \param[in] p    the first coordinate of the test point
    \return         the squared distance
*/
template<unsigned DIM, class InputIterator>
inline typename std::iterator_traits<InputIterator>::value_type
ray_distance2(InputIterator r1, InputIterator r2, InputIterator p) {
  typedef typename std::iterator_traits<InputIterator>::value_type value_type;

  value_type v[DIM]; // vector r1 --> r2
  value_type w[DIM]; // vector r1 --> p

  make_vector<DIM>(r1, r2, v);
  make_vector<DIM>(r1, p, w);

  value_type cv = dot<DIM>(v, v); // squared length of v
  value_type cw = dot<DIM>(w, v); // project w onto v

  if(cw <= 0) {
    // projection of w lies to the left of r1 (not on the ray)
    return point_distance2<DIM>(p, r1);
  }

  // avoid problems with divisions when value_type is an integer type
  float fraction = cv == 0 ? 0 : static_cast<float>(cw) / static_cast<float>(cv);

  value_type proj[DIM]; // p projected onto ray (r1, r2)
  interpolate<DIM>(r1, r2, fraction, proj);

  return point_distance2<DIM>(p, proj);
}

/*!
    \brief Computes various statistics for the range [first, last)

    \param[in] first   the first value
    \param[in] last    one beyond the last value
    \return            the calculated statistics
*/
template<class InputIterator>
inline Statistics
compute_statistics(InputIterator first, InputIterator last) {
  typedef typename std::iterator_traits<InputIterator>::value_type value_type;
  typedef typename std::iterator_traits<InputIterator>::difference_type diff_type;

  Statistics stats;

  diff_type count = std::distance(first, last);
  if(count == 0) {
    return stats;
  }

  value_type init = 0;
  stats.max = static_cast<double>(*std::max_element(first, last));
  stats.sum = static_cast<double>(std::accumulate(first, last, init));
  stats.mean = stats.sum / count;
  std::transform(first, last, first, std::bind(std::minus<value_type>(), stats.mean));
  stats.std = std::sqrt(static_cast<double>(std::inner_product(first, last, first, init)) / count);
  return stats;
}

} // namespace math

/*!
    \brief Provides various simplification algorithms for n-dimensional simple
   polylines.

    A polyline is simple when it is non-closed and non-selfintersecting. All
   algorithms operate on input iterators and output iterators. Note that
   unisgned integer types are NOT supported.
*/
template<unsigned DIM, class InputIterator, class OutputIterator> class PolylineSimplification {
  typedef typename std::iterator_traits<InputIterator>::difference_type diff_type;
  typedef typename std::iterator_traits<InputIterator>::value_type value_type;
  typedef typename std::iterator_traits<const value_type*>::difference_type ptr_diff_type;

public:
  /*!
      \brief Performs the nth point routine (NP).

      NP is an O(n) algorithm for polyline simplification. It keeps only the
     first, last and each nth point. As an example, consider any random line of
     8 points. Using n = 3 will always yield a simplification consisting of
     points: 1, 4, 7, 8

      \image html psimpl_np.png

      NP is applied to the range [first, last). The resulting simplified
     polyline is copied to the output range [result, result + m*DIM), where m is
     the number of vertices of the simplified polyline. The return value is the
     end of the output range: result + m*DIM.

      Input (Type) requirements:
      1- DIM is not 0, where DIM represents the dimension of the polyline
      2- The InputIterator type models the concept of a forward iterator
      3- The input iterator value type is convertible to a value type of the
     OutputIterator 4- The range [first, last) contains only vertex coordinates
     in multiples of DIM, f.e.: x, y, z, x, y, z, x, y, z when DIM = 3 5- The
     range [first, last) contains at least 2 vertices 6- n is not 0

      In case these requirements are not met, the entire input range [first,
     last) is copied to the output range [result, result + (last - first)) OR
     compile errors may occur.

      \param[in] first    the first coordinate of the first polyline point
      \param[in] last     one beyond the last coordinate of the last polyline
     point \param[in] n        specifies 'each nth point' \param[in] result
     destination of the simplified polyline \return             one beyond the
     last coordinate of the simplified polyline
  */
  OutputIterator
  nth_point(InputIterator first, InputIterator last, unsigned n, OutputIterator result) {
    diff_type coordCount = std::distance(first, last);
    diff_type pointCount = DIM // protect against zero DIM
                               ? coordCount / DIM
                               : 0;

    // validate input and check if simplification required
    if(coordCount % DIM || pointCount < 3 || n < 2) {
      return std::copy(first, last, result);
    }

    unsigned remaining = pointCount - 1; // the number of points remaining after key
    InputIterator key = first;           // indicates the current key

    // the first point is always part of the simplification
    copy_key(key, result);

    // copy each nth point
    while(Forward(key, n, remaining)) {
      copy_key(key, result);
    }

    return result;
  }

  /*!
      \brief Performs the (radial) distance between points routine (RD).

      RD is a brute-force O(n) algorithm for polyline simplification. It reduces
     successive vertices that are clustered too closely to a single vertex,
     called a key. The resulting keys form the simplified polyline.

      \image html psimpl_rd.png

      RD is applied to the range [first, last) using the specified tolerance
     tol. The resulting simplified polyline is copied to the output range
     [result, result + m*DIM), where m is the number of vertices of the
     simplified polyline. The return value is the end of the output range:
     result + m*DIM.

      Input (Type) requirements:
      1- DIM is not 0, where DIM represents the dimension of the polyline
      2- The InputIterator type models the concept of a forward iterator
      3- The input iterator value type is convertible to a value type of the
     output iterator 4- The range [first, last) contains only vertex coordinates
     in multiples of DIM, f.e.: x, y, z, x, y, z, x, y, z when DIM = 3 5- The
     range [first, last) contains at least 2 vertices 6- tol is not 0

      In case these requirements are not met, the entire input range [first,
     last) is copied to the output range [result, result + (last - first)) OR
     compile errors may occur.

      \param[in] first    the first coordinate of the first polyline point
      \param[in] last     one beyond the last coordinate of the last polyline
     point \param[in] tol      radial (point-to-point) distance tolerance
      \param[in] result   destination of the simplified polyline
      \return             one beyond the last coordinate of the simplified
     polyline
  */
  OutputIterator
  radial_distance(InputIterator first, InputIterator last, value_type tol, OutputIterator result) {
    diff_type coordCount = std::distance(first, last);
    diff_type pointCount = DIM // protect against zero DIM
                               ? coordCount / DIM
                               : 0;
    value_type tol2 = tol * tol; // squared distance tolerance

    // validate input and check if simplification required
    if(coordCount % DIM || pointCount < 3 || tol2 == 0) {
      return std::copy(first, last, result);
    }

    InputIterator current = first; // indicates the current key
    InputIterator next = first;    // used to find the next key

    // the first point is always part of the simplification
    copy_key_advance(next, result);

    // Skip first and last point, because they are always part of the
    // simplification
    for(diff_type index = 1; index < pointCount - 1; ++index) {
      if(math::point_distance2<DIM>(current, next) < tol2) {
        Advance(next);
        continue;
      }
      current = next;
      copy_key_advance(next, result);
    }
    // the last point is always part of the simplification
    copy_key_advance(next, result);

    return result;
  }

  /*!
      \brief Repeatedly performs the perpendicular distance routine (PD).

      The algorithm stops after calling the PD routine 'repeat' times OR when
     the simplification does not improve. Note that this algorithm will need to
     store up to two intermediate simplification results.

      \sa perpendicular_distance(InputIterator, InputIterator, value_type,
     OutputIterator)

      \param[in] first    the first coordinate of the first polyline point
      \param[in] last     one beyond the last coordinate of the last polyline
     point \param[in] tol      perpendicular (segment-to-point) distance
     tolerance \param[in] repeat   the number of times to successively apply the
     PD routine \param[in] result   destination of the simplified polyline
      \return             one beyond the last coordinate of the simplified
     polyline
  */
  OutputIterator
  perpendicular_distance(InputIterator first, InputIterator last, value_type tol, unsigned repeat, OutputIterator result) {
    if(repeat == 1) {
      // single pass
      return perpendicular_distance(first, last, tol, result);
    }
    // only validate repeat; other input is validated by
    // simplify_perpendicular_distance
    if(repeat < 1) {
      return std::copy(first, last, result);
    }
    diff_type coordCount = std::distance(first, last);

    // first pass: [first, last) --> temporary array 'temp_poly'
    util::scoped_array<value_type> temp_poly(coordCount);
    PolylineSimplification<DIM, InputIterator, value_type*> psimpl_to_array;
    diff_type tempCoordCount = std::distance(temp_poly.get(), psimpl_to_array.perpendicular_distance(first, last, tol, temp_poly.get()));

    // check if simplification did not improved
    if(coordCount == tempCoordCount) {
      return std::copy(temp_poly.get(), temp_poly.get() + coordCount, result);
    }
    std::swap(coordCount, tempCoordCount);
    --repeat;

    // intermediate passes: temporary array 'temp_poly' --> temporary array
    // 'temp_result'
    if(1 < repeat) {
      util::scoped_array<value_type> temp_result(coordCount);
      PolylineSimplification<DIM, value_type*, value_type*> psimpl_arrays;

      while(--repeat) {
        tempCoordCount =
            std::distance(temp_result.get(), psimpl_arrays.perpendicular_distance(temp_poly.get(), temp_poly.get() + coordCount, tol, temp_result.get()));

        // check if simplification did not improved
        if(coordCount == tempCoordCount) {
          return std::copy(temp_poly.get(), temp_poly.get() + coordCount, result);
        }
        util::swap(temp_poly, temp_result);
        std::swap(coordCount, tempCoordCount);
      }
    }

    // final pass: temporary array 'temp_poly' --> result
    PolylineSimplification<DIM, value_type*, OutputIterator> psimpl_from_array;
    return psimpl_from_array.perpendicular_distance(temp_poly.get(), temp_poly.get() + coordCount, tol, result);
  }

  /*!
      \brief Performs the perpendicular distance routine (PD).

      PD is an O(n) algorithm for polyline simplification. It computes the
     perpendicular distance of each point pi to the line segment S(pi-1, pi+1).
     Only when this distance is larger than the given tolerance will pi be part
     of the simpification. Note that the original polyline can only be reduced
     by a maximum of 50%. Multiple passes are required to achieve higher points
     reductions.

      \image html psimpl_pd.png

      PD is applied to the range [first, last) using the specified tolerance
     tol. The resulting simplified polyline is copied to the output range
     [result, result + m*DIM), where m is the number of vertices of the
     simplified polyline. The return value is the end of the output range:
     result + m*DIM.

      Input (Type) requirements:
      1- DIM is not 0, where DIM represents the dimension of the polyline
      2- The InputIterator type models the concept of a forward iterator
      3- The input iterator value type is convertible to a value type of the
     output iterator 4- The range [first, last) contains only vertex coordinates
     in multiples of DIM, f.e.: x, y, z, x, y, z, x, y, z when DIM = 3 5- The
     range [first, last) contains at least 2 vertices 6- tol is not 0

      In case these requirements are not met, the entire input range [first,
     last) is copied to the output range [result, result + (last - first)) OR
     compile errors may occur.

      \param[in] first    the first coordinate of the first polyline point
      \param[in] last     one beyond the last coordinate of the last polyline
     point \param[in] tol      perpendicular (segment-to-point) distance
     tolerance \param[in] result   destination of the simplified polyline
      \return             one beyond the last coordinate of the simplified
     polyline
  */
  OutputIterator
  perpendicular_distance(InputIterator first, InputIterator last, value_type tol, OutputIterator result) {
    diff_type coordCount = std::distance(first, last);
    diff_type pointCount = DIM // protect against zero DIM
                               ? coordCount / DIM
                               : 0;
    value_type tol2 = tol * tol; // squared distance tolerance

    // validate input and check if simplification required
    if(coordCount % DIM || pointCount < 3 || tol2 == 0) {
      return std::copy(first, last, result);
    }

    InputIterator p0 = first;
    InputIterator p1 = advance_copy(p0);
    InputIterator p2 = advance_copy(p1);

    // the first point is always part of the simplification
    copy_key(p0, result);

    while(p2 != last) {
      // test p1 against line segment S(p0, p2)
      if(math::segment_distance2<DIM>(p0, p2, p1) < tol2) {
        copy_key(p2, result);
        // move up by two points
        p0 = p2;
        Advance(p1, 2);
        if(p1 == last) {
          // protect against advancing p2 beyond last
          break;
        }
        Advance(p2, 2);
      } else {
        copy_key(p1, result);
        // move up by one point
        p0 = p1;
        p1 = p2;
        Advance(p2);
      }
    }
    // make sure the last point is part of the simplification
    if(p1 != last) {
      copy_key(p1, result);
    }

    return result;
  }

  /*!
      \brief Performs Reumann-Witkam approximation (RW).

      The O(n) RW routine uses a point-to-line (perpendicular) distance
     tolerance. It defines a line through the first two vertices of the original
     polyline. For each successive vertex vi its perpendicular distance to this
     line is calculated. A new key is found at vi-1, when this distance exceeds
     the specified tolerance. The vertices vi and vi+1 are then used to define a
     new line, and the process repeats itself.

      \image html psimpl_rw.png

      RW routine is applied to the range [first, last) using the specified
     perpendicular distance tolerance tol. The resulting simplified polyline is
     copied to the output range [result, result + m*DIM), where m is the number
     of vertices of the simplified polyline. The return value is the end of the
     output range: result + m*DIM.

      Input (Type) Requirements:
      1- DIM is not 0, where DIM represents the dimension of the polyline
      2- The InputIterator type models the concept of a forward iterator
      3- The input iterator value type is convertible to a value type of the
     output iterator 4- The range [first, last) contains vertex coordinates in
     multiples of DIM, f.e.: x, y, z, x, y, z, x, y, z when DIM = 3 5- The range
     [first, last) contains at least 2 vertices 6- tol is not 0

      In case these requirements are not met, the entire input range [first,
     last) is copied to the output range [result, result + (last - first)) OR
     compile errors may occur.

      \param[in] first    the first coordinate of the first polyline point
      \param[in] last     one beyond the last coordinate of the last polyline
     point \param[in] tol      perpendicular (point-to-line) distance tolerance
      \param[in] result   destination of the simplified polyline
      \return             one beyond the last coordinate of the simplified
     polyline
  */
  OutputIterator
  reumann_witkam(InputIterator first, InputIterator last, value_type tol, OutputIterator result) {
    diff_type coordCount = std::distance(first, last);
    diff_type pointCount = DIM // protect against zero DIM
                               ? coordCount / DIM
                               : 0;
    value_type tol2 = tol * tol; // squared distance tolerance

    // validate input and check if simplification required
    if(coordCount % DIM || pointCount < 3 || tol2 == 0) {
      return std::copy(first, last, result);
    }

    // define the line L(p0, p1)
    InputIterator p0 = first;               // indicates the current key
    InputIterator p1 = advance_copy(first); // indicates the next point after p0

    // keep track of two test points
    InputIterator pi = p1; // the previous test point
    InputIterator pj = p1; // the current test point (pi+1)

    // the first point is always part of the simplification
    copy_key(p0, result);

    // check each point pj against L(p0, p1)
    for(diff_type j = 2; j < pointCount; ++j) {
      pi = pj;
      Advance(pj);

      if(math::line_distance2<DIM>(p0, p1, pj) < tol2) {
        continue;
      }
      // found the next key at pi
      copy_key(pi, result);
      // define new line L(pi, pj)
      p0 = pi;
      p1 = pj;
    }
    // the last point is always part of the simplification
    copy_key(pj, result);

    return result;
  }

  /*!
      \brief Performs Opheim approximation (OP).

      The O(n) OP routine is very similar to the Reumann-Witkam (RW) routine,
     and can be seen as a constrained version of that RW routine. OP uses both a
     minimum and a maximum distance tolerance to constrain the search area. For
     each successive vertex vi, its radial distance to the current key vkey
     (initially v0) is calculated. The last point within the minimum distance
     tolerance is used to define a ray R (vkey, vi). If no such vi exists, the
     ray is defined as R(vkey, vkey+1). For each successive vertex vj beyond vi
     its perpendicular distance to the ray R is calculated. A new key is found
     at vj-1, when this distance exceeds the minimum tolerance Or when the
     radial distance between vj and the vkey exceeds the maximum tolerance.
     After a new key is found, the process repeats itself.

      \image html psimpl_op.png

      OP routine is applied to the range [first, last) using the specified
     distance tolerances min_tol and max_tol. The resulting simplified polyline
     is copied to the output range [result, result + m*DIM), where m is the
     number of vertices of the simplified polyline. The return value is the end
     of the output range: result + m*DIM.

      Input (Type) Requirements:
      1- DIM is not 0, where DIM represents the dimension of the polyline
      2- The InputIterator type models the concept of a forward iterator
      3- The input iterator value type is convertible to a value type of the
     output iterator 4- The range [first, last) contains vertex coordinates in
     multiples of DIM, f.e.: x, y, z, x, y, z, x, y, z when DIM = 3 5- The range
     [first, last) contains at least 2 vertices 6- min_tol is not 0 7- max_tol
     is not 0

      In case these requirements are not met, the entire input range [first,
     last) is copied to the output range [result, result + (last - first)) OR
     compile errors may occur.

      \param[in] first    the first coordinate of the first polyline point
      \param[in] last     one beyond the last coordinate of the last polyline
     point \param[in] min_tol  radial and perpendicular (point-to-ray) distance
     tolerance \param[in] max_tol  radial distance tolerance \param[in] result
     destination of the simplified polyline \return             one beyond the
     last coordinate of the simplified polyline
  */
  OutputIterator
  Opheim(InputIterator first, InputIterator last, value_type min_tol, value_type max_tol, OutputIterator result) {
    diff_type coordCount = std::distance(first, last);
    diff_type pointCount = DIM // protect against zero DIM
                               ? coordCount / DIM
                               : 0;
    value_type min_tol2 = min_tol * min_tol; // squared minimum distance tolerance
    value_type max_tol2 = max_tol * max_tol; // squared maximum distance tolerance

    // validate input and check if simplification required
    if(coordCount % DIM || pointCount < 3 || min_tol2 == 0 || max_tol2 == 0) {
      return std::copy(first, last, result);
    }

    // define the ray R(r0, r1)
    InputIterator r0 = first; // indicates the current key and start of the ray
    InputIterator r1 = first; // indicates a point on the ray
    bool rayDefined = false;

    // keep track of two test points
    InputIterator pi = r0; // the previous test point
    InputIterator pj =     // the current test point (pi+1)
        advance_copy(pi);

    // the first point is always part of the simplification
    copy_key(r0, result);

    for(diff_type j = 2; j < pointCount; ++j) {
      pi = pj;
      Advance(pj);

      if(!rayDefined) {
        // discard each point within minimum tolerance
        if(math::point_distance2<DIM>(r0, pj) < min_tol2) {
          continue;
        }
        // the last point within minimum tolerance pi defines the ray R(r0, r1)
        r1 = pi;
        rayDefined = true;
      }

      // check each point pj against R(r0, r1)
      if(math::point_distance2<DIM>(r0, pj) < max_tol2 && math::ray_distance2<DIM>(r0, r1, pj) < min_tol2) {
        continue;
      }
      // found the next key at pi
      copy_key(pi, result);
      // define new ray R(pi, pj)
      r0 = pi;
      rayDefined = false;
    }
    // the last point is always part of the simplification
    copy_key(pj, result);

    return result;
  }

  /*!
      \brief Performs Lang approximation (LA).

      The LA routine defines a fixed size search-region. The first and last
     points of that search region form a segment. This segment is used to
     calculate the perpendicular distance to each intermediate point. If any
     calculated distance is larger than the specified tolerance, the search
     region will be shrunk by excluding its last point. This process will
     continue untill all calculated distances fall below the specified tolerance
      , or there are no more intermediate points. At this point all intermediate
     points are removed and a new search region is defined starting at the last
     point from old search region. Note that the size of the search region
     (look_ahead parameter) controls the maximum amount of simplification, e.g.:
     a size of 20 will always result in a simplification that contains at least
     5% of the original points.

      \image html psimpl_la.png

      LA routine is applied to the range [first, last) using the specified
     tolerance and look ahead values. The resulting simplified polyline is
     copied to the output range [result, result + m*DIM), where m is the number
     of vertices of the simplified polyline. The return value is the end of the
     output range: result + m*DIM.

      Input (Type) Requirements:
      1- DIM is not 0, where DIM represents the dimension of the polyline
      2- The InputIterator type models the concept of a bidirectional iterator
      3- The InputIterator value type is convertible to a value type of the
     output iterator 4- The range [first, last) contains vertex coordinates in
     multiples of DIM, f.e.: x, y, z, x, y, z, x, y, z when DIM = 3 5- The range
     [first, last) contains at least 2 vertices 6- tol is not 0 7- look_ahead is
     not zero

      In case these requirements are not met, the entire input range [first,
     last) is copied to the output range [result, result + (last - first)) OR
     compile errors may occur.

      \param[in] first      the first coordinate of the first polyline point
      \param[in] last       one beyond the last coordinate of the last polyline
     point \param[in] tol        perpendicular (point-to-segment) distance
     tolerance \param[in] look_ahead defines the size of the search region
      \param[in] result     destination of the simplified polyline
      \return               one beyond the last coordinate of the simplified
     polyline
  */
  OutputIterator
  Lang(InputIterator first, InputIterator last, value_type tol, unsigned look_ahead, OutputIterator result) {
    diff_type coordCount = std::distance(first, last);
    diff_type pointCount = DIM // protect against zero DIM
                               ? coordCount / DIM
                               : 0;
    value_type tol2 = tol * tol; // squared minimum distance tolerance

    // validate input and check if simplification required
    if(coordCount % DIM || pointCount < 3 || look_ahead < 2 || tol2 == 0) {
      return std::copy(first, last, result);
    }

    InputIterator current = first; // indicates the current key
    InputIterator next = first;    // used to find the next key

    unsigned remaining = pointCount - 1; // the number of points remaining after current
    unsigned moved = Forward(next, look_ahead, remaining);

    // the first point is always part of the simplification
    copy_key(current, result);

    while(moved) {
      value_type d2 = 0;
      InputIterator p = advance_copy(current);

      while(p != next) {
        d2 = std::max(d2, math::segment_distance2<DIM>(current, next, p));
        if(tol2 < d2) {
          break;
        }
        Advance(p);
      }
      if(d2 < tol2) {
        current = next;
        copy_key(current, result);
        moved = Forward(next, look_ahead, remaining);
      } else {
        Backward(next, remaining);
      }
    }

    return result;
  }

  /*!
      \brief Performs Douglas-Peucker approximation (DP).

      The DP algorithm uses the radial_distance (RD) routine O(n) as a
     preprocessing step. After RD the algorithm is O (n m) in worst case and O(n
     log m) on average, where m < n (m is the number of points after RD).

      The DP algorithm starts with a simplification that is the single edge
     joining the first and last vertices of the polyline. The distance of the
     remaining vertices to that edge are computed. The vertex that is furthest
     away from theedge (called a key), and has a computed distance that is
     larger than a specified tolerance, will be added to the simplification.
     This process will recurse for each edge in the current simplification,
      untill all vertices of the original polyline are within tolerance.

      \image html psimpl_dp.png

      Note that this algorithm will create a copy of the input polyline during
     the vertex reduction step.

      RD followed by DP is applied to the range [first, last) using the
     specified tolerance tol. The resulting simplified polyline is copied to the
     output range [result, result + m*DIM), where m is the number of vertices of
     the simplified polyline. The return value is the end of the output range:
     result + m*DIM.

      Input (Type) requirements:
      1- DIM is not 0, where DIM represents the dimension of the polyline
      2- The InputIterator type models the concept of a forward iterator
      3- The InputIterator value type is convertible to a value type of the
     output iterator 4- The range [first, last) contains vertex coordinates in
     multiples of DIM, f.e.: x, y, z, x, y, z, x, y, z when DIM = 3 5- The range
     [first, last) contains at least 2 vertices 6- tol is not 0

      In case these requirements are not met, the entire input range [first,
     last) is copied to the output range [result, result + (last - first)) OR
     compile errors may occur.

      \param[in] first    the first coordinate of the first polyline point
      \param[in] last     one beyond the last coordinate of the last polyline
     point \param[in] tol      perpendicular (point-to-segment) distance
     tolerance \param[in] result   destination of the simplified polyline
      \return             one beyond the last coordinate of the simplified
     polyline
  */
  OutputIterator
  douglas_peucker(InputIterator first, InputIterator last, value_type tol, OutputIterator result) {
    diff_type coordCount = std::distance(first, last);
    diff_type pointCount = DIM // protect against zero DIM
                               ? coordCount / DIM
                               : 0;
    // validate input and check if simplification required
    if(coordCount % DIM || pointCount < 3 || tol == 0) {
      return std::copy(first, last, result);
    }
    // radial distance routine as preprocessing
    util::scoped_array<value_type> reduced(coordCount); // radial distance results
    PolylineSimplification<DIM, InputIterator, value_type*> psimpl_to_array;
    ptr_diff_type reducedCoordCount = std::distance(reduced.get(), psimpl_to_array.radial_distance(first, last, tol, reduced.get()));
    ptr_diff_type reducedPointCount = reducedCoordCount / DIM;

    // douglas-peucker approximation
    util::scoped_array<unsigned char> keys(pointCount); // douglas-peucker results
    DPHelper::Approximate(reduced.get(), reducedCoordCount, tol, keys.get());

    // copy all keys
    const value_type* current = reduced.get();
    for(ptr_diff_type p = 0; p < reducedPointCount; ++p, current += DIM) {
      if(keys[p]) {
        for(unsigned d = 0; d < DIM; ++d) {
          *result = current[d];
          ++result;
        }
      }
    }

    return result;
  }

  /*!
      \brief Performs a Douglas-Peucker approximation variant (DPn).

      This algorithm is a variation of the original implementation. Instead of
     considering one polyline segment at a time, all segments of the current
     simplified polyline are evaluated at each step. Only the vertex with the
     maximum distance from its edge is added to the simplification. This process
     will recurse untill the the simplification contains the desired amount of
     vertices.

      The algorithm, which does not use the (radial) distance between points
     routine as a preprocessing step, is O(n2) in worst case and O(n log n) on
     average.

      Note that this algorithm will create a copy of the input polyline for
     performance reasons.

      DPn is applied to the range [first, last). The resulting simplified
     polyline consists of count vertices and is copied to the output range
     [result, result + count). The return value is the end of the output range:
     result + count.

      Input (Type) requirements:
      1- DIM is not 0, where DIM represents the dimension of the polyline
      2- The InputIterator type models the concept of a forward iterator
      3- The InputIterator value type is convertible to a value type of the
     output iterator 4- The range [first, last) contains vertex coordinates in
     multiples of DIM, f.e.: x, y, z, x, y, z, x, y, z when DIM = 3 5- The range
     [first, last) contains a minimum of count vertices 6- count is at least 2

      In case these requirements are not met, the entire input range [first,
     last) is copied to the output range [result, result + (last - first)) OR
     compile errors may occur.

      \sa douglas_peucker

      \param[in] first    the first coordinate of the first polyline point
      \param[in] last     one beyond the last coordinate of the last polyline
     point \param[in] count    the maximum number of points of the simplified
     polyline \param[in] result   destination of the simplified polyline \return
     one beyond the last coordinate of the simplified polyline
  */
  OutputIterator
  douglas_peucker_n(InputIterator first, InputIterator last, unsigned count, OutputIterator result) {
    diff_type coordCount = std::distance(first, last);
    diff_type pointCount = DIM // protect against zero DIM
                               ? coordCount / DIM
                               : 0;
    // validate input and check if simplification required
    if(coordCount % DIM || pointCount <= static_cast<diff_type>(count) || count < 2) {
      return std::copy(first, last, result);
    }

    // copy coords
    util::scoped_array<value_type> coords(coordCount);
    for(ptr_diff_type c = 0; c < coordCount; ++c) {
      coords[c] = *first;
      ++first;
    }

    // douglas-peucker approximation
    util::scoped_array<unsigned char> keys(pointCount);
    DPHelper::approximate_n(coords.get(), coordCount, count, keys.get());

    // copy keys
    const value_type* current = coords.get();
    for(ptr_diff_type p = 0; p < pointCount; ++p, current += DIM) {
      if(keys[p]) {
        for(unsigned d = 0; d < DIM; ++d) {
          *result = current[d];
          ++result;
        }
      }
    }

    return result;
  }

  /*!
      \brief Computes the squared positional error between a polyline and its
     simplification.

      For each point in the range [original_first, original_last) the squared
     distance to the simplification [simplified_first, simplified_last) is
     calculated. Each positional error is copied to the output range [result,
     result + count), where count is the number of points in the original
     polyline. The return value is the end of the output range: result + count.

      Note that both the original and simplified polyline must be defined using
     the same value_type.

      \image html psimpl_pos_error.png

      Input (Type) requirements:
      1- DIM is not 0, where DIM represents the dimension of the polyline
      2- The InputIterator type models the concept of a forward iterator
      3- The InputIterator value type is convertible to a value type of the
     output iterator 4- The ranges [original_first, original_last) and
     [simplified_first, simplified_last) contain vertex coordinates in multiples
     of DIM, f.e.: x, y, z, x, y, z, x, y, z when DIM = 3 5- The ranges
     [original_first, original_last) and [simplified_first, simplified_last)
         contain a minimum of 2 vertices
      6- The range [simplified_first, simplified_last) represents a
     simplification of the range [original_first, original_last), meaning each
     point in the simplification has the exact same coordinates as some point
     from the original polyline

      In case these requirements are not met, the valid flag is set to false OR
      compile errors may occur.

      \param[in] original_first   the first coordinate of the first polyline
     point \param[in] original_last    one beyond the last coordinate of the
     last polyline point \param[in] simplified_first the first coordinate of the
     first simplified polyline point \param[in] simplified_last  one beyond the
     last coordinate of the last simplified polyline point \param[in] result
     destination of the squared positional errors \param[out] valid [optional]
     indicates if the computed positional errors are valid \return one beyond
     the last computed positional error
  */
  OutputIterator
  compute_positional_errors2(InputIterator original_first,
                             InputIterator original_last,
                             InputIterator simplified_first,
                             InputIterator simplified_last,
                             OutputIterator result,
                             bool* valid = 0) {
    diff_type original_coordCount = std::distance(original_first, original_last);
    diff_type original_pointCount = DIM // protect against zero DIM
                                        ? original_coordCount / DIM
                                        : 0;

    diff_type simplified_coordCount = std::distance(simplified_first, simplified_last);
    diff_type simplified_pointCount = DIM // protect against zero DIM
                                          ? simplified_coordCount / DIM
                                          : 0;

    // validate input
    if(original_coordCount % DIM || original_pointCount < 2 || simplified_coordCount % DIM || simplified_pointCount < 2 ||
       original_pointCount < simplified_pointCount || !math::equal<DIM>(original_first, simplified_first)) {
      if(valid) {
        *valid = false;
      }

      return result;
    }

    // define (simplified) line segment S(simplified_prev, simplified_first)
    InputIterator simplified_prev = simplified_first;
    std::advance(simplified_first, DIM);

    // process each simplified line segment
    while(simplified_first != simplified_last) {
      // process each original point until it equals the end of the line segment
      while(original_first != original_last && !math::equal<DIM>(original_first, simplified_first)) {
        *result = math::segment_distance2<DIM>(simplified_prev, simplified_first, original_first);
        ++result;
        std::advance(original_first, DIM);
      }
      // update line segment S
      simplified_prev = simplified_first;
      std::advance(simplified_first, DIM);
    }
    // check if last original point matched
    if(original_first != original_last) {
      *result = 0;
      ++result;
    }

    if(valid) {
      *valid = original_first != original_last;
    }

    return result;
  }

  /*!
      \brief Computes statistics for the positional errors between a polyline
     and its simplification.

      Various statistics (mean, max, sum, std) are calculated for the positional
     errors between the range [original_first, original_last) and its
     simplification the range [simplified_first, simplified_last).

      Input (Type) requirements:
      1- DIM is not 0, where DIM represents the dimension of the polyline
      2- The InputIterator type models the concept of a forward iterator
      3- The InputIterator value type is convertible to double
      4- The ranges [original_first, original_last) and [simplified_first,
     simplified_last) contain vertex coordinates in multiples of DIM, f.e.: x,
     y, z, x, y, z, x, y, z when DIM = 3 5- The ranges [original_first,
     original_last) and [simplified_first, simplified_last) contain a minimum of
     2 vertices 6- The range [simplified_first, simplified_last) represents a
     simplification of the range [original_first, original_last), meaning each
     point in the simplification has the exact same coordinates as some point
     from the original polyline

      In case these requirements are not met, the valid flag is set to false OR
      compile errors may occur.

      \sa compute_positional_errors2

      \param[in] original_first   the first coordinate of the first polyline
     point \param[in] original_last    one beyond the last coordinate of the
     last polyline point \param[in] simplified_first the first coordinate of the
     first simplified polyline point \param[in] simplified_last  one beyond the
     last coordinate of the last simplified polyline point \param[out] valid
     [optional] indicates if the computed statistics are valid \return the
     computed statistics
  */
  math::Statistics
  compute_positional_error_statistics(
      InputIterator original_first, InputIterator original_last, InputIterator simplified_first, InputIterator simplified_last, bool* valid = 0) {
    diff_type pointCount = std::distance(original_first, original_last) / DIM;
    util::scoped_array<double> errors(pointCount);
    PolylineSimplification<DIM, InputIterator, double*> ps;

    diff_type errorCount =
        std::distance(errors.get(), ps.compute_positional_errors2(original_first, original_last, simplified_first, simplified_last, errors.get(), valid));

    std::transform(errors.get(), errors.get() + errorCount, errors.get(), &sqrtl);

    return math::compute_statistics(errors.get(), errors.get() + errorCount);
  }

private:
  /*!
      \brief Copies the key to the output destination, and increments the
     iterator.

      \sa copy_key

      \param[in,out] key      the first coordinate of the key
      \param[in,out] result   destination of the copied key
  */
  inline void
  copy_key_advance(InputIterator& key, OutputIterator& result) {
    for(unsigned d = 0; d < DIM; ++d) {
      *result = *key;
      ++result;
      ++key;
    }
  }

  /*!
      \brief Copies the key to the output destination.

      \sa copy_key_advance

      \param[in]     key      the first coordinate of the key
      \param[in,out] result   destination of the copied key
  */
  inline void
  copy_key(InputIterator key, OutputIterator& result) {
    copy_key_advance(key, result);
  }

  /*!
      \brief Increments the iterator by n points.

      \sa advance_copy

      \param[in,out] it  iterator to be advanced
      \param[in]     n   number of points to advance
  */
  inline void
  Advance(InputIterator& it, diff_type n = 1) {
    std::advance(it, n * static_cast<diff_type>(DIM));
  }

  /*!
      \brief Increments a copy of the iterator by n points.

      \sa Advance

      \param[in] it   iterator to be advanced
      \param[in] n    number of points to advance
      \return         an incremented copy of the input iterator
  */
  inline InputIterator
  advance_copy(InputIterator it, diff_type n = 1) {
    Advance(it, n);
    return it;
  }

  /*!
      \brief Increments the iterator by n points if possible.

      If there are fewer than n point remaining the iterator will be incremented
     to the last point.

      \sa Backward

      \param[in,out] it           iterator to be advanced
      \param[in]     n            number of points to advance
      \param[in,out] remaining    number of points remaining after it
      \return                     the actual amount of points that the iterator
     advanced
  */
  inline unsigned
  Forward(InputIterator& it, unsigned n, unsigned& remaining) {
    n = std::min(n, remaining);
    Advance(it, n);
    remaining -= n;
    return n;
  }

  /*!
      \brief Decrements the iterator by 1 point.

      \sa Forward

      \param[in,out] it            iterator to be advanced
      \param[in,out] remaining     number of points remaining after it
  */
  inline void
  Backward(InputIterator& it, unsigned& remaining) {
    Advance(it, -1);
    ++remaining;
  }

private:
  /*!
      \brief Douglas-Peucker approximation helper class.

      Contains helper implentations for Douglas-Peucker approximation that
     operate solely on value_type arrays and value_type pointers. Note that the
     PolylineSimplification class only operates on iterators.
  */
  class DPHelper {
    //! \brief Defines a sub polyline.
    struct SubPoly {
      SubPoly(ptr_diff_type first = 0, ptr_diff_type last = 0) : first(first), last(last) {}

      ptr_diff_type first; //! coord index of the first point
      ptr_diff_type last;  //! coord index of the last point
    };

    //! \brief Defines the key of a polyline.
    struct KeyInfo {
      KeyInfo(ptr_diff_type index = 0, value_type dist2 = 0) : index(index), dist2(dist2) {}

      ptr_diff_type index; //! coord index of the key
      value_type dist2;    //! squared distance of the key to a segment
    };

    //! \brief Defines a sub polyline including its key.
    struct SubPolyAlt {
      SubPolyAlt(ptr_diff_type first = 0, ptr_diff_type last = 0) : first(first), last(last) {}

      ptr_diff_type first; //! coord index of the first point
      ptr_diff_type last;  //! coord index of the last point
      KeyInfo keyInfo;     //! key of this sub poly

      bool
      operator<(const SubPolyAlt& other) const {
        return keyInfo.dist2 < other.keyInfo.dist2;
      }
    };

  public:
    /*!
        \brief Performs Douglas-Peucker approximation.

        \param[in] coords       array of polyline coordinates
        \param[in] coordCount   number of coordinates in coords []
        \param[in] tol          approximation tolerance
        \param[out] keys        indicates for each polyline point if it is a key
    */
    static void
    Approximate(const value_type* coords, ptr_diff_type coordCount, value_type tol, unsigned char* keys) {
      value_type tol2 = tol * tol; // squared distance tolerance
      ptr_diff_type pointCount = coordCount / DIM;
      // zero out keys
      std::fill_n(keys, pointCount, 0);
      keys[0] = 1;              // the first point is always a key
      keys[pointCount - 1] = 1; // the last point is always a key

      typedef std::stack<SubPoly> Stack;
      Stack stack; // LIFO job-queue containing sub-polylines

      SubPoly sub_poly(0, coordCount - DIM);
      stack.push(sub_poly); // add complete poly

      while(!stack.empty()) {
        sub_poly = stack.top(); // take a sub poly
        stack.pop();            // and find its key
        KeyInfo keyInfo = find_key(coords, sub_poly.first, sub_poly.last);
        if(keyInfo.index && tol2 < keyInfo.dist2) {
          // store the key if valid
          keys[keyInfo.index / DIM] = 1;
          // split the polyline at the key and recurse
          stack.push(SubPoly(keyInfo.index, sub_poly.last));
          stack.push(SubPoly(sub_poly.first, keyInfo.index));
        }
      }
    }

    /*!
        \brief Performs Douglas-Peucker approximation.

        \param[in] coords       array of polyline coordinates
        \param[in] coordCount   number of coordinates in coords []
        \param[in] countTol     point count tolerance
        \param[out] keys        indicates for each polyline point if it is a key
    */
    static void
    approximate_n(const value_type* coords, ptr_diff_type coordCount, unsigned countTol, unsigned char* keys) {
      ptr_diff_type pointCount = coordCount / DIM;
      // zero out keys
      std::fill_n(keys, pointCount, 0);
      keys[0] = 1;              // the first point is always a key
      keys[pointCount - 1] = 1; // the last point is always a key
      unsigned keyCount = 2;

      if(countTol == 2) {
        return;
      }

      typedef std::priority_queue<SubPolyAlt> PriorityQueue;
      PriorityQueue queue; // sorted (max dist2) job queue containing sub-polylines

      SubPolyAlt sub_poly(0, coordCount - DIM);
      sub_poly.keyInfo = find_key(coords, sub_poly.first, sub_poly.last);
      queue.push(sub_poly); // add complete poly

      while(!queue.empty()) {
        sub_poly = queue.top(); // take a sub poly
        queue.pop();
        // store the key
        keys[sub_poly.keyInfo.index / DIM] = 1;
        // check point count tolerance
        keyCount++;
        if(keyCount == countTol) {
          break;
        }
        // split the polyline at the key and recurse
        SubPolyAlt left(sub_poly.first, sub_poly.keyInfo.index);
        left.keyInfo = find_key(coords, left.first, left.last);
        if(left.keyInfo.index) {
          queue.push(left);
        }
        SubPolyAlt right(sub_poly.keyInfo.index, sub_poly.last);
        right.keyInfo = find_key(coords, right.first, right.last);
        if(right.keyInfo.index) {
          queue.push(right);
        }
      }
    }

  private:
    /*!
        \brief Finds the key for the given sub polyline.

        Finds the point in the range [first, last] that is furthest away from
       the segment (first, last). This point is called the key.

        \param[in] coords   array of polyline coordinates
        \param[in] first    the first coordinate of the first polyline point
        \param[in] last     the first coordinate of the last polyline point
        \return             the index of the key and its distance, or last when
       a key could not be found
    */
    static KeyInfo
    find_key(const value_type* coords, ptr_diff_type first, ptr_diff_type last) {
      KeyInfo keyInfo;

      for(ptr_diff_type current = first + DIM; current < last; current += DIM) {
        value_type d2 = math::segment_distance2<DIM>(coords + first, coords + last, coords + current);
        if(d2 < keyInfo.dist2) {
          continue;
        }
        // update maximum squared distance and the point it belongs to
        keyInfo.index = current;
        keyInfo.dist2 = d2;
      }

      return keyInfo;
    }
  };
};

/*!
    \brief Performs the nth point routine (NP).

    This is a convenience function that provides template type deduction for
    PolylineSimplification::nth_point.

    \param[in] first    the first coordinate of the first polyline point
    \param[in] last     one beyond the last coordinate of the last polyline
   point \param[in] n        specifies 'each nth point' \param[in] result
   destination of the simplified polyline \return             one beyond the
   last coordinate of the simplified polyline
*/
template<unsigned DIM, class ForwardIterator, class OutputIterator>
OutputIterator
simplify_nth_point(ForwardIterator first, ForwardIterator last, unsigned n, OutputIterator result) {
  PolylineSimplification<DIM, ForwardIterator, OutputIterator> ps;
  return ps.nth_point(first, last, n, result);
}

/*!
    \brief Performs the (radial) distance between points routine (RD).

    This is a convenience function that provides template type deduction for
    PolylineSimplification::radial_distance.

    \param[in] first    the first coordinate of the first polyline point
    \param[in] last     one beyond the last coordinate of the last polyline
   point \param[in] tol      radial (point-to-point) distance tolerance
    \param[in] result   destination of the simplified polyline
    \return             one beyond the last coordinate of the simplified
   polyline
*/
template<unsigned DIM, class ForwardIterator, class OutputIterator>
OutputIterator
simplify_radial_distance(ForwardIterator first, ForwardIterator last, typename std::iterator_traits<ForwardIterator>::value_type tol, OutputIterator result) {
  PolylineSimplification<DIM, ForwardIterator, OutputIterator> ps;
  return ps.radial_distance(first, last, tol, result);
}

/*!
    \brief Repeatedly performs the perpendicular distance routine (PD).

    This is a convenience function that provides template type deduction for
    PolylineSimplification::perpendicular_distance.

    \param[in] first    the first coordinate of the first polyline point
    \param[in] last     one beyond the last coordinate of the last polyline
   point \param[in] tol      perpendicular (segment-to-point) distance tolerance
    \param[in] repeat   the number of times to successively apply the PD
   routine. \param[in] result   destination of the simplified polyline \return
   one beyond the last coordinate of the simplified polyline
*/
template<unsigned DIM, class ForwardIterator, class OutputIterator>
OutputIterator
simplify_perpendicular_distance(
    ForwardIterator first, ForwardIterator last, typename std::iterator_traits<ForwardIterator>::value_type tol, unsigned repeat, OutputIterator result) {
  PolylineSimplification<DIM, ForwardIterator, OutputIterator> ps;
  return ps.perpendicular_distance(first, last, tol, repeat, result);
}

/*!
    \brief Performs the perpendicular distance routine (PD).

    This is a convenience function that provides template type deduction for
    PolylineSimplification::perpendicular_distance.

    \param[in] first    the first coordinate of the first polyline point
    \param[in] last     one beyond the last coordinate of the last polyline
   point \param[in] tol      perpendicular (segment-to-point) distance tolerance
    \param[in] result   destination of the simplified polyline
    \return             one beyond the last coordinate of the simplified
   polyline
*/
template<unsigned DIM, class ForwardIterator, class OutputIterator>
OutputIterator
simplify_perpendicular_distance(ForwardIterator first,
                                ForwardIterator last,
                                typename std::iterator_traits<ForwardIterator>::value_type tol,
                                OutputIterator result) {
  PolylineSimplification<DIM, ForwardIterator, OutputIterator> ps;
  return ps.perpendicular_distance(first, last, tol, result);
}

/*!
    \brief Performs Reumann-Witkam polyline simplification (RW).

    This is a convenience function that provides template type deduction for
    PolylineSimplification::reumann_witkam.

    \param[in] first    the first coordinate of the first polyline point
    \param[in] last     one beyond the last coordinate of the last polyline
   point \param[in] tol      perpendicular (point-to-line) distance tolerance
    \param[in] result   destination of the simplified polyline
    \return             one beyond the last coordinate of the simplified  polyline
*/
template<unsigned DIM, class ForwardIterator, class OutputIterator>
OutputIterator
simplify_reumann_witkam(ForwardIterator first, ForwardIterator last, typename std::iterator_traits<ForwardIterator>::value_type tol, OutputIterator result) {
  PolylineSimplification<DIM, ForwardIterator, OutputIterator> ps;
  return ps.reumann_witkam(first, last, tol, result);
}

/*!
    \brief Performs Opheim polyline simplification (OP).

    This is a convenience function that provides template type deduction for
    PolylineSimplification::Opheim.

    \param[in] first    the first coordinate of the first polyline point
    \param[in] last     one beyond the last coordinate of the last polyline
   point \param[in] min_tol  minimum distance tolerance \param[in] max_tol
   maximum distance tolerance \param[in] result   destination of the simplified
   polyline \return             one beyond the last coordinate of the simplified
   polyline
*/
template<unsigned DIM, class ForwardIterator, class OutputIterator>
OutputIterator
simplify_opheim(ForwardIterator first,
                ForwardIterator last,
                typename std::iterator_traits<ForwardIterator>::value_type min_tol,
                typename std::iterator_traits<ForwardIterator>::value_type max_tol,
                OutputIterator result) {
  PolylineSimplification<DIM, ForwardIterator, OutputIterator> ps;
  return ps.Opheim(first, last, min_tol, max_tol, result);
}

/*!
    \brief Performs Lang polyline simplification (LA).

    This is a convenience function that provides template type deduction for
    PolylineSimplification::Lang.

    \param[in] first      the first coordinate of the first polyline point
    \param[in] last       one beyond the last coordinate of the last polyline
   point \param[in] tol        perpendicular (point-to-segment) distance
   tolerance \param[in] look_ahead defines the size of the search region
    \param[in] result     destination of the simplified polyline
    \return               one beyond the last coordinate of the simplified
   polyline
*/
template<unsigned DIM, class BidirectionalIterator, class OutputIterator>
OutputIterator
simplify_lang(BidirectionalIterator first,
              BidirectionalIterator last,
              typename std::iterator_traits<BidirectionalIterator>::value_type tol,
              unsigned look_ahead,
              OutputIterator result) {
  PolylineSimplification<DIM, BidirectionalIterator, OutputIterator> ps;
  return ps.Lang(first, last, tol, look_ahead, result);
}

/*!
    \brief Performs Douglas-Peucker polyline simplification (DP).

    This is a convenience function that provides template type deduction for
    PolylineSimplification::douglas_peucker.

    \param[in] first    the first coordinate of the first polyline point
    \param[in] last     one beyond the last coordinate of the last polyline
   point \param[in] tol      perpendicular (point-to-segment) distance tolerance
    \param[in] result   destination of the simplified polyline
    \return             one beyond the last coordinate of the simplified
   polyline
*/
template<unsigned DIM, class ForwardIterator, class OutputIterator>
OutputIterator
simplify_douglas_peucker(ForwardIterator first, ForwardIterator last, typename std::iterator_traits<ForwardIterator>::value_type tol, OutputIterator result) {
  PolylineSimplification<DIM, ForwardIterator, OutputIterator> ps;
  return ps.douglas_peucker(first, last, tol, result);
}

/*!
    \brief Performs a variant of Douglas-Peucker polyline simplification (DPn).

    This is a convenience function that provides template type deduction for
    PolylineSimplification::DouglasPeuckerAlt.

    \param[in] first    the first coordinate of the first polyline point
    \param[in] last     one beyond the last coordinate of the last polyline
   point \param[in] count    the maximum number of points of the simplified
   polyline \param[in] result   destination of the simplified polyline \return
   one beyond the last coordinate of the simplified polyline
*/
template<unsigned DIM, class ForwardIterator, class OutputIterator>
OutputIterator
simplify_douglas_peucker_n(ForwardIterator first, ForwardIterator last, unsigned count, OutputIterator result) {
  PolylineSimplification<DIM, ForwardIterator, OutputIterator> ps;
  return ps.douglas_peucker_n(first, last, count, result);
}

/*!
    \brief Computes the squared positional error between a polyline and its
   simplification.

    This is a convenience function that provides template type deduction for
    PolylineSimplification::compute_positional_errors2.

    \param[in] original_first   the first coordinate of the first polyline point
    \param[in] original_last    one beyond the last coordinate of the last
   polyline point \param[in] simplified_first the first coordinate of the first
   simplified polyline point \param[in] simplified_last  one beyond the last
   coordinate of the last simplified polyline point \param[in] result
   destination of the squared positional errors \param[out] valid [optional]
   indicates if the computed positional errors are valid \return one beyond the
   last computed positional error
*/
template<unsigned DIM, class ForwardIterator, class OutputIterator>
OutputIterator
compute_positional_errors2(ForwardIterator original_first,
                           ForwardIterator original_last,
                           ForwardIterator simplified_first,
                           ForwardIterator simplified_last,
                           OutputIterator result,
                           bool* valid = 0) {
  PolylineSimplification<DIM, ForwardIterator, OutputIterator> ps;
  return ps.compute_positional_errors2(original_first, original_last, simplified_first, simplified_last, result, valid);
}

/*!
    \brief Computes statistics for the positional errors between a polyline and
   its simplification.

    This is a convenience function that provides template type deduction for
    PolylineSimplification::compute_positional_error_statistics.

    \param[in] original_first   the first coordinate of the first polyline point
    \param[in] original_last    one beyond the last coordinate of the last
   polyline point \param[in] simplified_first the first coordinate of the first
   simplified polyline point \param[in] simplified_last  one beyond the last
   coordinate of the last simplified polyline point \param[out] valid [optional]
   indicates if the computed statistics are valid \return the computed
   statistics
*/
template<unsigned DIM, class ForwardIterator>
math::Statistics
compute_positional_error_statistics(
    ForwardIterator original_first, ForwardIterator original_last, ForwardIterator simplified_first, ForwardIterator simplified_last, bool* valid = 0) {
  PolylineSimplification<DIM, ForwardIterator, ForwardIterator> ps;
  return ps.compute_positional_error_statistics(original_first, original_last, simplified_first, simplified_last, valid);
}

} // namespace psimpl

#endif // PSIMPL_GENERIC
#endif // defined PSIMPL_HPP