File: rectangular_3_center_2.h

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// Copyright (c) 1998-2003  ETH Zurich (Switzerland).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
// You can redistribute it and/or modify it under the terms of the GNU
// General Public License as published by the Free Software Foundation,
// either version 3 of the License, or (at your option) any later version.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL: svn+ssh://scm.gforge.inria.fr/svn/cgal/branches/next/Matrix_search/include/CGAL/rectangular_3_center_2.h $
// $Id: rectangular_3_center_2.h 67117 2012-01-13 18:14:48Z lrineau $
// 
//
// Author(s)     : Michael Hoffmann <hoffmann@inf.ethz.ch>

#ifndef CGAL_RECTANGULAR_3_CENTER_2_H
#define CGAL_RECTANGULAR_3_CENTER_2_H 1

#include <CGAL/basic.h>
#include <CGAL/Optimisation/assertions.h>
#include <CGAL/algorithm.h>
#include <CGAL/Rectangular_p_center_traits_2.h>
#include <algorithm>
#include <vector>
#include <boost/bind.hpp>
#include <boost/function.hpp>

namespace CGAL {

template < class ForwardIterator, class OutputIterator,
           class FT, class Traits >
OutputIterator
rectangular_2_center_2(
  ForwardIterator f,
  ForwardIterator l,
  OutputIterator o,
  FT& r,
  Traits& t)
{
  using std::pair;
  using std::greater;
  using std::less;

  typedef typename Traits::Iso_rectangle_2        Rectangle;
  typedef typename Traits::Point_2                Point;
  typedef typename Traits::Infinity_distance_2    Dist;
  typedef typename Traits::Construct_vertex_2     CVertex;
  typedef typename Traits::Construct_point_2_above_right_implicit_point_2
    P_above_right;
  typedef typename Traits::Construct_point_2_above_left_implicit_point_2
    P_above_left;
  typedef typename Traits::Construct_point_2_below_right_implicit_point_2
    P_below_right;
  typedef typename Traits::Construct_point_2_below_left_implicit_point_2
    P_below_left;
  typedef boost::function1<FT, Point>              Gamma;

  // fetch function objects from traits class
  CVertex       v      = t.construct_vertex_2_object();
  Dist          dist   = t.infinity_distance_2_object();
  P_above_left  pt_a_l =
    t.construct_point_2_above_left_implicit_point_2_object();
  P_above_right pt_a_r =
    t.construct_point_2_above_right_implicit_point_2_object();
  P_below_left  pt_b_l =
    t.construct_point_2_below_left_implicit_point_2_object();
  P_below_right pt_b_r =
    t.construct_point_2_below_right_implicit_point_2_object();

  // compute bounding box
  Rectangle bb = bounding_box_2(f, l, t);

  // two cases: top-left & bottom-right or top-right & bottom-left
  Min< FT > minft;
  Gamma gamma1 =
    boost::bind(minft, boost::bind(dist, v(bb, 0), _1), boost::bind(dist, v(bb, 2), _1));
  Gamma gamma2 =
    boost::bind(minft, boost::bind(dist, v(bb, 1), _1), boost::bind(dist, v(bb, 3), _1));

  pair< ForwardIterator, ForwardIterator > cand =
    min_max_element(f, l,
                    boost::bind(greater<FT>(), boost::bind(gamma1, _1), boost::bind(gamma1, _2)),
                    boost::bind(less<FT>(), boost::bind(gamma2, _1), boost::bind(gamma2, _2)));

  // return the result
  if (gamma1(*cand.first) < gamma2(*cand.second)) {
    r = gamma1(*cand.first);
    *o++ = pt_a_r(v(bb, 0), v(bb, 0), r / FT(2));
    *o++ = pt_b_l(v(bb, 2), v(bb, 2), r / FT(2));
  } else {
    r = gamma2(*cand.second);
    *o++ = pt_a_l(v(bb, 2), v(bb, 0), r / FT(2));
    *o++ = pt_b_r(v(bb, 0), v(bb, 2), r / FT(2));
  }
  return o;
}
template < class RandomAccessIterator,
           class OutputIterator,
           class Traits >
OutputIterator
rectangular_3_center_2_type1(
  RandomAccessIterator f,
  RandomAccessIterator l,
  const typename Traits::Iso_rectangle_2& r,
  OutputIterator o,
  typename Traits::FT& rad,
  Traits& t)
{
  using std::max;
  using std::less;
  using std::nth_element;

  typedef typename Traits::FT                         FT;
  typedef typename Traits::Iso_rectangle_2            Rectangle;
  typedef typename Traits::Point_2                    Point;
  typedef typename Traits::Infinity_distance_2        Dist;
  typedef typename Traits::Signed_infinity_distance_2 Sdist;
  typedef typename Traits::Construct_iso_rectangle_2  Rect;
  typedef typename Traits::Construct_vertex_2         CVertex;
  typedef typename Traits::Construct_point_2_above_right_implicit_point_2
    P_above_right;
  typedef typename Traits::Construct_point_2_above_left_implicit_point_2
    P_above_left;
  typedef typename Traits::Construct_point_2_below_right_implicit_point_2
    P_below_right;
  typedef typename Traits::Construct_point_2_below_left_implicit_point_2
    P_below_left;
  typedef boost::function1<FT, Point>                 Gamma;

  // fetch function objects from traits class
  Rect          rect   = t.construct_iso_rectangle_2_object();
  CVertex       v      = t.construct_vertex_2_object();
  Dist          dist   = t.infinity_distance_2_object();
  Sdist         sdist  = t.signed_infinity_distance_2_object();
  P_above_left  pt_a_l =
    t.construct_point_2_above_left_implicit_point_2_object();
  P_above_right pt_a_r =
    t.construct_point_2_above_right_implicit_point_2_object();
  P_below_left  pt_b_l =
    t.construct_point_2_below_left_implicit_point_2_object();
  P_below_right pt_b_r =
    t.construct_point_2_below_right_implicit_point_2_object();

  // initialize best radius so far
  rad = sdist(v(r, 2), v(r, 0));
  // init to prevent default constructor requirement
  Point bestpoint = *f;
  // (initialisation avoids warning)
  unsigned int bestrun = 0;

  // two cases: top-left & bottom-right or top-right & bottom-left

  // init to prevent default constructor requirement
  Rectangle b = rect(*f, *f);
  for (unsigned int i = 0; i < 2; ++i) {

    // the range [s, e) defines the point set Pt
    RandomAccessIterator s = f;
    RandomAccessIterator e = l;
    bool b_empty = true;
    Min< FT > minft;
    Gamma gamma = boost::bind(minft, 
		       boost::bind(dist, v(r, i), _1),
		       boost::bind(dist, v(r, 2 + i), _1));
    
    while (e - s > 1) {
      // step (a)
      RandomAccessIterator m = s + (e - s - 1) / 2;
      nth_element(s, m, e, boost::bind(less<FT>(), boost::bind(gamma, _1), boost::bind(gamma, _2)));
      
      // step (b)
      Rectangle b_prime = bounding_box_2(m + 1, e, t);
      if (!b_empty)
        b_prime = construct_bounding_box_union_2(b, b_prime, t);

      // step (c) / (d)
      if (sdist(v(b_prime, 2), v(b_prime, 0)) > gamma(*m))
        s = m + 1;
      else {
        e = m + 1;
        b_empty = false;
        b = b_prime;
      }
    }

    // check whether s or (s-1) is the solution
    Rectangle b_prime = b_empty ?
      rect(*s, *s) : construct_bounding_box_union_2(b, rect(*s, *s), t);
    FT b_prime_size = sdist(v(b_prime, 2), v(b_prime, 0));
    if (b_prime_size < gamma(*s)) {
      if (b_prime_size < rad) {
        rad = b_prime_size;
        bestpoint = midpoint(v(b_prime, 0), v(b_prime, 2));
        bestrun = i;
      }
    } else if (gamma(*s) < rad) {
      rad = gamma(*s);
      bestpoint = midpoint(v(b, 0), v(b, 2));
      bestrun = i;
    }
  }

  // return the result
  *o++ = bestpoint;
  if (bestrun == 1) {
    *o++ = pt_a_l(v(r, 2), v(r, 0), rad / FT(2));
    *o++ = pt_b_r(v(r, 0), v(r, 2), rad / FT(2));
  } else {
    *o++ = pt_a_r(v(r, 0), v(r, 0), rad / FT(2));
    *o++ = pt_b_l(v(r, 2), v(r, 2), rad / FT(2));
  }
  return o;
}

template < class R >
struct Rectangular_3_center_2_type2_operations_base {

  typedef typename R::FT                         FT;
  typedef typename R::Point_2                    Point_2;
  typedef typename R::Iso_rectangle_2            Iso_rectangle_2;
  typedef typename R::Infinity_distance_2        Infinity_distance_2;
  typedef typename R::Less_x_2                   Less_x_2;
  typedef typename R::Less_y_2                   Less_y_2;
  typedef boost::function2<bool,Point_2,Point_2> Greater_x_2;
  typedef boost::function2<bool,Point_2,Point_2> Greater_y_2;
  typedef Min< Point_2, Less_x_2 >               Min_x_2;
  typedef Max< Point_2, Less_x_2 >               Max_x_2;
  typedef Min< Point_2, Less_y_2 >               Min_y_2;
  typedef Max< Point_2, Less_y_2 >               Max_y_2;
  typedef typename R::Construct_vertex_2         Construct_vertex_2;
  typedef typename R::Construct_iso_rectangle_2  Construct_iso_rectangle_2;
  typedef typename R::Construct_point_2_above_right_implicit_point_2
    Construct_point_2_above_right_implicit_point_2;
  typedef typename R::Construct_point_2_above_left_implicit_point_2
    Construct_point_2_above_left_implicit_point_2;
  typedef typename R::Construct_point_2_below_right_implicit_point_2
    Construct_point_2_below_right_implicit_point_2;
  typedef typename R::Construct_point_2_below_left_implicit_point_2
    Construct_point_2_below_left_implicit_point_2;
  typedef boost::function1<FT,Point_2>           Delta;
  
  Delta  delta() const { return delta_; }
  Less_x_2  less_x_2_object() const { return r_.less_x_2_object(); }
  Less_y_2  less_y_2_object() const { return r_.less_y_2_object(); }
  Greater_x_2  greater_x_2_object() const
  { return boost::bind(less_x_2_object(),_2,_1); }
  Greater_y_2  greater_y_2_object() const
  { return boost::bind(less_y_2_object(),_2,_1); }
  Infinity_distance_2  distance() const
  { return r_.infinity_distance_2_object(); }
  Construct_vertex_2  construct_vertex_2_object() const
  { return r_.construct_vertex_2_object(); }
  Construct_iso_rectangle_2 construct_iso_rectangle_2_object() const
  { return r_.construct_iso_rectangle_2_object(); }
  
  Construct_point_2_below_left_implicit_point_2
  pt_b_l() const
  { return r_.construct_point_2_below_left_implicit_point_2_object(); }
  Construct_point_2_above_left_implicit_point_2
  pt_a_l() const
  { return r_.construct_point_2_above_left_implicit_point_2_object(); }
  Construct_point_2_below_right_implicit_point_2
  pt_b_r() const
  { return r_.construct_point_2_below_right_implicit_point_2_object(); }
  Construct_point_2_above_right_implicit_point_2
  pt_a_r() const
  { return r_.construct_point_2_above_right_implicit_point_2_object(); }
  
  Min_x_2 minx() const { return Min_x_2(less_x_2_object()); }
  Min_y_2 miny() const { return Min_y_2(less_y_2_object()); }
  Max_x_2 maxx() const { return Max_x_2(less_x_2_object()); }
  Max_y_2 maxy() const { return Max_y_2(less_y_2_object()); }
  
  private:
    R& r_;
    Delta delta_;

public:

  Rectangular_3_center_2_type2_operations_base(R& r, const Point_2& p)
    : r_(r), delta_(boost::bind(r.infinity_distance_2_object(), p, _1))
  {}

};
template < class R >
struct Rectangular_3_center_2_type2_operations0
: public Rectangular_3_center_2_type2_operations_base< R >
{
  typedef Rectangular_3_center_2_type2_operations0< R >      This;
  typedef Rectangular_3_center_2_type2_operations_base< R >  Base;
  typedef typename Base::FT                   FT;
  typedef typename Base::Point_2              Point;
  typedef typename Base::Iso_rectangle_2      Rectangle;
  typedef typename Base::Less_x_2             X_compare;
  typedef typename Base::Greater_y_2          Y_compare;
  typedef typename Base::Infinity_distance_2  Distance;
  typedef typename Base::Construct_iso_rectangle_2
                                              Construct_iso_rectangle_2;
  typedef typename Base::Construct_vertex_2   Construct_vertex_2;

  using Base::less_x_2_object;
  using Base::greater_y_2_object;
  using Base::construct_iso_rectangle_2_object;
  using Base::construct_vertex_2_object;
  using Base::minx;
  using Base::miny;
  using Base::distance;
  using Base::pt_b_l;
  using Base::pt_b_r;
  using Base::pt_a_l;
  using Base::pt_a_r;

  Rectangular_3_center_2_type2_operations0(R& r, const Point& p)
  : Rectangular_3_center_2_type2_operations_base< R >(r, p)
  {}

  X_compare  compare_x() const { return less_x_2_object(); }
  Y_compare  compare_y() const { return greater_y_2_object(); }
  
  Point place_x_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Point& first_uncovered,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(constraint_empty ? first_uncovered :
                      minx()(first_uncovered, v(constraint, 0)),
                    v(bbox, 2)),
               3);
    }
  
    Point place_x_square(bool constraint_empty,
                         const Rectangle& constraint,
                         const Rectangle& bbox) const
    {
      Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
      Construct_vertex_2        v    = construct_vertex_2_object();
      return v(rect(constraint_empty ? v(bbox, 2) : v(constraint, 0),
                    v(bbox, 2)),
               3);
    }
  
    Point place_x_square(const Point& so_far,
                         const Rectangle& bbox,
                         FT radius) const
    {
      Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
      Construct_vertex_2        v    = construct_vertex_2_object();
      return v(rect(minx()(pt_b_l()(v(bbox, 2), v(bbox, 2), radius), so_far),
                    so_far),
             3);
    }
  
    Point place_y_square(bool constraint_empty,
                         const Rectangle& constraint,
                         const Point& first_uncovered,
                         const Rectangle& bbox) const
    {
      Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
      Construct_vertex_2        v    = construct_vertex_2_object();
      return v(rect(v(bbox, 2),
                    constraint_empty ? first_uncovered :
                      miny()(first_uncovered, v(constraint, 0))),
               1);
    }
  
    Point place_y_square(bool constraint_empty,
                         const Rectangle& constraint,
                         const Rectangle& bbox) const
    {
      Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
      Construct_vertex_2        v    = construct_vertex_2_object();
      return v(rect(v(bbox, 2),
                    constraint_empty ? v(bbox, 2) : v(constraint, 0)),
               1);
    }
  
    Point place_y_square(const Point& so_far,
                         const Rectangle& bbox,
                         FT radius) const
    {
      Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
      Construct_vertex_2        v    = construct_vertex_2_object();
      return v(rect(so_far,
                    miny()(pt_b_l()(v(bbox, 2), v(bbox, 2), radius),
                           so_far)),
               1);
    }
  
    Point update_x_square(const Point& s, const Point& newp) const
    {
      Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
      Construct_vertex_2        v    = construct_vertex_2_object();
  
      return v(rect(minx()(s, newp), s), 3);
    }
  
    Point update_y_square(const Point& s, const Point& newp) const {
      Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
      Construct_vertex_2        v    = construct_vertex_2_object();
  
      return v(rect(s, miny()(s, newp)), 1);
    }
  
    FT compute_x_distance(const Point& extreme,
                          const Rectangle& constraint) const
    { return distance()(extreme, construct_vertex_2_object()(constraint, 1)); }
  
    FT compute_y_distance(const Point& extreme,
                          const Rectangle& constraint) const
    { return distance()(extreme, construct_vertex_2_object()(constraint, 3)); }
  
    Point construct_corner_square(const Rectangle& bbox, FT r) const
    { return pt_a_r()(construct_vertex_2_object()(bbox, 0),
                      construct_vertex_2_object()(bbox, 0), r); }
  
    Point construct_x_square(const Point& p, FT r) const
    { return pt_b_r()(p, p, r); }
  
    Point construct_y_square(const Point& p, FT r) const
    { return pt_a_l()(p, p, r); }
};

template < class R >
struct Rectangular_3_center_2_type2_operations1
: public Rectangular_3_center_2_type2_operations_base< R >
{
  typedef Rectangular_3_center_2_type2_operations_base< R >  Base;
  typedef typename Base::FT                   FT;
  typedef typename Base::Point_2              Point;
  typedef typename Base::Iso_rectangle_2      Rectangle;
  typedef typename Base::Infinity_distance_2  Distance;
  typedef typename Base::Greater_x_2          X_compare;
  typedef typename Base::Greater_y_2          Y_compare;
  typedef typename Base::Construct_iso_rectangle_2
                                              Construct_iso_rectangle_2;
  typedef typename Base::Construct_vertex_2   Construct_vertex_2;

  using Base::greater_x_2_object;
  using Base::greater_y_2_object;
  using Base::construct_iso_rectangle_2_object;
  using Base::construct_vertex_2_object;
  using Base::maxx;
  using Base::miny;
  using Base::pt_a_r;
  using Base::pt_b_l;
  using Base::pt_a_l;
  using Base::distance;

  Rectangular_3_center_2_type2_operations1(R& r, const Point& p)
  : Rectangular_3_center_2_type2_operations_base< R >(r, p)
  {}

  X_compare  compare_x() const { return greater_x_2_object(); }
  Y_compare  compare_y() const { return greater_y_2_object(); }
  
  Point place_x_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Point& first_uncovered,
                       const Rectangle& bbox) const
  {
      Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
      Construct_vertex_2        v    = construct_vertex_2_object();
      return v(rect(constraint_empty ? first_uncovered :
                    maxx()(first_uncovered, v(constraint, 2)),
                  v(bbox, 3)),
             2);
  }
  
  Point place_x_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(constraint_empty ? v(bbox, 0) : v(constraint, 2),
                  v(bbox, 3)),
             2);
  }
  
  Point place_x_square(const Point& so_far,
                       const Rectangle& bbox,
                       FT radius) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(maxx()(so_far, pt_a_r()(v(bbox, 0),
                                          v(bbox, 0), radius)),
                  so_far),
             2);
  }
  
  Point place_y_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Point& first_uncovered,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(v(bbox, 3),
                  constraint_empty ? first_uncovered :
                    miny()(first_uncovered, v(constraint, 0))),
             0);
  }
  
  Point place_y_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(v(bbox, 3),
                  constraint_empty ? v(bbox, 2) : v(constraint, 0)),
             0);
  }
  
  Point place_y_square(const Point& so_far,
                       const Rectangle& bbox,
                       FT radius) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(so_far,
                  miny()(so_far, pt_b_l()(v(bbox, 2),
                                          v(bbox, 2), radius))),
             0);
  }
  
  Point update_x_square(const Point& s, const Point& newp) const {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(maxx()(s, newp), s), 2);
  }
  
  Point update_y_square(const Point& s, const Point& newp) const {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(s, miny()(s, newp)), 0);
  }
  
  FT compute_x_distance(const Point& extreme,
                        const Rectangle& constraint) const
  { return distance()(extreme, construct_vertex_2_object()(constraint, 0)); }
  
  FT compute_y_distance(const Point& extreme,
                        const Rectangle& constraint) const
  { return distance()(extreme, construct_vertex_2_object()(constraint, 2)); }
  
  Point construct_corner_square(const Rectangle& bbox, FT r) const
  { return pt_a_l()(construct_vertex_2_object()(bbox, 2),
                    construct_vertex_2_object()(bbox, 0), r); }
  
  Point construct_x_square(const Point& p, FT r) const
  { return pt_b_l()(p, p, r); }
  
  Point construct_y_square(const Point& p, FT r) const
  { return pt_a_r()(p, p, r); }
};

template < class R >
struct Rectangular_3_center_2_type2_operations2
: public Rectangular_3_center_2_type2_operations_base< R >
{
  typedef Rectangular_3_center_2_type2_operations_base< R >  Base;
  typedef typename Base::FT                   FT;
  typedef typename Base::Point_2              Point;
  typedef typename Base::Iso_rectangle_2      Rectangle;
  typedef typename Base::Infinity_distance_2  Distance;
  typedef typename Base::Greater_x_2          X_compare;
  typedef typename Base::Less_y_2             Y_compare;
  typedef typename Base::Construct_iso_rectangle_2
                                              Construct_iso_rectangle_2;
  typedef typename Base::Construct_vertex_2   Construct_vertex_2;

  using Base::greater_x_2_object;
  using Base::less_y_2_object;
  using Base::construct_iso_rectangle_2_object;
  using Base::construct_vertex_2_object;
  using Base::distance;
  using Base::maxx;
  using Base::maxy;
  using Base::pt_a_r;
  using Base::pt_a_l;
  using Base::pt_b_r;
  using Base::pt_b_l;

  Rectangular_3_center_2_type2_operations2(R& r, const Point& p)
  : Rectangular_3_center_2_type2_operations_base< R >(r, p)
  {}

  X_compare  compare_x() const { return greater_x_2_object(); }
  Y_compare  compare_y() const { return less_y_2_object(); }
  
  Point place_x_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Point& first_uncovered,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(constraint_empty ? first_uncovered :
                    maxx()(first_uncovered, v(constraint, 2)),
                  v(bbox, 0)),
             1);
  }
  
  Point place_x_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(constraint_empty ? v(bbox, 0) : v(constraint, 2),
                  v(bbox, 0)),
             1);
  }
  
  Point place_x_square(const Point& so_far,
                       const Rectangle& bbox,
                       FT radius) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(maxx()(so_far, pt_a_r()(v(bbox, 0),
                                          v(bbox, 0), radius)),
                  so_far),
             1);
  }
  
  Point place_y_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Point& first_uncovered,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(v(bbox, 0),
                  constraint_empty ? first_uncovered :
                    maxy()(first_uncovered, v(constraint, 2))),
             3);
  }
  
  Point place_y_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(v(bbox, 0),
                  constraint_empty ? v(bbox, 0) : v(constraint, 2)),
             3);
  }
  
  Point place_y_square(const Point& so_far,
                       const Rectangle& bbox,
                       FT radius) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(so_far,
                  maxy()(pt_a_r()(v(bbox, 0),
                                  v(bbox, 0), radius), so_far)),
             3);
  }
  
  Point update_x_square(const Point& s, const Point& newp) const {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(maxx()(s, newp), s), 1);
  }
  
  Point update_y_square(const Point& s, const Point& newp) const {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(s, maxy()(s, newp)), 3);
  }
  
  FT compute_x_distance(const Point& extreme,
                        const Rectangle& constraint) const
  { return distance()(extreme, construct_vertex_2_object()(constraint, 3)); }
  
  FT compute_y_distance(const Point& extreme,
                        const Rectangle& constraint) const
  { return distance()(extreme, construct_vertex_2_object()(constraint, 1)); }
  
  Point construct_corner_square(const Rectangle& bbox, FT r) const
  { return pt_b_l()(construct_vertex_2_object()(bbox, 2),
                    construct_vertex_2_object()(bbox, 2), r); }
  
  Point construct_x_square(const Point& p, FT r) const
  { return pt_a_l()(p, p, r); }
  
  Point construct_y_square(const Point& p, FT r) const
  { return pt_b_r()(p, p, r); }
};

template < class R >
struct Rectangular_3_center_2_type2_operations3
: public Rectangular_3_center_2_type2_operations_base< R >
{
  typedef Rectangular_3_center_2_type2_operations_base< R >  Base;
  typedef typename Base::FT                   FT;
  typedef typename Base::Point_2              Point;
  typedef typename Base::Iso_rectangle_2      Rectangle;
  typedef typename Base::Infinity_distance_2  Distance;
  typedef typename Base::Less_x_2             X_compare;
  typedef typename Base::Less_y_2             Y_compare;
  typedef typename Base::Construct_iso_rectangle_2
                                              Construct_iso_rectangle_2;
  typedef typename Base::Construct_vertex_2   Construct_vertex_2;

  using Base::less_x_2_object;
  using Base::less_y_2_object;
  using Base::construct_iso_rectangle_2_object;
  using Base::construct_vertex_2_object;
  using Base::distance;
  using Base::minx;
  using Base::maxy;
  using Base::pt_b_l;
  using Base::pt_b_r;
  using Base::pt_a_r;

  Rectangular_3_center_2_type2_operations3(R& r, const Point& p)
  : Rectangular_3_center_2_type2_operations_base< R >(r, p)
  {}

  X_compare  compare_x() const { return less_x_2_object(); }
  Y_compare  compare_y() const { return less_y_2_object(); }
  
  Point place_x_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Point& first_uncovered,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(constraint_empty ? first_uncovered :
                    minx()(first_uncovered, v(constraint, 0)),
                  v(bbox, 1)),
             0);
  }
  
  Point place_x_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(constraint_empty ? v(bbox, 2) : v(constraint, 0),
                  v(bbox, 1)),
             0);
  }
  
  Point place_x_square(const Point& so_far,
                       const Rectangle& bbox,
                       FT radius) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(minx()(pt_b_l()(v(bbox, 2),
                                  v(bbox, 2), radius), so_far),
                  so_far),
             0);
  }
  
  Point place_y_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Point& first_uncovered,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(v(bbox, 1),
                  constraint_empty ? first_uncovered :
                    maxy()(first_uncovered, v(constraint, 2))),
             2);
  }
  
  Point place_y_square(bool constraint_empty,
                       const Rectangle& constraint,
                       const Rectangle& bbox) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(v(bbox, 1),
                  constraint_empty ? v(bbox, 0) : v(constraint, 2)),
             2);
  }
  
  Point place_y_square(const Point& so_far,
                       const Rectangle& bbox,
                       FT radius) const
  {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(so_far,
                  maxy()(pt_a_r()(v(bbox, 0),
                                  v(bbox, 0), radius), so_far)),
             2);
  }
  
  Point update_x_square(const Point& s, const Point& newp) const {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(minx()(s, newp), s), 0);
  }
  
  Point update_y_square(const Point& s, const Point& newp) const {
    Construct_iso_rectangle_2 rect = construct_iso_rectangle_2_object();
    Construct_vertex_2        v    = construct_vertex_2_object();
    return v(rect(s, maxy()(s, newp)), 2);
  }
  
  FT compute_x_distance(const Point& extreme,
                        const Rectangle& constraint) const
  { return distance()(extreme, construct_vertex_2_object()(constraint, 2)); }
  
  FT compute_y_distance(const Point& extreme,
                        const Rectangle& constraint) const
  { return distance()(extreme, construct_vertex_2_object()(constraint, 0)); }
  
  Point construct_corner_square(const Rectangle& bbox, FT r) const
  { return pt_b_r()(construct_vertex_2_object()(bbox, 0),
                    construct_vertex_2_object()(bbox, 2), r); }
  
  Point construct_x_square(const Point& p, FT r) const
  { return pt_a_r()(p, p, r); }
  
  Point construct_y_square(const Point& p, FT r) const
  { return pt_b_l()(p, p, r); }
};


template < class RandomAccessIterator,
           class Rectangle,
           class OutputIterator,
           class FT,
           class Operations >
OutputIterator
rectangular_3_center_2_type2(
  RandomAccessIterator f,
  RandomAccessIterator l,
  const Rectangle& r,
  OutputIterator o,
  FT& rad,
  Operations op)
{
  BOOST_USING_STD_MAX();
  using std::less;
  using std::greater;
  using std::greater_equal;
  using std::not_equal_to;
  using std::logical_and;
  using std::max_element;
  using std::nth_element;
  using std::find_if;
  using std::sort;
  using std::partition;
  using std::pair;

  typedef typename Operations::Point                       Point;
  typedef typename Operations::Distance                 Distance;
  typedef pair< RandomAccessIterator, RandomAccessIterator >  IP;

  typename Operations::Construct_iso_rectangle_2
  rect = op.construct_iso_rectangle_2_object();

  typename Operations::Construct_vertex_2
  v = op.construct_vertex_2_object();

  typename Operations::Less_x_2 less_x_2 = op.less_x_2_object();
  typename Operations::Less_y_2 less_y_2 = op.less_y_2_object();

  // constant fraction to be excluded on every iteration (1/.)
  const unsigned int fraction = 7;

  // the range [s, e) defines the point set P
  RandomAccessIterator s = f;
  RandomAccessIterator e = l;
  // a flag to indicate whether we need to search the radius
  // in the final brute-force step or not
  bool radius_is_known = false;

  // the bounding boxes of assigned points
  Rectangle Q_t, Q_r;
  bool Q_t_empty = true, Q_r_empty = true;

  // lower bound for the diameter (2 * radius)
  // also store the corresponding positions of q_t and q_r
  FT rho_max = 0, rho_min = -1, q_t_q_r_cover_at_rho_min = 0;
  Point q_t_at_rho_max, q_r_at_rho_max, q_t_at_rho_min, q_r_at_rho_min;
  RandomAccessIterator s_at_rho_min = s, e_at_rho_min = s;

#ifndef CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST
  {
    // First try whether the best radius so far can be reached at all
    RandomAccessIterator m =
      partition(f, l, boost::bind(greater< FT >(), rad, boost::bind(op.delta(), _1)));
    IP pos = min_max_element(m, l, op.compare_x(), op.compare_y());
    // extreme points of the two other squares
    Point q_t =
      op.place_x_square(op.place_x_square(Q_t_empty, Q_t, *pos.first, r),
                        r,
                        rad);
    Point q_r =
      op.place_y_square(op.place_y_square(Q_r_empty, Q_r, *pos.second, r),
                        r,
                        rad);
    boost::function1<bool,FT> le_rad = boost::bind(greater_equal<FT>(), rad, _1);
    RandomAccessIterator b1 =
      partition(m, l, boost::bind(le_rad, boost::bind(op.distance(), q_t, _1)));
    RandomAccessIterator b2 =
      partition(b1, l, boost::bind(le_rad, boost::bind(op.distance(), q_r, _1)));

    if (b2 != l)
      return o;
  }
#endif // ! CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST

#ifndef CGAL_3COVER_NO_PREFILTER
  // Prefiltering heuristic
  while (e - s > 6) {
    int cutoff = (e - s) / 2;
    RandomAccessIterator m = s + cutoff - 1;
    nth_element(s, m, e, 
		boost::bind(less<FT>(), boost::bind(op.delta(), _1), boost::bind(op.delta(), _2)));
    
    // step (b)
    IP pos = min_max_element(m + 1, e, op.compare_x(), op.compare_y());
    // extreme points of the two other squares
    // (consider Q_i and pos [move as far as possible])
    Point q_t_afap = op.place_x_square(Q_t_empty, Q_t, *pos.first, r);
    Point q_r_afap = op.place_y_square(Q_r_empty, Q_r, *pos.second, r);
    // now consider also that we must not leave the bbox r
    Point q_t = op.place_x_square(q_t_afap, r, op.delta()(*m));
    Point q_r = op.place_y_square(q_r_afap, r, op.delta()(*m));

    // check for covering
    boost::function1<bool,FT>
      le_delta_m = boost::bind(greater_equal<FT>(), op.delta()(*m), _1);
    RandomAccessIterator b1 =
      partition(m + 1, e,
                boost::bind(le_delta_m, boost::bind(op.distance(), q_t, _1)));
    RandomAccessIterator b2 =
      partition(b1, e, boost::bind(le_delta_m, boost::bind(op.distance(), q_r, _1)));

    if (b2 != e)
      s = m;
    else
      break;
  }
#endif // ! CGAL_3COVER_NO_PREFILTER


  while (e - s > 6) {
    /*
    cerr << e - s << " points (" << e - s - (e - s) / fraction
         << ")" << endl;
    */
    // step (a)
    int cutoff = (e - s) / fraction;
    RandomAccessIterator m = s + cutoff - 1;
    nth_element(s, m, e, 
		boost::bind(less<FT>(), boost::bind(op.delta(), _1), boost::bind(op.delta(), _2)));
    
    // step (b)
    IP pos = min_max_element(m + 1, e, op.compare_x(), op.compare_y());
    // extreme points of the two other squares
    // (consider Q_i and pos [move as far as possible])
    Point q_t_afap = op.place_x_square(Q_t_empty, Q_t, *pos.first, r);
    Point q_r_afap = op.place_y_square(Q_r_empty, Q_r, *pos.second, r);
    // now consider also that we must not leave the bbox r
    Point q_t = op.place_x_square(q_t_afap, r, op.delta()(*m));
    Point q_r = op.place_y_square(q_r_afap, r, op.delta()(*m));


#ifdef CGAL_3COVER_CHECK
    // check whether the points in [e,l) which have been assigned
    // to Q_t and Q_r are covered by q_t and q_r
    if ((Q_t_empty || op.compute_x_distance(q_t, Q_t) <= op.delta()(*m)) &&
        (Q_r_empty || op.compute_y_distance(q_r, Q_r) <= op.delta()(*m))) {
      boost::function1<bool,FT> 
        greater_delta_m = boost::bind(less< FT >(), op.delta()(*m));
      CGAL_optimisation_assertion_code(RandomAccessIterator iii =)
        find_if(e,
                l,
                boost::bind(logical_and< bool >(),
		     boost::bind(greater_delta_m,
			  boost::bind(op.distance(), q_t, _1)),
		     boost::bind(greater_delta_m,
			  boost::bind(op.distance(), q_r, _1))));
        CGAL_optimisation_assertion(iii == l);
    }
    // check whether the points in [f,s) are covered
    {
      boost::function1<bool,FT> 
	le_delta_m = boost::bind(greater_equal<FT>(), op.delta()(*m));
      RandomAccessIterator iii =
        partition(f, s, boost::bind(le_delta_m, boost::bind(op.delta(), _1)));
      iii = partition(iii, s,
                      boost::bind(le_delta_m, boost::bind(op.distance(), q_t, _1)));
      iii = partition(iii, s,
                      boost::bind(le_delta_m, boost::bind(op.distance(), q_r, _1)));
      CGAL_optimisation_assertion(iii == s);
    }
#endif // CGAL_3COVER_CHECK

    // partition the range [m+1, e) into ranges
    // [m+1, b1), [b1, b2),   [b2, b3) and [b3, e)
    //     R      G cap q_t  G cap q_r      none
    boost::function1<bool,FT> 
      le_delta_m = boost::bind(greater_equal<FT>(), op.delta()(*m), _1);
    RandomAccessIterator b2 =
      partition(m + 1, e, boost::bind(le_delta_m, boost::bind(op.distance(), q_t, _1)));
    RandomAccessIterator b1 =
      partition(m + 1, b2,
                boost::bind(le_delta_m, boost::bind(op.distance(), q_r, _1)));
    RandomAccessIterator b3 =
      partition(b2, e, boost::bind(le_delta_m, boost::bind(op.distance(), q_r, _1)));


    // step (c)
    if (b3 != e ||
        (!Q_t_empty && op.compute_x_distance(q_t, Q_t) > op.delta()(*m)) ||
        (!Q_r_empty && op.compute_y_distance(q_r, Q_r) > op.delta()(*m)))
      {
        // not covered
        s = b1;
        rho_min = op.delta()(*m);
        q_t_q_r_cover_at_rho_min = 0;
        if (!Q_t_empty)
          q_t_q_r_cover_at_rho_min =
            max BOOST_PREVENT_MACRO_SUBSTITUTION (q_t_q_r_cover_at_rho_min,
                           op.compute_x_distance(q_t, Q_t));
        if (!Q_r_empty)
          q_t_q_r_cover_at_rho_min =
            max BOOST_PREVENT_MACRO_SUBSTITUTION (q_t_q_r_cover_at_rho_min,
                           op.compute_y_distance(q_r, Q_r));
        q_t_at_rho_min = q_t, q_r_at_rho_min = q_r;
        s_at_rho_min = s, e_at_rho_min = e;
        continue;
      }

    // step (d) [covered]
    if (b3 - b1 >= cutoff) { // enough points in G
      e = b1;
      // adjust Q_t
      if (b1 != b2) {
        if (Q_t_empty) {
          Q_t = bounding_box_2(b1, b2, op);
          Q_t_empty = false;
        } else
          Q_t =
            construct_bounding_box_union_2(
              Q_t, bounding_box_2(b1, b2, op), op);
      }
      // adjust Q_r
      if (b2 != b3) {
        if (Q_r_empty) {
          Q_r = bounding_box_2(b2, b3, op);
          Q_r_empty = false;
        } else
          Q_r =
            construct_bounding_box_union_2(
              Q_r, bounding_box_2(b2, b3, op), op);
      }
      continue;
    }

    // step (e) [not enough points in G]
    CGAL_optimisation_assertion(b1 - (m + 1) >= 5 * cutoff);

    // compute the four cutting lines for R
    nth_element(m + 1, m + 1 + cutoff, b1, less_x_2);
    Point x_min_cutoff = *(m + 1 + cutoff);
    nth_element(m + 1, m + 1 + cutoff, b1, op.greater_x_2_object());
    Point x_max_cutoff = *(m + 1 + cutoff);
    nth_element(m + 1, m + 1 + cutoff, b1, less_y_2);
    Point y_min_cutoff = *(m + 1 + cutoff);
    nth_element(m + 1, m + 1 + cutoff, b1, op.greater_y_2_object());
    Point y_max_cutoff = *(m + 1 + cutoff);

    Point Pmin = v(rect(x_min_cutoff, y_min_cutoff),
                   less_x_2(x_min_cutoff, y_min_cutoff) ?
                   less_y_2(x_min_cutoff, y_min_cutoff) ? 3 : 0
                   : less_y_2(x_min_cutoff, y_min_cutoff) ? 2 : 1);
    Point Pmax = v(rect(x_max_cutoff, y_max_cutoff),
                   less_x_2(x_max_cutoff, y_max_cutoff) ?
                   less_y_2(x_max_cutoff, y_max_cutoff) ? 1 : 2
                   : less_y_2(x_max_cutoff, y_max_cutoff) ? 0 : 3);

    Rectangle B = rect(Pmin, Pmax);

    // Algorithm search_E:

    // the range [s_b, s_e) defines the point set S
    RandomAccessIterator s_b = s;
    RandomAccessIterator s_e = m + 1;

    while (s_e - s_b > 1) {
      // step 1
      RandomAccessIterator s_m = s_b + (s_e - s_b - 1) / 2;
      nth_element(s_b, s_m, s_e,
                  boost::bind(less<FT>(), 
		       boost::bind(op.delta(), _1), 
		       boost::bind(op.delta(), _2)));

      // step 2 (as above)
      Point q_t_m = q_t_afap;
      Point q_r_m = q_r_afap;
      if (s_m + 1 != s_e) {
        pos = min_max_element(s_m + 1, s_e, op.compare_x(), op.compare_y());
        q_t_m = op.update_x_square(q_t_m, *pos.first);
        q_r_m = op.update_y_square(q_r_m, *pos.second);
      }

      // step 3/4
      if (op.compute_x_distance(
          op.place_x_square(q_t_m, r, op.delta()(*s_m)), B) <=
          op.delta()(*s_m) &&
          op.compute_y_distance(
            op.place_y_square(q_r_m, r, op.delta()(*s_m)), B) <=
          op.delta()(*s_m)) {
        s_e = s_m + 1;
        q_t_afap = q_t_m;
        q_r_afap = q_r_m;
      } else
        s_b = s_m + 1;
    }

    // now s_b corresponds to the first moment in [s, m+1)
    // where q_t and q_r cover B

    // place q_t and q_r
    q_t = op.place_x_square(q_t_afap, r, op.delta()(*s_b));
    q_r = op.place_y_square(q_r_afap, r, op.delta()(*s_b));

    // partition the range [s_b+1, e) into ranges
    // [s_b+1, b1), [b1, b2),   [b2, b3) and [b3, e)
    //     R      G cap q_t  G cap q_r      none
    boost::function1<bool,FT>
      le_delta_sb = boost::bind(greater_equal<FT>(), op.delta()(*s_b), _1);
    b2 = partition(s_b + 1, e, boost::bind(le_delta_sb,
				    boost::bind(op.distance(), q_t, _1)));
    b1 = partition(s_b + 1, b2, boost::bind(le_delta_sb,
				     boost::bind(op.distance(), q_r, _1)));
    b3 = partition(b2, e,
                   boost::bind(le_delta_sb, boost::bind(op.distance(), q_r, _1)));

    if (b3 != e ||
        (!Q_t_empty && op.compute_x_distance(q_t, Q_t) > op.delta()(*s_b)) ||
        (!Q_r_empty && op.compute_y_distance(q_r, Q_r) > op.delta()(*s_b))) {
      // no covering
      CGAL_optimisation_assertion(b1 - s >= cutoff);
      s = b1;
      rho_min = op.delta()(*s_b);
      q_t_at_rho_min = q_t, q_r_at_rho_min = q_r;
      q_t_q_r_cover_at_rho_min = 0;
      if (!Q_t_empty)
        q_t_q_r_cover_at_rho_min =
          max BOOST_PREVENT_MACRO_SUBSTITUTION (q_t_q_r_cover_at_rho_min,
                         op.compute_x_distance(q_t, Q_t));
      if (!Q_r_empty)
        q_t_q_r_cover_at_rho_min =
          max BOOST_PREVENT_MACRO_SUBSTITUTION (q_t_q_r_cover_at_rho_min,
                         op.compute_y_distance(q_r, Q_r));
      s_at_rho_min = s, e_at_rho_min = e;
      continue;
    }

    // we still have a covering

    if (s_b == s) {
      CGAL_optimisation_expensive_assertion_code(
        std::vector< Point > tmppts(f, l);
        RandomAccessIterator ii =
          partition(tmppts.begin(), tmppts.end(),
                    boost::bind(le_delta_sb, boost::bind(op.delta(), _1)));
        IP tmppos = min_max_element(ii, tmppts.end(),
                                    op.compare_x(), op.compare_y());
        )
      CGAL_optimisation_expensive_assertion(
        !op.compare_x()(*tmppos.first, q_t));
      CGAL_optimisation_expensive_assertion(
        !op.compare_y()(q_r, *tmppos.second));

      // we are done
      rho_max = op.delta()(*s);
      q_t_at_rho_max = q_t, q_r_at_rho_max = q_r;
      radius_is_known = true;
      break;
    }

    // if there are enough points in G ...
    if (b3 - b1 >= cutoff) {
      e = b1;
      // adjust Q_t
      if (b1 != b2) {
        if (Q_t_empty) {
          Q_t = bounding_box_2(b1, b2, op);
          Q_t_empty = false;
        } else
          Q_t =
            construct_bounding_box_union_2(
              Q_t, bounding_box_2(b1, b2, op), op);
      }
      // adjust Q_r
      if (b2 != b3) {
        if (Q_r_empty) {
          Q_r = bounding_box_2(b2, b3, op);
          Q_r_empty = false;
        } else
          Q_r =
            construct_bounding_box_union_2(
              Q_r, bounding_box_2(b2, b3, op), op);
      }
      continue;
    }

    // we have to take the next smaller radius
    RandomAccessIterator next =
      max_element_if(s, s_b,
                     boost::bind(less<FT>(), 
			  boost::bind(op.delta(), _1), 
			  boost::bind(op.delta(), _2)),
                     boost::bind(not_equal_to<FT>(),
			  op.delta()(*s_b),
			  boost::bind(op.delta(), _1)));
    rho_max = op.delta()(*s_b);
    q_t_at_rho_max = q_t, q_r_at_rho_max = q_r;
    CGAL_optimisation_assertion(op.delta()(*next) < op.delta()(*s_b));
    q_t_afap = op.update_x_square(q_t_afap, *s_b);
    q_r_afap = op.update_y_square(q_r_afap, *s_b);
    q_t = op.place_x_square(q_t_afap, r, op.delta()(*next));
    q_r = op.place_y_square(q_r_afap, r, op.delta()(*next));

    // again check for covering
    boost::function1<bool,FT>
      le_delta_next = boost::bind(greater_equal<FT>(), op.delta()(*next), _1);
    b2 = partition(s_b, e,
                   boost::bind(le_delta_next, boost::bind(op.distance(), q_t, _1)));
    b1 = partition(s_b, b2,
                   boost::bind(le_delta_next, boost::bind(op.distance(), q_r, _1)));
    b3 = partition(b2, e,
                   boost::bind(le_delta_next, boost::bind(op.distance(), q_r, _1)));

    if (b3 != e ||
        (!Q_t_empty && op.compute_x_distance(q_t, Q_t) > op.delta()(*next)) ||
        (!Q_r_empty && op.compute_y_distance(q_r, Q_r) > op.delta()(*next))) {
      // no covering
      rho_min = op.delta()(*next);
      q_t_q_r_cover_at_rho_min = 0;
      if (!Q_t_empty)
        q_t_q_r_cover_at_rho_min =
          max BOOST_PREVENT_MACRO_SUBSTITUTION (q_t_q_r_cover_at_rho_min,
                         op.compute_x_distance(q_t, Q_t));
      if (!Q_r_empty)
        q_t_q_r_cover_at_rho_min =
          max BOOST_PREVENT_MACRO_SUBSTITUTION (q_t_q_r_cover_at_rho_min,
                         op.compute_y_distance(q_r, Q_r));
      q_t_at_rho_min = q_t, q_r_at_rho_min = q_r;
      s_at_rho_min = b3, e_at_rho_min = e;
      radius_is_known = true;
      break;
    }


    CGAL_optimisation_assertion(b3 - b1 >= cutoff);
    e = b1;
    // adjust Q_t
    if (b1 != b2) {
      if (Q_t_empty) {
        Q_t = bounding_box_2(b1, b2, op);
        Q_t_empty = false;
      } else
        Q_t =
          construct_bounding_box_union_2(
            Q_t, bounding_box_2(b1, b2, op), op);
    }
    // adjust Q_r
    if (b2 != b3) {
      if (Q_r_empty) {
        Q_r = bounding_box_2(b2, b3, op);
        Q_r_empty = false;
      } else
        Q_r =
          construct_bounding_box_union_2(
            Q_r, bounding_box_2(b2, b3, op), op);
    }

  } // while (e - s > 6)

  // compute the solution brute-force
  if (!radius_is_known) {
    RandomAccessIterator t = e;
    Point q_t_afap = op.place_x_square(Q_t_empty, Q_t, r);
    Point q_r_afap = op.place_y_square(Q_r_empty, Q_r, r);
    if (s != e) {
      sort(s, e, boost::bind(less<FT>(), boost::bind(op.delta(), _1), boost::bind(op.delta(), _2)));
      rho_max = op.delta()(*--t);
    } else
      rho_max = rho_min;
    Point q_t = op.place_x_square(q_t_afap, r, rho_max);
    Point q_r = op.place_y_square(q_r_afap, r, rho_max);


    if ((!Q_t_empty && op.compute_x_distance(q_t, Q_t) > rho_max) ||
        (!Q_r_empty && op.compute_y_distance(q_r, Q_r) > rho_max)) {
      rho_max = max BOOST_PREVENT_MACRO_SUBSTITUTION (op.compute_x_distance(q_t, Q_t),
                               op.compute_y_distance(q_r, Q_r));
#ifndef CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST
      CGAL_optimisation_assertion(rho_max <= rad);
#endif // ! CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST
      rad = rho_max;
      *o++ = op.construct_corner_square(r, rad / FT(2));
      *o++ = op.construct_x_square(q_t, rad / FT(2));
      *o++ = op.construct_y_square(q_r, rad / FT(2));
      return o;
    }
    CGAL_optimisation_assertion(s != e);

    // find the first diameter where covering is possible
    for (;;) {
      q_t_at_rho_max = q_t, q_r_at_rho_max = q_r;

      if (t == s)
        break;

      // these get uncovered now
      do {
        q_t_afap = op.update_x_square(q_t_afap, *t);
        q_r_afap = op.update_y_square(q_r_afap, *t);
      } while (t != s && op.delta()(*--t) == rho_max);

      // try the next possible diameter value
      FT try_rho = op.delta()(*t);
      CGAL_optimisation_assertion(t == s || try_rho < rho_max);
      q_t = op.place_x_square(q_t_afap, r, try_rho);
      q_r = op.place_y_square(q_r_afap, r, try_rho);

      // check for covering
      boost::function1<bool,FT>
        greater_rho_max = boost::bind(less<FT>(), try_rho, _1);
      if ((!Q_t_empty && op.compute_x_distance(q_t, Q_t) > try_rho) ||
          (!Q_r_empty && op.compute_y_distance(q_r, Q_r) > try_rho) ||
          e != find_if(
            t + 1,
            e,
            boost::bind(logical_and<bool>(),
		 boost::bind(greater_rho_max, boost::bind(op.distance(), q_t, _1)),
		 boost::bind(greater_rho_max, boost::bind(op.distance(), q_r, _1)))))
        {
          rho_min = try_rho;
          q_t_q_r_cover_at_rho_min = 0;
          if (!Q_t_empty)
            q_t_q_r_cover_at_rho_min =
              max BOOST_PREVENT_MACRO_SUBSTITUTION (q_t_q_r_cover_at_rho_min,
                  op.compute_x_distance(q_t, Q_t));
          if (!Q_r_empty)
            q_t_q_r_cover_at_rho_min =
              max BOOST_PREVENT_MACRO_SUBSTITUTION (q_t_q_r_cover_at_rho_min,
                  op.compute_y_distance(q_r, Q_r));
          q_t_at_rho_min = q_t, q_r_at_rho_min = q_r;
          s_at_rho_min = t + 1, e_at_rho_min = e;
          break;
        }
        rho_max = try_rho;
    } // for (;;)
  } // if (!radius_is_known)

  // now we have the following:
  // rho_min corresponds to a non-covering with
  //   - q_t_at_rho_min is the corr. position of q_t,
  //   - q_r_at_rho_min is the corr. position of q_r and
  //   - q_t_q_r_cover_at_rho_min is the radius needed to
  //     cover Q_t and Q_r
  //   - the range [s_at_rho_min, e_at_rho_min) contains the points
  //     still to be covered
  // rho_max corresponds to a covering with
  //   - q_t_at_rho_max is the corr. position of q_t and
  //   - q_r_at_rho_max is the corr. position of q_r.

  // try rho_min
  CGAL_optimisation_assertion(rho_min <= rho_max);
  CGAL_optimisation_assertion(rho_min >= 0);
  FT rad_2 = q_t_q_r_cover_at_rho_min;
  if (s_at_rho_min != e_at_rho_min) {
    boost::function1<FT,Point>
      mydist = boost::bind(Min<FT>(),
		    boost::bind(op.distance(), q_t_at_rho_min, _1),
		    boost::bind(op.distance(), q_r_at_rho_min, _1));
    rad_2 =
      max BOOST_PREVENT_MACRO_SUBSTITUTION (
        rad_2,
        mydist(*max_element(s_at_rho_min, e_at_rho_min,
                            boost::bind(less< FT >(), 
				 boost::bind(mydist, _1), 
				 boost::bind(mydist, _2)))));
  }
  CGAL_optimisation_assertion(rad_2 == 0 || rad_2 > rho_min);

  // if a covering with rho == 0 is possible,
  // it will be catched in the type1 functions
  Point q_t, q_r;
  if (rad_2 > rho_max || rho_min == -1) {
    // it is rho_max ...
    q_t = q_t_at_rho_max, q_r = q_r_at_rho_max;
    rad_2 = rho_max;
  } else
    q_t = q_t_at_rho_min, q_r = q_r_at_rho_min;

#ifndef CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST
  CGAL_optimisation_assertion(rad_2 <= rad);
#endif // ! CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST
  rad = rad_2;
  *o++ = op.construct_corner_square(r, rad / FT(2));
  *o++ = op.construct_x_square(q_t, rad / FT(2));
  *o++ = op.construct_y_square(q_r, rad / FT(2));
  return o;
} // rectangular_3_center_2_type2( ... )
template < class ForwardIterator, class OutputIterator, class Traits >
OutputIterator
rectangular_3_center_2(
  ForwardIterator f,
  ForwardIterator l,
  OutputIterator o,
  typename Traits::FT& r,
  Traits& t)
{
  CGAL_optimisation_precondition(f != l);
  typedef typename Traits::FT                                    FT;
  typedef typename Traits::Point_2                            Point;
  typedef typename Traits::Iso_rectangle_2                Rectangle;
  typedef Rectangular_3_center_2_type2_operations0< Traits >    Op0;
  typedef Rectangular_3_center_2_type2_operations1< Traits >    Op1;
  typedef Rectangular_3_center_2_type2_operations2< Traits >    Op2;
  typedef Rectangular_3_center_2_type2_operations3< Traits >    Op3;

  std::vector< Point > points(f, l);
  Rectangle bb = bounding_box_2(points.begin(), points.end(), t);

  // try to place two squares at opposite corners of bb
  Point ptst[3];
  rectangular_3_center_2_type1(
    points.begin(), points.end(), bb, ptst, r, t);

  // try to place one square at a corner and the others
  // at the two remaining sides of bb
  Point pts0[3];
  Point pts1[3];
  Point pts2[3];
  Point pts3[3];
  Point* pts = ptst;
  FT rmin = r;

  rectangular_3_center_2_type2(
    points.begin(), points.end(), bb, pts0, r, Op0(t, bb[0]));
  if (r < rmin)
    pts = pts0, rmin = r;
#ifdef CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST
  else
    r = rmin;
#endif // CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST


  rectangular_3_center_2_type2(
    points.begin(), points.end(), bb, pts1, r, Op1(t, bb[1]));
  if (r < rmin)
    pts = pts1, rmin = r;
#ifdef CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST
  else
    r = rmin;
#endif // CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST


  rectangular_3_center_2_type2(
    points.begin(), points.end(), bb, pts2, r, Op2(t, bb[2]));
  if (r < rmin)
    pts = pts2, rmin = r;
#ifdef CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST
  else
    r = rmin;
#endif // CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST


  rectangular_3_center_2_type2(
    points.begin(), points.end(), bb, pts3, r, Op3(t, bb[3]));
  if (r < rmin)
    pts = pts3;
#ifdef CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST
  else
    r = rmin;
#endif // CGAL_3COVER_NO_CHECK_OPTIMUM_FIRST

  *o++ = pts[0];
  *o++ = pts[1];
  *o++ = pts[2];
  return o;

} // rectangular_3_center_2( ... )

} //namespace CGAL

#endif // ! (CGAL_RECTANGULAR_3_CENTER_2_H)