# Types Of Refugees Essaytyper

- Authors
- Hans Tangelder and Andreas Fabri

## Introduction

The spatial searching package implements exact and approximate distance browsing by providing implementations of algorithms supporting

both nearest and furthest neighbor searching

both exact and approximate searching

(approximate) range searching

(approximate) -nearest and -furthest neighbor searching

(approximate) incremental nearest and incremental furthest neighbor searching

query items representing points and spatial objects.

In these searching problems a set of data points in -dimensional space is given. The points can be represented by Cartesian coordinates or homogeneous coordinates. These points are preprocessed into a tree data structure, so that given any query item the points of can be browsed efficiently. The approximate spatial searching package is designed for data sets that are small enough to store the search structure in main memory (in contrast to approaches from databases that assume that the data reside in secondary storage).

## Neighbor Searching

Spatial searching supports browsing through a collection of -dimensional spatial objects stored in a spatial data structure on the basis of their distances to a query object. The query object may be a point or an arbitrary spatial object, e.g, a -dimensional sphere. The objects in the spatial data structure are -dimensional points.

Often the number of the neighbors to be computed is not know beforehand, e.g., because the number may depend on some properties of the neighbors (for example when querying for the nearest city to Paris with population greater than a million) or the distance to the query point. The conventional approach is -nearest neighbor searching that makes use of a -nearest neighbor algorithm, where is known prior to the invocation of the algorithm. Hence, the number of nearest neighbors has to be guessed. If the guess is too large redundant computations are performed. If the number is too small the computation has to be re-invoked for a larger number of neighbors, thereby performing redundant computations. Therefore, Hjaltason and Samet [5] introduced *incremental nearest neighbor searching* in the sense that having obtained the nearest neighbors, the st neighbor can be obtained without having to calculate the nearest neighbor from scratch.

Spatial searching typically consists of a preprocessing phase and a searching phase. In the preprocessing phase one builds a search structure and in the searching phase one makes the queries. In the preprocessing phase the user builds a tree data structure storing the spatial data. In the searching phase the user invokes a searching method to browse the spatial data.

With relatively minor modifications, nearest neighbor searching algorithms can be used to find the furthest object from the query object. Therefore, *furthest neighbor searching* is also supported by the spatial searching package.

The execution time for exact neighbor searching can be reduced by relaxing the requirement that the neighbors should be computed exactly. If the distances of two objects to the query object are approximately the same, instead of computing the nearest/furthest neighbor exactly, one of these objects may be returned as the approximate nearest/furthest neighbor. I.e., given some non-negative constant \( \epsilon\) the distance of an object returned as an approximate -nearest neighbor must not be larger than \( (1+\epsilon)r\), where \( r\) denotes the distance to the real th nearest neighbor. Similar the distance of an approximate -furthest neighbor must not be smaller than \( r/(1+\epsilon)\). Obviously, for \( \epsilon=0\) we get the exact result, and the larger \( \epsilon\) is, the less exact the result.

While searching the nearest neighbor the algorithm descends the kd-tree and has to decide two things for each node : Which child node should be visited first and could there be possible nearest neighbors in the other child. This basically comes down to computing the distance to the further child, because the distance to the closer child is the same as the one to the parent. There are two options now:

- In general, we compute the distance with the given metric. This is the
*k-neighbor search*with a*general distance*class. - For point queries we can "update" the distance, because it is only changed in one dimension at a time. This is the
*orthogonal k-neighbor search*with an*orthogonal distance*class. The following example shows the orthogonal distance computation in detail:

Figure 75.1 Orthogonal distance computation technique

Assume we are searching the nearest neighbor, descending the kd-tree, with \( R_{p} \) as the parent rectangle and \( R_{lo} \) and \( R_{hi}\) as its childs in the current step. Further assume \( R_{lo} \) is closer to query point \(q\). Let \(cd\) denote the cutting dimension and let \(cv\) denote the cutting value. At this point we already know the distance \(rd_{p}\) to the parent rectangle and need to check if \(R_{hi}\) could contain nearest neighbors. Because \(R_{lo}\) is the closer rectangle, its distance to \(q\), \(rd_{lo}\), is the same as \(rd_{p}\). Notice that for each dimension \(i \neq cd \), \( \mathrm{dists}_{lo}[i] = \mathrm{dists}_{hi}[i]\), since these coordinates are not affected by the current cut. So the new distance along the cutting dimension is \( \mathrm{dists}_{hi}[cd] = cv - q[cd]\). Now we can compute \(rd_{hi}\) in constant time (independent of dimension) with \(rd_{hi} = rd_{p} - \mathrm{dists}_{lo}[cd]^2 + (cv - q[cd])^2\).

This strategy can be used if and only if the distance changes only in one dimension at a time, which is the case for point queries.

The following two classes implement the standard search strategy for orthogonal distances like the weighted Minkowski distance. The second one is a specialization for incremental neighbor searching and distance browsing. Both require extendes nodes.

The other two classes implement the standard search strategy for general distances like the Manhattan distance for iso-rectangle queries. Again, the second one is a specialization for incremental neighbor searching and distance browsing .

## Range Searching

*Exact range searching* and *approximate range searching* are supported using exact or fuzzy -dimensional objects enclosing a region. The fuzziness of the query object is specified by a parameter \( \epsilon\) used to define inner and outer approximations of the query object. Points in the interior of the inner approximation are always reported and points that are not in the closure of the outer approximation are never reported. Other points may or may not be reported. For exact range searching, the fuzziness parameter \( \epsilon\) is set to zero.

The class implements range searching in the method , which is a template method with an output iterator and a model of the concept as or . For range searching of large data sets, the user may set the parameter used in building the tree to a large value (e.g. 100), because in general the query time will be less than using the default value.

## Splitting Rules

Instead of using the default splitting rule described below, a user may, depending upon the data, select one from the following splitting rules, which determine how a separating hyperplane is computed. Every splitter has degenerated worst cases, which may lead to a linear tree and a stack overflow. Switching the splitting rule to one of a different kind will solve the problem.

This splitting rule cuts a rectangle through its midpoint orthogonal to the longest side.

This splitting rule cuts a rectangle through \( (\mathrm{Mind}+\mathrm{Maxd})/2\) orthogonal to the dimension with the maximum point spread \( [\mathrm{Mind},\mathrm{Maxd}]\).

This is a modification of the midpoint of rectangle splitting rule. It first attempts to perform a midpoint of rectangle split as described above. If data points lie on both sides of the separating plane the sliding midpoint rule computes the same separator as the midpoint of rectangle rule. If the data points lie only on one side it avoids this by sliding the separator, computed by the midpoint of rectangle rule, to the nearest data point.

As all the midpoint rules cut the bounding box in the middle of the longest side, the tree will become linear for a dataset with exponential increasing distances in one dimension.

Figure 75.2 Midpoint worst case point set in 2d.

The splitting dimension is the dimension of the longest side of the rectangle. The splitting value is defined by the median of the coordinates of the data points along this dimension.

The splitting dimension is the dimension of the longest side of the rectangle. The splitting value is defined by the median of the coordinates of the data points along this dimension.

The tree can become linear for the median rules, if many points are collinear in a dimension which is not the cutting dimension.

Figure 75.3 Median worst case point set in 2d.

a is longer than b, so this will be the cutting dimension.

This splitting rule is a compromise between the median of rectangle splitting rule and the midpoint of rectangle splitting rule. This splitting rule maintains an upper bound on the maximal allowed ratio of the longest and shortest side of a rectangle (the value of this upper bound is set in the constructor of the fair splitting rule). Among the splits that satisfy this bound, it selects the one in which the points have the largest spread. It then splits the points in the most even manner possible, subject to maintaining the bound on the ratio of the resulting rectangles.

This splitting rule is a compromise between the fair splitting rule and the sliding midpoint rule. Sliding fair-split is based on the theory that there are two types of splits that are good: balanced splits that produce fat rectangles, and unbalanced splits provided the rectangle with fewer points is fat.

Also, this splitting rule maintains an upper bound on the maximal allowed ratio of the longest and shortest side of a rectangle (the value of this upper bound is set in the constructor of the fair splitting rule). Among the splits that satisfy this bound, it selects the one one in which the points have the largest spread. It then considers the most extreme cuts that would be allowed by the aspect ratio bound. This is done by dividing the longest side of the rectangle by the aspect ratio bound. If the median cut lies between these extreme cuts, then we use the median cut. If not, then consider the extreme cut that is closer to the median. If all the points lie to one side of this cut, then we slide the cut until it hits the first point. This may violate the aspect ratio bound, but will never generate empty cells.

## Example Programs

We give seven examples. The first example illustrates k nearest neighbor searching, and the second example incremental neighbor searching. The third is an example of approximate furthest neighbor searching using a -dimensional iso-rectangle as an query object. Approximate range searching is illustrated by the fourth example. The fifth example illustrates k neighbor searching for a user defined point class. The sixth example shows how to choose another splitting rule in the tree that is used as search tree. The last example shows two worst-case scenarios for different splitter types.

## Example for K Neighbor Searching

The first example illustrates k neighbor searching with an Euclidean distance and 2-dimensional points. The generated random data points are inserted in a search tree. We then initialize the k neighbor search object with the origin as query. Finally, we obtain the result of the computation in the form of an iterator range. The value of the iterator is a pair of a point and its square distance to the query point. We use square distances, or *transformed distances* for other distance classes, as they are computationally cheaper.

**File**Spatial_searching/nearest_neighbor_searching.cpp

#include <CGAL/Simple_cartesian.h>

#include <CGAL/point_generators_2.h>

#include <CGAL/Orthogonal_k_neighbor_search.h>

#include <CGAL/Search_traits_2.h>

#include <list>

#include <cmath>

typedefCGAL::Simple_cartesian<double> K;

typedef K::Point_2 Point_d;

typedefCGAL::Search_traits_2<K> TreeTraits;

typedefCGAL::Orthogonal_k_neighbor_search<TreeTraits> Neighbor_search;

typedef Neighbor_search::Tree Tree;

int main() {

constunsignedint N = 1;

std::list<Point_d> points;

points.push_back(Point_d(0,0));

Tree tree(points.begin(), points.end());

Point_d query(0,0);

Neighbor_search search(tree, query, N);

for(Neighbor_search::iterator it = search.begin(); it != search.end(); ++it){

std::cout << it->first << " "<< std::sqrt(it->second) << std::endl;

}

return 0;

}

## Example for Incremental Searching

This example program illustrates incremental searching for the closest point with a positive first coordinate. We can use the orthogonal incremental neighbor search class, as the query is also a point and as the distance is the Euclidean distance.

As for the neighbor search, we first initialize the search tree with the data. We then create the search object, and finally obtain the iterator with the method. Note that the iterator is of the input iterator category, that is one can make only one pass over the data.

**File**Spatial_searching/distance_browsing.cpp

#include <CGAL/Simple_cartesian.h>

#include <CGAL/Orthogonal_incremental_neighbor_search.h>

#include <CGAL/Search_traits_2.h>

typedefCGAL::Simple_cartesian<double> K;

typedef K::Point_2 Point_d;

typedefCGAL::Search_traits_2<K> TreeTraits;

typedefCGAL::Orthogonal_incremental_neighbor_search<TreeTraits> NN_incremental_search;

typedef NN_incremental_search::iterator NN_iterator;

typedef NN_incremental_search::Tree Tree;

struct X_not_positive {

bool operator()(const NN_iterator& it) { return ((*it).first)[0]<0; }

};

typedefCGAL::Filter_iterator<NN_iterator, X_not_positive> NN_positive_x_iterator;

int main() {

Tree tree;

tree.insert(Point_d(0,0));

tree.insert(Point_d(1,1));

tree.insert(Point_d(0,1));

tree.insert(Point_d(10,110));

tree.insert(Point_d(45,0));

tree.insert(Point_d(0,2340));

tree.insert(Point_d(0,30));

Point_d query(0,0);

NN_incremental_search NN(tree, query);

NN_positive_x_iterator it(NN.end(), X_not_positive(), NN.begin()), end(NN.end(), X_not_positive());

std::cout << "The first 5 nearest neighbours with positive x-coord are: " << std::endl;

for (int j=0; (j < 5)&&(it!=end); ++j,++it)

std::cout << (*it).first << " at squared distance = " << (*it).second << std::endl;

return 0;

}

## Example for General Neighbor Searching

This example program illustrates approximate nearest and furthest neighbor searching using 4-dimensional Cartesian coordinates. Five approximate furthest neighbors of the query rectangle \( [0.1,0.2]^4\) are computed. Because the query object is a rectangle we cannot use the orthogonal neighbor search. As in the previous examples we first initialize a search tree, create the search object with the query, and obtain the result of the search as iterator range.

**File**Spatial_searching/general_neighbor_searching.cpp

#include <CGAL/Epick_d.h>

#include <CGAL/point_generators_d.h>

#include <CGAL/Manhattan_distance_iso_box_point.h>

#include <CGAL/K_neighbor_search.h>

#include <CGAL/Search_traits_d.h>

typedefCGAL::Epick_d<CGAL::Dimension_tag<4> > Kernel;

typedef Kernel::Point_d Point_d;

typedef CGAL::Random_points_in_cube_d<Point_d> Random_points_iterator;

typedef Kernel::Iso_box_d Iso_box_d;

typedef Kernel TreeTraits;

typedefCGAL::Manhattan_distance_iso_box_point<TreeTraits> Distance;

typedefCGAL::K_neighbor_search<TreeTraits, Distance> Neighbor_search;

typedef Neighbor_search::Tree Tree;

int main() {

constint N = 1000;

constunsignedint K = 10;

Tree tree;

Random_points_iterator rpit(4,1000.0);

for(int i = 0; i < N; i++){

tree.insert(*rpit++);

}

Point_d pp(0.1,0.1,0.1,0.1);

Point_d qq(0.2,0.2,0.2,0.2);

Iso_box_d query(pp,qq);

Distance tr_dist;

Neighbor_search N1(tree, query, 5, 10.0, false);

std::cout << "For query rectangle = [0.1, 0.2]^4 " << std::endl

<< "the " << K << " approximate furthest neighbors are: " << std::endl;

for (Neighbor_search::iterator it = N1.begin();it != N1.end();it++) {

std::cout << " Point " << it->first << " at distance " << tr_dist.inverse_of_transformed_distance(it->second) << std::endl;

}

return 0;

}

## Example for a Range Query

This example program illustrates approximate range querying for 4-dimensional fuzzy iso-rectangles and spheres using the higher dimensional kernel . The range queries are member functions of the tree class.

**File**Spatial_searching/fuzzy_range_query.cpp

#include <CGAL/Epick_d.h>

#include <CGAL/point_generators_d.h>

#include <CGAL/Kd_tree.h>

#include <CGAL/Fuzzy_sphere.h>

#include <CGAL/Fuzzy_iso_box.h>

#include <CGAL/Search_traits_d.h>

constint D = 4;

typedefCGAL::Epick_d<CGAL::Dimension_tag<D> > K;

typedef K::Point_d Point_d;

typedefCGAL::Search_traits_d<K,CGAL::Dimension_tag<D> > Traits;

typedef CGAL::Random_points_in_cube_d<Point_d> Random_points_iterator;

typedefCGAL::Counting_iterator<Random_points_iterator> N_Random_points_iterator;

typedefCGAL::Kd_tree<Traits> Tree;

typedefCGAL::Fuzzy_sphere<Traits> Fuzzy_sphere;

typedefCGAL::Fuzzy_iso_box<Traits> Fuzzy_iso_box;

int main() {

constint N = 1000;

Random_points_iterator rpit(4, 1000.0);

Tree tree(N_Random_points_iterator(rpit,0),

N_Random_points_iterator(rpit,N));

double pcoord[D] = { 300, 300, 300, 300 };

double qcoord[D] = { 900.0, 900.0, 900.0, 900.0 };

Point_d p(D, pcoord+0, pcoord+D);

Point_d q(D, qcoord+0, qcoord+D);

Fuzzy_sphere fs(p, 700.0, 100.0);

Fuzzy_iso_box fib(p, q, 100.0);

std::cout << "points approximately in fuzzy spherical range query" << std::endl;

std::cout << "with center (300, 300, 300, 300)" << std::endl;

std::cout << "and fuzzy radius [600, 800] are:" << std::endl;

tree.search(std::ostream_iterator<Point_d>(std::cout, "\n"), fs);

std::cout << "points approximately in fuzzy rectangular range query ";

std::cout << "[[200, 400], [800,1000]]^4 are:" << std::endl;

tree.search(std::ostream_iterator<Point_d>(std::cout, "\n"), fib);

return 0;

}

## Example for User Defined Point and Distance Class

The neighbor searching works with all CGAL kernels, as well as with user defined points and distance classes. In this example we assume that the user provides the following 3-dimensional points class.

**File**Spatial_searching/Point.h

struct Point {

double vec[3];

Point() { vec[0]= vec[1] = vec[2] = 0; }

Point (double x, double y, double z) { vec[0]=x; vec[1]=y; vec[2]=z; }

double x() const { return vec[ 0 ]; }

double y() const { return vec[ 1 ]; }

double z() const { return vec[ 2 ]; }

double& x() { return vec[ 0 ]; }

double& y() { return vec[ 1 ]; }

double& z() { return vec[ 2 ]; }

bool operator==(const Point& p) const

{

return (x() == p.x()) && (y() == p.y()) && (z() == p.z()) ;

}

bool operator!=(const Point& p) const { return ! (*this == p); }

};

struct Construct_coord_iterator {

typedefconstdouble* result_type;

constdouble* operator()(const Point& p) const

{ returnstatic_cast<const double*>(p.vec); }

constdouble* operator()(const Point& p, int) const

{ returnstatic_cast<const double*>(p.vec+3); }

};

We have put the glue layer in this file as well, that is a class that allows to iterate over the Cartesian coordinates of the point, and a class to construct such an iterator for a point. We next need a distance class **File**Spatial_searching/Distance.h

struct Distance {

typedef Point Query_item;

typedefdouble FT;

typedefCGAL::Dimension_tag<3> D;

double transformed_distance(const Point& p1, const Point& p2) const {

double distx= p1.x()-p2.x();

double disty= p1.y()-p2.y();

double distz= p1.z()-p2.z();

return distx*distx+disty*disty+distz*distz;

}

double min_distance_to_rectangle(const Point& p,

constCGAL::Kd_tree_rectangle<FT,D>& b) const {

double distance(0.0), h = p.x();

if (h < b.min_coord(0)) distance += (b.min_coord(0)-h)*(b.min_coord(0)-h);

if (h > b.max_coord(0)) distance += (h-b.max_coord(0))*(h-b.max_coord(0));

h=p.y();

if (h < b.min_coord(1)) distance += (b.min_coord(1)-h)*(b.min_coord(1)-h);

if (h > b.max_coord(1)) distance += (h-b.max_coord(1))*(h-b.min_coord(1));

h=p.z();

if (h < b.min_coord(2)) distance += (b.min_coord(2)-h)*(b.min_coord(2)-h);

if (h > b.max_coord(2)) distance += (h-b.max_coord(2))*(h-b.max_coord(2));

return distance;

}

double min_distance_to_rectangle(const Point& p,

constCGAL::Kd_tree_rectangle<FT,D>& b,std::vector<double>& dists){

double distance(0.0), h = p.x();

if (h < b.min_coord(0)){

dists[0] = (b.min_coord(0)-h);

distance += dists[0]*dists[0];

}

if (h > b.max_coord(0)){

dists[0] = (h-b.max_coord(0));

distance += dists[0]*dists[0];

}

h=p.y();

if (h < b.min_coord(1)){

dists[1] = (b.min_coord(1)-h);

distance += dists[1]*dists[1];

}

if (h > b.max_coord(1)){

dists[1] = (h-b.max_coord(1));

distance += dists[1]*dists[1];

}

h=p.z();

if (h < b.min_coord(2)){

dists[2] = (b.min_coord(2)-h);

distance += dists[2]*dists[2];

}

if (h > b.max_coord(2)){

dists[2] = (h-b.max_coord(2));

distance += dists[2]*dists[2];

}

return distance;

}

double max_distance_to_rectangle(const Point& p,

constCGAL::Kd_tree_rectangle<FT,D>& b) const {

double h = p.x();

double d0 = (h >= (b.min_coord(0)+b.max_coord(0))/2.0) ?

(h-b.min_coord(0))*(h-b.min_coord(0)) : (b.max_coord(0)-h)*(b.max_coord(0)-h);

h=p.y();

double d1 = (h >= (b.min_coord(1)+b.max_coord(1))/2.0) ?

(h-b.min_coord(1))*(h-b.min_coord(1)) : (b.max_coord(1)-h)*(b.max_coord(1)-h);

h=p.z();

double d2 = (h >= (b.min_coord(2)+b.max_coord(2))/2.0) ?

(h-b.min_coord(2))*(h-b.min_coord(2)) : (b.max_coord(2)-h)*(b.max_coord(2)-h);

return d0 + d1 + d2;

}

double max_distance_to_rectangle(const Point& p,

constCGAL::Kd_tree_rectangle<FT,D>& b,std::vector<double>& dists){

double h = p.x();

dists[0] = (h >= (b.min_coord(0)+b.max_coord(0))/2.0) ?

(h-b.min_coord(0)) : (b.max_coord(0)-h);

h=p.y();

dists[1] = (h >= (b.min_coord(1)+b.max_coord(1))/2.0) ?

(h-b.min_coord(1)) : (b.max_coord(1)-h);

h=p.z();

dists[2] = (h >= (b.min_coord(2)+b.max_coord(2))/2.0) ?

(h-b.min_coord(2)) : (b.max_coord(2)-h);

return dists[0] * dists[0] + dists[1] * dists[1] + dists[2] * dists[2];

}

double new_distance(double& dist, double old_off, double new_off,

int) const {

return dist + new_off*new_off - old_off*old_off;

}

double transformed_distance(double d) const { return d*d; }

double inverse_of_transformed_distance(double d) { returnstd::sqrt(d); }

};

We are ready to put the pieces together. The class ,which you see in the next file, is a mere wrapper for all our defined types. The searching itself works exactly as for CGAL kernels.

**File**Spatial_searching/user_defined_point_and_distance.cpp

#include <CGAL/Search_traits.h>

#include <CGAL/point_generators_3.h>

#include <CGAL/Orthogonal_k_neighbor_search.h>

#include "Point.h"

#include "Distance.h"

typedefCGAL::Creator_uniform_3<double,Point> Point_creator;

typedef CGAL::Random_points_in_cube_3<Point, Point_creator> Random_points_iterator;

typedefCGAL::Counting_iterator<Random_points_iterator> N_Random_points_iterator;

typedefCGAL::Dimension_tag<3> D;

typedefCGAL::Search_traits<double, Point, const double*, Construct_coord_iterator, D> Traits;

typedefCGAL::Orthogonal_k_neighbor_search<Traits, Distance> K_neighbor_search;

typedefK_neighbor_search::Tree Tree;

int main() {

constint N = 1000;

constunsignedint K = 5;

Random_points_iterator rpit( 1.0);

Tree tree(N_Random_points_iterator(rpit,0),

N_Random_points_iterator(N));

Point query(0.0, 0.0, 0.0);

Distance tr_dist;

K_neighbor_search search(tree, query, K);

for(K_neighbor_search::iterator it = search.begin(); it != search.end(); it++){

std::cout << " d(q, nearest neighbor)= "

<< tr_dist.inverse_of_transformed_distance(it->second) << std::endl;

}

K_neighbor_search search2(tree, query, K, 0.0, false);

for(K_neighbor_search::iterator it = search2.begin(); it != search2.end(); it++){

std::cout << " d(q, furthest neighbor)= "

<< tr_dist.inverse_of_transformed_distance(it->second) << std::endl;

}

return 0;

}

## Examples for Using an Arbitrary Point Type with Point Property Maps

The following four example programs illustrate how to use the classes and to store in the kd-tree objects of an arbitrary key type. Points are accessed through a point property map. This enables to associate information to a point or to reduce the size of the search structure.

### Using a Point and an Integer as Key Type

In this example program, the search tree stores tuples of point and integer. The value type of the iterator of the neighbor searching algorithm is this tuple type.

**File**Spatial_searching/searching_with_point_with_info.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>

#include <CGAL/Search_traits_3.h>

#include <CGAL/Search_traits_adapter.h>

#include <CGAL/point_generators_3.h>

#include <CGAL/Orthogonal_k_neighbor_search.h>

#include <CGAL/property_map.h>

#include <boost/iterator/zip_iterator.hpp>

#include <utility>

typedefCGAL::Exact_predicates_inexact_constructions_kernel Kernel;

typedefKernel::Point_3 Point_3;

typedef boost::tuple<Point_3,int> Point_and_int;

typedef CGAL::Random_points_in_cube_3<Point_3> Random_points_iterator;

typedefCGAL::Search_traits_3<Kernel> Traits_base;

typedefCGAL::Search_traits_adapter<Point_and_int,

CGAL::Nth_of_tuple_property_map<0, Point_and_int>,

Traits_base> Traits;

typedefCGAL::Orthogonal_k_neighbor_search<Traits> K_neighbor_search;

typedefK_neighbor_search::Tree Tree;

typedefK_neighbor_search::Distance Distance;

int main() {

constunsignedint K = 5;

Random_points_iterator rpit( 1.0);

std::vector<Point_3> points;

std::vector<int> indices;

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

indices.push_back(0);

indices.push_back(1);

indices.push_back(2);

indices.push_back(3);

indices.push_back(4);

indices.push_back(5);

indices.push_back(6);

Tree tree(

boost::make_zip_iterator(boost::make_tuple( points.begin(),indices.begin() )),

boost::make_zip_iterator(boost::make_tuple( points.end(),indices.end() ) )

);

Point_3 query(0.0, 0.0, 0.0);

Distance tr_dist;

K_neighbor_search search(tree, query, K);

for(K_neighbor_search::iterator it = search.begin(); it != search.end(); it++){

std::cout << " d(q, nearest neighbor)= "

<< tr_dist.inverse_of_transformed_distance(it->second) << " "

<< boost::get<0>(it->first)<< " " << boost::get<1>(it->first) << std::endl;

}

return 0;

}

### Using an Integer as Key Type

In this example program, the search tree stores only integers that refer to points stored within a user vector. The point type of the search traits is .

**File**Spatial_searching/searching_with_point_with_info_inplace.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>

#include <CGAL/Search_traits_3.h>

#include <CGAL/Search_traits_adapter.h>

#include <CGAL/point_generators_3.h>

#include <CGAL/Orthogonal_k_neighbor_search.h>

#include <CGAL/boost/iterator/counting_iterator.hpp>

#include <utility>

typedefCGAL::Exact_predicates_inexact_constructions_kernel Kernel;

typedefKernel::Point_3 Point_3;

class My_point_property_map{

const std::vector<Point_3>& points;

public:

typedef Point_3 value_type;

typedefconst value_type& reference;

typedef std::size_t key_type;

typedef boost::lvalue_property_map_tag category;

My_point_property_map(const std::vector<Point_3>& pts):points(pts){}

reference operator[](key_type k) const {return points[k];}

friend reference get(const My_point_property_map& ppmap,key_type i)

{return ppmap[i];}

};

typedef CGAL::Random_points_in_cube_3<Point_3> Random_points_iterator;

typedefCGAL::Search_traits_3<Kernel> Traits_base;

typedefCGAL::Search_traits_adapter<std::size_t,My_point_property_map,Traits_base> Traits;

typedefCGAL::Orthogonal_k_neighbor_search<Traits> K_neighbor_search;

typedefK_neighbor_search::Tree Tree;

typedef Tree::Splitter Splitter;

typedefK_neighbor_search::Distance Distance;

int main() {

constunsignedint K = 5;

Random_points_iterator rpit( 1.0);

std::vector<Point_3> points;

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

points.push_back(Point_3(*rpit++));

My_point_property_map ppmap(points);

Tree tree(

boost::counting_iterator<std::size_t>(0),

boost::counting_iterator<std::size_t>(points.size()),

Splitter(),

Traits(ppmap)

);

Point_3 query(0.0, 0.0, 0.0);

Distance tr_dist(ppmap);

K_neighbor_search search(tree, query, K,0,true,tr_dist);

for(K_neighbor_search::iterator it = search.begin(); it != search.end(); it++){

std::cout << " d(q, nearest neighbor)= "

<< tr_dist.inverse_of_transformed_distance(it->second) << " "

<< points[it->first] << " " << it->first << std::endl;

}

return 0;

}

### Using a Model of L-value Property Map Concept

This example programs uses a model of . Points are read from a . The search tree stores integers of type . The value type of the iterator of the neighbor searching algorithm is .

**File**Spatial_searching/searching_with_point_with_info_pmap.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>

#include <CGAL/Search_traits_3.h>

#include <CGAL/Search_traits_adapter.h>

#include <CGAL/point_generators_3.h>

#include <CGAL/Orthogonal_k_neighbor_search.h>

#include <CGAL/boost/iterator/counting_iterator.hpp>

#include <utility>

typedefCGAL::Exact_predicates_inexact_constructions_kernel Kernel;

typedefKernel::Point_3 Point_3;

typedef boost::const_associative_property_map<std::map<std::size_t,Point_3> > My_point_property_map;

typedef CGAL::Random_points_in_cube_3<Point_3> Random_points_iterator;

typedefCGAL::Search_traits_3<Kernel> Traits_base;

typedefCGAL::Search_traits_adapter<std::size_t,My_point_property_map,Traits_base> Traits;

typedefCGAL::Orthogonal_k_neighbor_search<Traits> K_neighbor_search;

typedefK_neighbor_search::Tree Tree;

typedef Tree::Splitter Splitter;

typedefK_neighbor_search::Distance Distance;

int main() {

constunsignedint K = 5;

Random_points_iterator rpit( 1.0);

std::map<std::size_t,Point_3> points;

points[0]=Point_3(*rpit++);

points[1]=Point_3(*rpit++);

points[2]=Point_3(*rpit++);

points[3]=Point_3(*rpit++);

points[4]=Point_3(*rpit++);

points[5]=Point_3(*rpit++);

points[6]=Point_3(*rpit++);

My_point_property_map ppmap(points);

Tree tree(

boost::counting_iterator<std::size_t>(0),

boost::counting_iterator<std::size_t>(points.size()),

Splitter(),

Traits(ppmap)

);

Point_3 query(0.0, 0.0, 0.0);

Distance tr_dist(ppmap);

K_neighbor_search search(tree, query, K,0,true,tr_dist);

for(K_neighbor_search::iterator it = search.begin(); it != search.end(); it++){

std::cout << " d(q, nearest neighbor)= "

<< tr_dist.inverse_of_transformed_distance(it->second) << " "

<< points[it->first] << " " << it->first << std::endl;

}

return 0;

}

### Using a Point Property Map of a Polygonal Mesh

This example programs shows how to search the closest vertices of a or, quite similar, of a . Points are stored in the polygonal mesh. The search tree stores vertex descriptors. The value type of the iterator of the neighbor searching algorithm is .

**File**Spatial_searching/searching_surface_mesh_vertices.cpp

Global immigration remains a burning socio-political issue, prompting our need for **World Refugee Day**. On June 20th, we recognize all types of refugees in crisis as 65 million refugees and displaced people seek asylum across the globe. Read more about how the United Nations Refugee Agency (UNHCR) and countless other groups raise awareness for these displaced people around the world.

In this time of relentless refugees crisis, governments, aid workers, celebrities, and civilians draw public attention to those who have fled their homes due to war, conflict, and persecution.

World Refugee Day Tuesday, June 20th marks a day in which we honor the strength, courage, spirit, and resilience of millions of refugees. By raising awareness about the injustices against migrants and asylum seekers, we encourage people to think about the lives of refugees and their fight for human rights.

**World Refugee Day: How Do You Define Refugee? **

There are several types of refugees and forcibly displaced persons:

### Refugees

According to the United Nations 1951 Refugee Convention, a refugee is someone who has fled their home and country due to “a well-founded fear of persecution because of his/her race, religion, nationality, membership in a particular social group, or political opinion”.

### Asylum seekers

Asylum seekers are similar to refugees in their status, but they have not been formally declared as refugees by the country to which they fled. Many struggle through this long asylum application process due to local prejudice, leaving them in appalling living conditions in refugee camps while others end up on the streets. **Internally Displaced Persons**

Internally Displaced Persons (IDPs) are people who have had to leave their homes but have not left their own country. As a result, they often remain in a state of distress while searching for food, shelter, and other necessities.

### Stateless Persons

Stateless persons are commonly discriminated against by certain groups, making them unrecognized as a nationality or belonging to a specific country. They are not in possession of identification or citizen certification, excluding them from access to basic human rights, including healthcare, education, employment, and other government services.

### Returnees

Returnees are former refugees who have returned to their home countries after fleeing. Although this can be a positive outcome, returnees still require support and reintegration in order to reestablish their lives at home.

Regardless of their official title, it is clear that all types of refugees need our help. Whether it’s finding a new home or as simple as having access to food, clean water, and shelter, all types of refugees deserve support.

**1951 Refugee Convention and its 1967 Protocol**

The 1951 Refugee Convention and its 1967 Protocol are the only orders in place protecting refugees. It simply states that all types of refugees deserve the same standards of treatment as other foreign nationals. An obvious, yet surprising widely unmet standard.

The orders also state refugees should not be returned to a country where they face threats to their life or freedom. This keeps countries like Spain from being able to automatically reject asylum seekers who attempt to cross their borders through its North African enclaves.

This order, along with other basic rights, applies to all types of refugees unless the refugee is considered a danger to the community or security of the country.

The rights contained in the 1951 Convention include:

- The right not to be expelled, except under certain, strictly defined conditions;
- The right not to be punished for illegal entry into the territory of a Contracting State;
- The right to work;
- The right to housing;
- The right to education;
- The right to public relief and assistance;
- The right to freedom of religion;
- The right to access the courts;
- The right to freedom of movement within the territory;
- The right to be issued identity and travel documents.

The accumulation of additional rights increase the longer the refugees in the host country.

**The Situation for All Types of Refugees **

Sixty-five million people have fled their homes to escape war and violence. This staggering number grows by thousands every single day as victims flee areas such as Syria, Burundi, Central African Republic, Iraq, Myanmar, Nigeria, South Sudan, Ukraine, Yemen and beyond.

Read just a fraction of their stories, shared by UNHCR here:

Further, take a moment to read the following graphics:

**What You Can Do**

On this day, we hope to raise awareness about these delicate types of refugees. We recognize these vulnerable people and their ability to survive these impossible circumstances. We focus on how we can become involved and help those who are suffering around the world. It is time to show that the global public stands with refugees.

In 2016, the UN Refugee Agency launched the**#WithRefugees** petition promoting collaborative governments and leaders to work together and provide proper support for all types of refugees.

“I’ve met so many who have lost so much. But they never lose their dreams for their children or their desire to better our world. They ask for little in return – only our support in their time of greatest need” — UN Secretary-General, António Guterres

Those who have left everything behind in hopes of a safer future deserve safety and security.

By stating your support for the #WithRefugees movement, you will stand with a petition that asks governments to:

- Ensure every refugee child gets an education.
- Ensure every refugee family has somewhere safe to live.
- Ensure every refugee can work or learn new skills to make a positive contribution to their community.

Get behind the global movement and **stand #WithRefugees today. ****Sign the petition here.**

We also invite you to join** #UnblocktheWeb**,our partnership with the human rights crowdsourcing platform Movements.org. Through this initiative, we provide activists and dissidents in closed societies with free VPN access, so that they can browse the Internet safe from government monitoring and exercise their right to freedom of expression online.

Over the course of our work with Movements.org, we’ve met many brave individuals, like notable journalist and Human Rights activist Kaveh Taheri, who’ve had no other choice but to flee their home countries due to lack of basic freedoms, safety and rights.

After he was arrested by Iran’s intelligence service for the contents of his *personal* blogs, and then subjected to 10 months of unthinkable psychological and physical torture in prison, Taheri, whom we recently interviewed, fled his homeland of Iran to seek safety and political refuge in Turkey.

Today, Taheri continues to speak on behalf of the voiceless on ACUNews, a unique site that strives to shed light on untold stories on Human Rights issues, provided by a team of independent investigative journalists and Human rights activists.

“As a freelance journalist and human rights activist who has spent his life fighting for freedom and justice, I call on all global villagers to come out in support of those who have been persecuted for simply exercising their rights to freedom of information and speech, ” says Taheri.

It is freedom fighters like Taheri and initiatives like #WithRefugees and #UnblockTheWeb that facilitate the necessary steps forward in an urgent fight for freedom for all types of refugees. Today, on World Refugee Day, we honor this fight.

World Refugee Day: Recognizing All Types of Refugees

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