Hans Tangelder and Andreas Fabri
The spatial searching package implements exact and approximate distance browsing by providing implementations of algorithms supporting
In these searching problems a set P of data points in d-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 q the points of P 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).
Spatial searching supports browsing through a collection of d-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 d-dimensional sphere. The objects in the spatial data structure are d-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 k-nearest neighbor searching that makes use of a k-nearest neighbor algorithm, where k 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 [HS95] introduced incremental nearest neighbor searching in the sense that having obtained the k nearest neighbors, the k + 1st neighbor can be obtained without having to calculate the k + 1 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 ε the distance of an object returned as an approximate k-nearest neighbor must not be larger than (1+ε)r, where r denotes the distance to the real kth nearest neighbor. Similar the distance of an approximate k-furthest neighbor must not be smaller than r/(1+ε). Obviously, for ε=0 we get the exact result, and the larger ε is, the less exact the result.
Neighbor searching is implemented by the following four classes.
The class CGAL::Orthogonal_k_neighbor_search<Traits, OrthogonalDistance, Splitter, SpatialTree> implements the standard search strategy for orthogonal distances like the weighted Minkowski distance. It requires the use of extended nodes in the spatial tree and supports only k neighbor searching for point queries.
The class CGAL::K_neighbor_search<Traits, GeneralDistance, Splitter, SpatialTree> implements the standard search strategy for general distances like the Manhattan distance for iso-rectangles. It does not require the use of extended nodes in the spatial tree and supports only k neighbor searching for queries defined by points or spatial objects.
The class Orthogonal_incremental_neighbor_search<Traits, GeneralDistance, Splitter, SpatialTree> implements the incremental search strategy for general distances like the weighted Minkowski distance. It requires the use of extended nodes in the spatial tree and supports incremental neighbor searching and distance browsing for point queries.
The class CGAL::Incremental_neighbor_search<Traits, GeneralDistance, Splitter, SpatialTree> implements the incremental search strategy for general distances like the Manhattan distance for iso-rectangles. It does not require the use of extended nodes in the spatial tree and supports incremental neighbor searching and distance browsing for queries defined by points or spatial objects.
Exact range searching and approximate range searching is supported using exact or fuzzy d-dimensional objects enclosing a region. The fuzziness of the query object is specified by a parameter ε denoting a maximal allowed distance to the boundary of a query object. If the distance to the boundary is at least ε, points inside the object are always reported and points outside the object are never reported. Points within distance ε to the boundary may be or may be not reported. For exact range searching the fuzziness parameter ε is set to zero.
The class Kd_tree implements range searching in the method search, which is a template method with an output iterator and a model of the concept FuzzyQueryItem as CGAL::Fuzzy_iso_box_d or CGAL::Fuzzy_sphere_d. For range searching of large data sets the user may set the parameter bucket_size used in building the k-d tree to a large value (e.g. 100), because in general the query time will be less then using the default value.
Instead of using the default splitting rule Sliding_midpoint described below, a user may, depending upon the data, select one from the following splitting rules, which determine how a separating hyperplane is computed:
This splitting rule cuts a rectangle through its midpoint orthogonal to the longest side.
This splitting rule cuts a rectangle through (Mind+Maxd)/2 orthogonal to the dimension with the maximum point spread [Mind,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.
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.
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.
We give six 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 d-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 last example shows how to choose another splitting rule in the k-d tree that is used as search tree.
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: examples/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>
typedef CGAL::Simple_cartesian<double> K;
typedef K::Point_2 Point_d;
typedef CGAL::Search_traits_2<K> TreeTraits;
typedef CGAL::Orthogonal_k_neighbor_search<TreeTraits> Neighbor_search;
typedef Neighbor_search::Tree Tree;
int main() {
const unsigned int 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);
// Initialize the search structure, and search all N points
Neighbor_search search(tree, query, N);
// report the N nearest neighbors and their distance
// This should sort all N points by increasing distance from origin
for(Neighbor_search::iterator it = search.begin(); it != search.end(); ++it){
std::cout << it->first << " "<< std::sqrt(it->second) << std::endl;
}
return 0;
}
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 k neighbor search, we first initialize the search tree with the data. We then create the search object, and finally obtain the iterator with the begin() method. Note that the iterator is of the input iterator category, that is one can make only one pass over the data.
File: examples/Spatial_searching/distance_browsing.cpp
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Orthogonal_incremental_neighbor_search.h>
#include <CGAL/Search_traits_2.h>
typedef CGAL::Simple_cartesian<double> K;
typedef K::Point_2 Point_d;
typedef CGAL::Search_traits_2<K> TreeTraits;
typedef CGAL::Orthogonal_incremental_neighbor_search<TreeTraits> NN_incremental_search;
typedef NN_incremental_search::iterator NN_iterator;
typedef NN_incremental_search::Tree Tree;
// A functor that returns true, iff the x-coordinate of a dD point is not positive
struct X_not_positive {
bool operator()(const NN_iterator& it) { return ((*it).first)[0]<0; }
};
// An iterator that only enumerates dD points with positive x-coordinate
typedef CGAL::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;
}
This example program illustrates approximate nearest and furthest neighbor searching using 4-dimensional Cartesian coordinates. Five approximate nearest 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: examples/Spatial_searching/general_neighbor_searching.cpp
#include <CGAL/Cartesian_d.h>
#include <CGAL/point_generators_2.h>
#include <CGAL/Manhattan_distance_iso_box_point.h>
#include <CGAL/K_neighbor_search.h>
#include <CGAL/Search_traits_2.h>
typedef CGAL::Cartesian_d<double> K;
typedef K::Point_d Point_d;
typedef CGAL::Random_points_in_square_2<Point_d> Random_points_iterator;
typedef K::Iso_box_d Iso_box_d;
typedef K TreeTraits;
typedef CGAL::Manhattan_distance_iso_box_point<TreeTraits> Distance;
typedef CGAL::K_neighbor_search<TreeTraits, Distance> Neighbor_search;
typedef Neighbor_search::Tree Tree;
int main() {
const int N = 1000;
const unsigned int K = 10;
Tree tree;
Random_points_iterator rpg;
for(int i = 0; i < N; i++){
tree.insert(*rpg++);
}
Point_d pp(0.1,0.1);
Point_d qq(0.2,0.2);
Iso_box_d query(pp,qq);
Distance tr_dist;
Neighbor_search N1(tree, query, K, 0.0, false); // eps=10.0, nearest=false
std::cout << "For query rectange = [0.1,0.2]^2 " << 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;
}
This example program illustrates approximate range querying for 4-dimensional fuzzy iso-rectangles and spheres using homogeneous coordinates. The range queries are member functions of the k-d tree class.
File: examples/Spatial_searching/fuzzy_range_query.cpp
#include <CGAL/Cartesian_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>
typedef CGAL::Cartesian_d<double> K;
typedef K::Point_d Point_d;
typedef CGAL::Search_traits_d<K> Traits;
typedef CGAL::Random_points_in_iso_box_d<Point_d> Random_points_iterator;
typedef CGAL::Counting_iterator<Random_points_iterator> N_Random_points_iterator;
typedef CGAL::Kd_tree<Traits> Tree;
typedef CGAL::Fuzzy_sphere<Traits> Fuzzy_sphere;
typedef CGAL::Fuzzy_iso_box<Traits> Fuzzy_iso_box;
int main() {
const int D = 4;
const int N = 1000;
// generator for random data points in the square ( (-1000,-1000), (1000,1000) )
Random_points_iterator rpit(4, 1000.0);
// Insert N points in the tree
Tree tree(N_Random_points_iterator(rpit,0),
N_Random_points_iterator(N));
// define range query objects
double pcoord[D] = { 300, 300, 300, 300 };
double qcoord[D] = { 900.0, 900.0, 900.0, 900.0 };
Point_d p(D, pcoord, pcoord+D);
Point_d q(D, qcoord, qcoord+D);
Fuzzy_sphere fs(p, 700.0, 100.0);
Fuzzy_iso_box fib(p, q, 100.0);
std::cout << "points approximately in fuzzy range query" << std::endl;
std::cout << "with center (300.0, 300.0, 300.0, 300.0)" << std::endl;
std::cout << "and fuzzy radius <200.0,400.0> are:" << std::endl;
tree.search(std::ostream_iterator<Point_d>(std::cout, "\n"), fs);
std::cout << "points approximately in fuzzy range query ";
std::cout << "[<200,4000>,<800,1000>]]^4 are:" << std::endl;
tree.search(std::ostream_iterator<Point_d>(std::cout, "\n"), fib);
return 0;
}
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: examples/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); }
}; //end of class
namespace CGAL {
template <>
struct Kernel_traits<Point> {
struct Kernel {
typedef double FT;
typedef double RT;
};
};
}
struct Construct_coord_iterator {
const double* operator()(const Point& p) const
{ return static_cast<const double*>(p.vec); }
const double* operator()(const Point& p, int) const
{ return static_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: examples/Spatial_searching/Distance.h
struct Distance {
typedef Point Query_item;
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;
}
template <class TreeTraits>
double min_distance_to_rectangle(const Point& p,
const CGAL::Kd_tree_rectangle<TreeTraits>& 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;
}
template <class TreeTraits>
double max_distance_to_rectangle(const Point& p,
const CGAL::Kd_tree_rectangle<TreeTraits>& 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 new_distance(double& dist, double old_off, double new_off,
int /* cutting_dimension */) 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) { return std::sqrt(d); }
}; // end of struct Distance
We are ready to put the pieces together. The class Search_traits<..> which you see in the next file is then a mere wrapper for all these types. The searching itself works exactly as for Cgal kernels.
File: examples/Spatial_searching/user_defined_point_and_distance.cpp
#include <CGAL/basic.h>
#include <CGAL/Search_traits.h>
#include <CGAL/point_generators_3.h>
#include <CGAL/Orthogonal_k_neighbor_search.h>
#include "Point.h" // defines types Point, Construct_coord_iterator
#include "Distance.h"
typedef CGAL::Random_points_in_cube_3<Point> Random_points_iterator;
typedef CGAL::Counting_iterator<Random_points_iterator> N_Random_points_iterator;
typedef CGAL::Search_traits<double, Point, const double*, Construct_coord_iterator> Traits;
typedef CGAL::Orthogonal_k_neighbor_search<Traits, Distance> K_neighbor_search;
typedef K_neighbor_search::Tree Tree;
int main() {
const int N = 1000;
const unsigned int K = 5;
// generator for random data points in the cube ( (-1,-1,-1), (1,1,1) )
Random_points_iterator rpit( 1.0);
// Insert number_of_data_points in the tree
Tree tree(N_Random_points_iterator(rpit,0),
N_Random_points_iterator(N));
Point query(0.0, 0.0, 0.0);
Distance tr_dist;
// search K nearest neighbours
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;
}
// search K furthest neighbour searching, with eps=0, search_nearest=false
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;
}
This example program illustrates selecting a splitting rule and setting the maximal allowed bucket size. The only differences with the first example are the declaration of the Fair splitting rule, needed to set the maximal allowed bucket size.
File: examples/Spatial_searching/using_fair_splitting_rule.cpp
#include <CGAL/Simple_cartesian.h>
#include <CGAL/point_generators_2.h>
#include <CGAL/Search_traits_2.h>
#include <CGAL/Orthogonal_k_neighbor_search.h>
#include <cmath>
typedef CGAL::Simple_cartesian<double> R;
typedef R::Point_2 Point_d;
typedef CGAL::Random_points_in_square_2<Point_d> Random_points_iterator;
typedef CGAL::Counting_iterator<Random_points_iterator> N_Random_points_iterator;
typedef CGAL::Search_traits_2<R> Traits;
typedef CGAL::Euclidean_distance<Traits> Distance;
typedef CGAL::Fair<Traits> Fair;
typedef CGAL::Orthogonal_k_neighbor_search<Traits,Distance,Fair> Neighbor_search;
typedef Neighbor_search::Tree Tree;
int main() {
const unsigned int N = 1000;
// generator for random data points in the square ( (-1,-1), (1,1) )
Random_points_iterator rpit( 1.0);
Fair fair(5); // bucket size=5
// Insert number_of_data_points in the tree
Tree tree(N_Random_points_iterator(rpit,0),
N_Random_points_iterator(N),
fair);
Point_d query(0,0);
// Initialize the search structure, and search all N points
Neighbor_search search(tree, query, N);
// report the N nearest neighbors and their distance
// This should sort all N points by increasing distance from origin
for(Neighbor_search::iterator it = search.begin(); it != search.end(); ++it){
std::cout << it->first << " "<< std::sqrt(it->second) << std::endl;
}
return 0;
}
Bentley [Ben75] introduced the k-d tree as a generalization of the binary search tree in higher dimensions. k-d trees hierarchically decompose space into a relatively small number of rectangles such that no rectangle contains too many input objects. For our purposes, a rectangle in real d dimensional space, ℝd, is the product of d closed intervals on the coordinate axes. k-d trees are obtained by partitioning point sets in ℝd using (d-1)-dimensional hyperplanes. Each node in the tree is split into two children by one such separating hyperplane. Several splitting rules (see Section 57.2 can be used to compute a separating (d-1)-dimensional hyperplane.
Each internal node of the k-d tree is associated with a rectangle and a hyperplane orthogonal to one of the coordinate axis, which splits the rectangle into two parts. Therefore, such a hyperplane, defined by a splitting dimension and a splitting value, is called a separator. These two parts are then associated with the two child nodes in the tree. The process of partitioning space continues until the number of data points in the rectangle falls below some given threshold. The rectangles associated with the leaf nodes are called buckets, and they define a subdivision of the space into rectangles. Data points are only stored in the leaf nodes of the tree, not in the internal nodes.
Friedmann, Bentley and Finkel [FBF77] described the standard search algorithm to find the kth nearest neighbor by searching a k-d tree recursively.
When encountering a node of the tree, the algorithm first visits the child that is closest to the query point. On return, if the rectangle containing the other child lies within 1/ (1+ε) times the distance to the kth nearest neighbors so far, then the other child is visited recursively. Priority search [AM93b] visits the nodes in increasing order of distance from the queue with help of a priority queue. The search stops when the distance of the query point to the nearest nodes exceeds the distance to the nearest point found with a factor 1/ (1+ε). Priority search supports next neighbor search, standard search does not.
In order to speed-up the internal distance computations in nearest neighbor searching in high dimensional space, the approximate searching package supports orthogonal distance computation. Orthogonal distance computation implements the efficient incremental distance computation technique introduced by Arya and Mount [AM93a]. This technique works only for neighbor queries with query items represented as points and with a quadratic form distance, defined by dA(x,y)= (x-y)A(x-y)T, where the matrix A is positive definite, i.e. dA(x,y) ≥ 0. An important class of quadratic form distances are weighted Minkowski distances. Given a parameter p>0 and parameters wi ≥ 0, 0 < i ≤ d, the weighted Minkowski distance is defined by lp(w)(r,q)= (Σi=1i=d wi(ri-qi)p)1/p for 0 < p <∞ and defined by l∞(w)(r,q)=max {wi |ri-qi| | 1 ≤ i ≤ d}. The Manhattan distance (p=1, wi=1) and the Euclidean distance (p=2, wi=1) are examples of a weighted Minkowski metric.
To speed up distance computations also transformed distances are used instead of the distance itself. For instance for the Euclidean distance, to avoid the expensive computation of square roots, squared distances are used instead of the Euclidean distance itself.