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CGAL 4.5 - dD Spatial Searching
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CGAL 4.5 - dD Spatial Searching
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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) k-nearest and k-furthest neighbor searching
(approximate) incremental nearest and incremental furthest neighbor searching
query items representing points and spatial objects.
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 [5] 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 \( \epsilon\) the distance of an object returned as an approximate k-nearest neighbor must not be larger than \( (1+\epsilon)r\), 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 \( r/(1+\epsilon)\). Obviously, for \( \epsilon=0\) we get the exact result, and the larger \( \epsilon\) is, the less exact the result.
Neighbor searching is implemented by the following four classes.
The class 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 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 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 \( \epsilon\) denoting a maximal allowed distance to the boundary of a query object. If the distance to the boundary is at least \( \epsilon\), points inside the object are always reported and points outside the object are never reported. Points within distance \( \epsilon\) to the boundary may be or may be not reported. For exact range searching the fuzziness parameter \( \epsilon\) 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 Fuzzy_iso_box or Fuzzy_sphere. 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:
Midpoint_of_rectangleThis splitting rule cuts a rectangle through its midpoint orthogonal to the longest side.
Midpoint_of_max_spreadThis splitting rule cuts a rectangle through \( (\mathrm{Mind}+\mathrm{Maxd})/2\) orthogonal to the dimension with the maximum point spread \( [\mathrm{Mind},\mathrm{Maxd}]\).
Sliding_midpointThis 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.
Median_of_rectangleThe 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.
Median_of_max_spreadThe 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.
FairThis 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.
Sliding_fairThis 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 Spatial_searching/nearest_neighbor_searching.cpp
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 Spatial_searching/distance_browsing.cpp
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 Spatial_searching/general_neighbor_searching.cpp
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 Spatial_searching/fuzzy_range_query.cpp
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
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
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 Spatial_searching/user_defined_point_and_distance.cpp
The following three example programs illustrate how to use the classes Search_traits_adapter<Key,PointPropertyMap,BaseTraits> and Distance_adapter<Key,PointPropertyMap,Base_distance> to store in the kd-tree objects of an arbitrary key type. Points are accessed through a point property map. This is really convenient to associate information to a point or to reduce the size of the search structure.
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
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 std::size_t.
File Spatial_searching/searching_with_point_with_info_inplace.cpp
This example programs uses a model of LvaluePropertyMap. Points are read from a std::map. The search tree stores integers of type std::size_t. The value type of the iterator of the neighbor searching algorithm is std::size_t.
File Spatial_searching/searching_with_point_with_info_pmap.cpp
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 Spatial_searching/using_fair_splitting_rule.cpp
Bentley [3] 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, \( \mathbb{R}^d\), is the product of d closed intervals on the coordinate axes. k-d trees are obtained by partitioning point sets in \( \mathbb{R}^d\) 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 Splitting Rules 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 [4] 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+ \( \epsilon\)) times the distance to the kth nearest neighbors so far, then the other child is visited recursively. Priority search [2] 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+ \( \epsilon\)). 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 [1]. This technique works only for neighbor queries with query items represented as points and with a quadratic form distance, defined by \( d_A(x,y)= (x-y)A(x-y)^T\), where the matrix \( A\) is positive definite, i.e. \( d_A(x,y) \geq 0\). An important class of quadratic form distances are weighted Minkowski distances. Given a parameter \( p>0\) and parameters \( w_i \geq 0, 0 < i \leq d\), the weighted Minkowski distance is defined by \( l_p(w)(r,q)= ({\Sigma_{i=1}^{i=d} \, w_i(r_i-q_i)^p})^{1/p}\) for \( 0 < p <\infty\) and defined by \( l_{\infty}(w)(r,q)=max \{w_i |r_i-q_i| \mid 1 \leq i \leq d\}\). The Manhattan distance ( \( p=1\), \( w_i=1\)) and the Euclidean distance ( \( p=2\), \( w_i=1\)) 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.
1.8.4