class TKDTree<int,double>: public TObject

                      kd-tree and its implementation in TKDTree

 Contents:
 1. What is kd-tree
 2. How to cosntruct kdtree - Pseudo code
 3. Using TKDTree
    a. Creating the kd-tree and setting the data
    b. Navigating the kd-tree
 4. TKDTree implementation - technical details
    a. The order of nodes in internal arrays
    b. Division algorithm
    c. The order of nodes in boundary related arrays



 1. What is kdtree ? ( http://en.wikipedia.org/wiki/Kd-tree )

 In computer science, a kd-tree (short for k-dimensional tree) is a space-partitioning data structure
 for organizing points in a k-dimensional space. kd-trees are a useful data structure for several
 applications, such as searches involving a multidimensional search key (e.g. range searches and
 nearest neighbour searches). kd-trees are a special case of BSP trees.

 A kd-tree uses only splitting planes that are perpendicular to one of the coordinate system axes.
 This differs from BSP trees, in which arbitrary splitting planes can be used.
 In addition, in the typical definition every node of a kd-tree, from the root to the leaves, stores a point.
 This differs from BSP trees, in which leaves are typically the only nodes that contain points
 (or other geometric primitives). As a consequence, each splitting plane must go through one of
 the points in the kd-tree. kd-trees are a variant that store data only in leaf nodes.

 2. Constructing a classical kd-tree ( Pseudo code)

 Since there are many possible ways to choose axis-aligned splitting planes, there are many different ways
 to construct kd-trees. The canonical method of kd-tree construction has the following constraints:

     * As one moves down the tree, one cycles through the axes used to select the splitting planes.
      (For example, the root would have an x-aligned plane, the root's children would both have y-aligned
       planes, the root's grandchildren would all have z-aligned planes, and so on.)
     * At each step, the point selected to create the splitting plane is the median of the points being
       put into the kd-tree, with respect to their coordinates in the axis being used. (Note the assumption
       that we feed the entire set of points into the algorithm up-front.)

 This method leads to a balanced kd-tree, in which each leaf node is about the same distance from the root.
 However, balanced trees are not necessarily optimal for all applications.
 The following pseudo-code illustrates this canonical construction procedure (NOTE, that the procedure used
 by the TKDTree class is a bit different, the following pseudo-code is given as a simple illustration of the
 concept):

 function kdtree (list of points pointList, int depth)
 {
     if pointList is empty
         return nil;
     else
     {
         // Select axis based on depth so that axis cycles through all valid values
         var int axis := depth mod k;

         // Sort point list and choose median as pivot element
         select median from pointList;

         // Create node and construct subtrees
         var tree_node node;
         node.location := median;
         node.leftChild := kdtree(points in pointList before median, depth+1);
         node.rightChild := kdtree(points in pointList after median, depth+1);
         return node;
     }
 }

 Our construction method is optimized to save memory, and differs a bit from the constraints above.
 In particular, the division axis is chosen as the one with the biggest spread, and the point to create the
 splitting plane is chosen so, that one of the two subtrees contains exactly 2^k terminal nodes and is a
 perfectly balanced binary tree, and, while at the same time, trying to keep the number of terminal nodes
 in the 2 subtrees as close as possible. The following section gives more details about our implementation.

 3. Using TKDTree

 3a. Creating the tree and setting the data
     The interface of the TKDTree, that allows to set input data, has been developped to simplify using it
     together with TTree::Draw() functions. That's why the data has to be provided column-wise. For example:
     {
     TTree *datatree = ...

     datatree->Draw("x:y:z", "selection", "goff");
     //now make a kd-tree on the drawn variables
     TKDTreeID *kdtree = new TKDTreeID(npoints, 3, 1);
     kdtree->SetData(0, datatree->GetV1());
     kdtree->SetData(1, datatree->GetV2());
     kdtree->SetData(2, datatree->GetV3());
     kdtree->Build();
     }
     NOTE, that this implementation of kd-tree doesn't support adding new points after the tree has been built
     Of course, it's not necessary to use TTree::Draw(). What is important, is to have data columnwise.
     An example with regular arrays:
     {
     Int_t npoints = 100000;
     Int_t ndim = 3;
     Int_t bsize = 1;
     Double_t xmin = -0.5;
     Double_t xmax = 0.5;
     Double_t *data0 = new Double_t[npoints];
     Double_t *data1 = new Double_t[npoints];
     Double_t *data2 = new Double_t[npoints];
     Double_t *y     = new Double_t[npoints];
     for (Int_t i=0; i<npoints; i++){
        data0[i]=gRandom->Uniform(xmin, xmax);
        data1[i]=gRandom->Uniform(xmin, xmax);
        data2[i]=gRandom->Uniform(xmin, xmax);
     }
     TKDTreeID *kdtree = new TKDTreeID(npoints, ndim, bsize);
     kdtree->SetData(0, data0);
     kdtree->SetData(1, data1);
     kdtree->SetData(2, data2);
     kdtree->Build();
     }

     By default, the kd-tree doesn't own the data and doesn't delete it with itself. If you want the
     data to be deleted together with the kd-tree, call TKDTree::SetOwner(kTRUE).

     Most functions of the kd-tree don't require the original data to be present after the tree
     has been built. Check the functions documentation for more details.

 3b. Navigating the kd-tree

     Nodes of the tree are indexed top to bottom, left to right. The root node has index 0. Functions
     TKDTree::GetLeft(Index inode), TKDTree::GetRight(Index inode) and TKDTree::GetParent(Index inode)
     allow to find the children and the parent of a given node.

     For a given node, one can find the indexes of the original points, contained in this node,
     by calling the GetNodePointsIndexes(Index inode) function. Additionally, for terminal nodes,
     there is a function GetPointsIndexes(Index inode) that returns a pointer to the relevant
     part of the index array. To find the number of point in the node
     (not only terminal), call TKDTree::GetNpointsNode(Index inode).

 4.  TKDtree implementation details - internal information, not needed to use the kd-tree.
     4a. Order of nodes in the node information arrays:

 TKDtree is optimized to minimize memory consumption.
 Nodes of the TKDTree do not store pointers to the left and right children or to the parent node,
 but instead there are several 1-d arrays of size fNNodes with information about the nodes.
 The order of the nodes information in the arrays is described below. It's important to understand
 it, if one's class needs to store some kind of additional information on the per node basis, for
 example, the fit function parameters.

 Drawback:   Insertion to the TKDtree is not supported.
 Advantage:  Random access is supported

 As noted above, the construction of the kd-tree involves choosing the axis and the point on
 that axis to divide the remaining points approximately in half. The exact algorithm for choosing
 the division point is described in the next section. The sequence of divisions is
 recorded in the following arrays:
 fAxix[fNNodes]  - Division axis (0,1,2,3 ...)
 fValue[fNNodes] - Division value

 Given the index of a node in those arrays, it's easy to find the indices, corresponding to
 children nodes or the parent node:
 Suppose, the parent node is stored under the index inode. Then:
 Left child index  = inode*2+1
 Right child index =  (inode+1)*2
 Suppose, that the child node is stored under the index inode. Then:
 Parent index = inode/2

 Number of division nodes and number of terminals :
 fNNodes = (fNPoints/fBucketSize)

 The nodes are filled always from left side to the right side:
 Let inode be the index of a node, and irow - the index of a row
 The TKDTree looks the following way:
 Ideal case:
 Number of _terminal_ nodes = 2^N,  N=3

            INode
 irow 0     0                                                                   -  1 inode
 irow 1     1                              2                                    -  2 inodes
 irow 2     3              4               5               6                    -  4 inodes
 irow 3     7       8      9      10       11     12       13      14           -  8 inodes


 Non ideal case:
 Number of _terminal_ nodes = 2^N+k,  N=3  k=1

           INode
 irow 0     0                                                                   - 1 inode
 irow 1     1                              2                                    - 2 inodes
 irow 2     3              4               5               6                    - 3 inodes
 irow 3     7       8      9      10       11     12       13      14           - 8 inodes
 irow 4     15  16                                                              - 2 inodes


 3b. The division algorithm:

 As described above, the kd-tree is built by repeatingly dividing the given set of points into
 2 smaller sets. The cut is made on the axis with the biggest spread, and the value on the axis,
 on which the cut is performed, is chosen based on the following formula:
 Suppose, we want to divide n nodes into 2 groups, left and right. Then the left and right
 will have the following number of nodes:

 n=2^k+rest

 Left  = 2^k-1 +  ((rest>2^k-2) ?  2^k-2      : rest)
 Right = 2^k-1 +  ((rest>2^k-2) ?  rest-2^k-2 : 0)

 For example, let n_nodes=67. Then, the closest 2^k=64, 2^k-1=32, 2^k-2=16.
 Left node gets 32+3=35 sub-nodes, and the right node gets 32 sub-nodes

 The division process continues until all the nodes contain not more than a predefined number
 of points.

 3c. The order of nodes in boundary-related arrays

 Some kd-tree based algorithms need to know the boundaries of each node. This information can
 be computed by calling the TKDTree::MakeBoundaries() function. It fills the following arrays:

 fRange : array containing the boundaries of the domain:
 | 1st dimension (min + max) | 2nd dimension (min + max) | ...
 fBoundaries : nodes boundaries
 | 1st node {1st dim * 2 elements | 2nd dim * 2 elements | ...} | 2nd node {...} | ...
 The nodes are arranged in the order described in section 3a.


 Note: the storage of the TKDTree in a file which include also the contained data is not
       supported. One must store the data separatly in a file (e.g. using a TTree) and then
       re-creating the TKDTree from the data, after having read them from the file