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20.3. KD Trees

20.3.1. KD Trees

The kd tree is a modification to the BST that allows for efficient processing of multi-dimensional search keys. The kd tree differs from the BST in that each level of the kd tree makes branching decisions based on a particular search key associated with that level, called the discriminator. In principle, the kd tree could be used to unify key searching across any arbitrary set of keys such as name and zipcode. But in practice, it is nearly always used to support search on multi-dimensional coordinates, such as locations in 2D or 3D space. We define the discriminator at level \(i\) to be \(i\) mod \(k\) for \(k\) dimensions. For example, assume that we store data organized by \(xy\)-coordinates. In this case, \(k\) is 2 (there are two coordinates), with the \(x\)-coordinate field arbitrarily designated key 0, and the \(y\)-coordinate field designated key 1. At each level, the discriminator alternates between \(x\) and \(y\). Thus, a node \(N\) at level 0 (the root) would have in its left subtree only nodes whose \(x\) values are less than \(N_x\) (because \(x\) is search key 0, and \(0 \mod 2 = 0\)). The right subtree would contain nodes whose \(x\) values are greater than \(N_x\). A node \(M\) at level 1 would have in its left subtree only nodes whose \(y\) values are less than \(M_y\). There is no restriction on the relative values of \(M_x\) and the \(x\) values of \(M\) 's descendants, because branching decisions made at \(M\) are based solely on the \(y\) coordinate. Figure 20.3.1 shows an example of how a collection of two-dimensional points would be stored in a kd tree.

Example of a kd tree

Figure 20.3.1: Example of a kd tree. (a) The kd tree decomposition for a \(128 \times 128\)-unit region containing seven data points. (b) The kd tree for the region of (a).

In Figure 20.3.1, the region containing the points is (arbitrarily) restricted to a \(128 \times 128\) square, and each internal node splits the search space. Each split is shown by a line, vertical for nodes with \(x\) discriminators and horizontal for nodes with \(y\) discriminators. The root node splits the space into two parts; its children further subdivide the space into smaller parts. The children's split lines do not cross the root's split line. Thus, each node in the kd tree helps to decompose the space into rectangles that show the extent of where nodes can fall in the various subtrees.

Searching a kd tree for the record with a specified xy-coordinate is like searching a BST, except that each level of the kd tree is associated with a particular discriminator.

Example 20.3.1

Consider searching the kd tree for a record located at \(P = (69, 50)\). First compare \(P\) with the point stored at the root (record \(A\) in Figure 20.3.1). If \(P\) matches the location of :math:A`, then the search is successful. In this example the positions do not match (\(A\) 's location (40, 45) is not the same as (69, 50)), so the search must continue. The \(x\) value of \(A\) is compared with that of \(P\) to determine in which direction to branch. Because \(A_x\)'s value of 40 is less than \(P\)'s \(x\) value of 69, we branch to the right subtree (all cities with \(x\) value greater than or equal to 40 are in the right subtree). \(A_y\) does not affect the decision on which way to branch at this level. At the second level, \(P\) does not match record \(C\)'s position, so another branch must be taken. However, at this level we branch based on the relative \(y\) values of point \(P\) and record \(C\) (because \(1 \mod 2 = 1\), which corresponds to the \(y\)-coordinate). Because \(C_y\)'s value of 10 is less than \(P_y\)'s value of 50, we branch to the right. At this point, \(P\) is compared against the position of \(D\). A match is made and the search is successful.

If the search process reaches a NULL pointer, then that point is not contained in the tree. Here is a kd tree search implementation, equivalent to the findhelp function of the BST class. KD class private member D stores the key's dimension.

private E findhelp(KDNode<E> rt, int[] key, int level) {
  if (rt == null) return null;
  E it = rt.element();
  int[] itkey = rt.key();
  if ((itkey[0] == key[0]) && (itkey[1] == key[1]))
    return rt.element();
  if (itkey[level] > key[level])
    return findhelp(rt.left(), key, (level+1)%D);
  else
    return findhelp(rt.right(), key, (level+1)%D);
}

Inserting a new node into the kd tree is similar to BST insertion. The kd tree search procedure is followed until a NULL pointer is found, indicating the proper place to insert the new node.

Example 20.3.2

Inserting a record at location (10, 50) in the kd tree of Figure 20.3.1 first requires a search to the node containing record \(B\). At this point, the new record is inserted into \(B\)'s left subtree.

Deleting a node from a kd tree is similar to deleting from a BST, but slightly harder. As with deleting from a BST, the first step is to find the node (call it \(N\)) to be deleted. It is then necessary to find a descendant of \(N\) which can be used to replace \(N\) in the tree. If \(N\) has no children, then \(N\) is replaced with a NULL pointer. Note that if \(N\) has one child that in turn has children, we cannot simply assign \(N\)'s parent to point to \(N\)'s child as would be done in the BST. To do so would change the level of all nodes in the subtree, and thus the discriminator used for a search would also change. The result is that the subtree would no longer be a kd tree because a node's children might now violate the BST property for that discriminator.

Similar to BST deletion, the record stored in \(N\) should be replaced either by the record in \(N\)'s right subtree with the least value of <var>N</var>'s discriminator, or by the record in \(N\)'s left subtree with the greatest value for this discriminator. Assume that \(N\) was at an odd level and therefore \(y\) is the discriminator. \(N\) could then be replaced by the record in its right subtree with the least \(y\) value (call it \(Y_{min}\)). The problem is that <var>Y</var><sub>min</sub> is not necessarily the leftmost node, as it would be in the BST. A modified search procedure to find the least \(y\) value in the left subtree must be used to find it instead. The implementation for findmin is shown next. A recursive call to the delete routine will then remove :math`Y_{min}` from the tree. Finally, \(Y_{min}\)'s record is substituted for the record in node \(N\).

private KDNode<E>
findmin(KDNode<E> rt, int descrim, int level) {
  KDNode<E> temp1, temp2;
  int[] key1 = null;
  int[] key2 = null;
  if (rt == null) return null;
  temp1 = findmin(rt.left(), descrim, (level+1)%D);
  if (temp1 != null) key1 = temp1.key();
  if (descrim != level) {
    temp2 = findmin(rt.right(), descrim, (level+1)%D);
    if (temp2 != null) key2 = temp2.key();
    if ((temp1 == null) || ((temp2 != null) &&
                   (key1[descrim] > key2[descrim])))
    temp1 = temp2;
    key1 = key2;
  } // Now, temp1 has the smaller value
  int[] rtkey = rt.key();
  if ((temp1 == null) || (key1[descrim] > rtkey[descrim]))
    return rt;
  else
    return temp1;
}

In findmin, on levels using the minimum value's discriminator, branching is to the left. On other levels, both children's subtrees must be visited. Helper function min takes two nodes and a discriminator as input, and returns the node with the smaller value in that discriminator.

Note that we can replace the node to be deleted with the least-valued node from the right subtree only if the right subtree exists. If it does not, then a suitable replacement must be found in the left subtree. Unfortunately, it is not satisfactory to replace \(N\)'s record with the record having the greatest value for the discriminator in the left subtree, because this new value might be duplicated. If so, then we would have equal values for the discriminator in \(N\)'s left subtree, which violates the ordering rules for the kd tree. Fortunately, there is a simple solution to the problem. We first move the left subtree of node \(N\) to become the right subtree (i.e., we simply swap the values of \(N\)'s left and right child pointers). At this point, we proceed with the normal deletion process, replacing the record of <var>N</var> to be deleted with the record containing the least value of the discriminator from what is now \(N\)'s right subtree.

Assume that we want to print out a list of all records that are within a certain distance \(d\) of a given point \(P\). We will use Euclidean distance, that is, point \(P\) is defined to be within distance \(d\) of point \(N\) if \(\sqrt{(P_x - N_x)^2 + (P_y - N_y)^2} \leq d.\) [1]

If the search process reaches a node whose key value for the discriminator is more than \(d\) above the corresponding value in the search key, then it is not possible that any record in the right subtree can be within distance \(d\) of the search key because all key values in that dimension are always too great. Similarly, if the current node's key value in the discriminator is \(d\) less than that for the search key value, then no record in the left subtree can be within the radius. In~such cases, the subtree in question need not be searched, potentially saving much time. In the average case, the number of nodes that must be visited during a range query is linear on the number of data records that fall within the query circle.

Example 20.3.3

We will now find all cities in the kd tree of Figure 20.3.2 within 25 units of the point (25, 65). The search begins with the root node, which contains record \(A\). Because (40, 45) is exactly 25 units from the search point, it will be reported. The search procedure then determines which branches of the tree to take. The search circle extends to both the left and the right of \(A\)'s (vertical) dividing line, so both branches of the tree must be searched. The left subtree is processed first. Here, record \(B\) is checked and found to fall within the search circle. Because the node storing \(B\) has no children, processing of the left subtree is complete. Processing of \(A<\)'s right subtree now begins. The coordinates of record \(C\) are checked and found not to fall within the circle. Thus, it should not be reported. However, it is possible that cities within \(C\)'s subtrees could fall within the search circle even if \(C\) does not. As \(C\) is at level 1, the discriminator at this level is the \(y\)-coordinate. Because \(65-25 > 10\), no record in \(C\)'s left subtree (i.e., records above \(C\)) could possibly be in the search circle. Thus, \(C\)'s left subtree (if it had one) need not be searched. However, cities in \(C\)'s right subtree could fall within the circle. Thus, search proceeds to the node containing record \(D\). Again, \(D\) is outside the search circle. Because \(25+25 < 69\), no record in \(D<\)'s right subtree could be within the search circle. Thus, only \(D\)'s left subtree need be searched. This leads to comparing record \(E\)'s coordinates against the search circle. Record \(E\) falls outside the search circle, and processing is complete. So we see that we only search subtrees whose rectangles fall within the search circle.

Example of searching in a kd tree

Figure 20.3.2: Searching in the kd tree of Figure 20.3.1. (a) The kd tree decomposition for a \(128 \times 128\)-unit region containing seven data points. (b) The kd tree for the region of (a).

Here is an implementation for the region search method.

private void rshelp(KDNode<E> rt, int[] point,
                    int radius, int lev) {
  if (rt == null) return;
  int[] rtkey = rt.key();
  if (InCircle(point, radius, rtkey))
    System.out.println(rt.element());
  if (rtkey[lev] > (point[lev] - radius))
    rshelp(rt.left(), point, radius, (lev+1)%D);
  if (rtkey[lev] < (point[lev] + radius))
    rshelp(rt.right(), point, radius, (lev+1)%D);
}

When a node is visited, function InCircle is used to check the Euclidean distance between the node's record and the query point. It is not enough to simply check that the differences between the \(x\)- and \(y\)-coordinates are each less than the query distances because the the record could still be outside the search circle, as illustrated by Figure 20.3.3.

Euclidean distance checking

Figure 20.3.3: Function InCircle must check the Euclidean distance between a record and the query point. It is possible for a record \(A\) to have \(x\)- and \(y\)-coordinates each within the query distance of the query point \(C\), yet have \(A\) itself lie outside the query circle.

Here is a visualization of building a kd-tree.

Todo

type:AV

Add version where you can click to get the node inserted.

Todo

type:exercise

We need a proficiency exercise.

[1]A more efficient computation is \((P_x - N_x)^2 + (P_y - N_y)^{2} \leq d^{2}\). This avoids performing a square root function.

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