kd-tree

 

kd树是做多维数据索引时候用到的一种数据结构：

k-d树是二叉检索树的扩展，k-d树的每一层将空间分成两个。树的顶层结点按一维进行划分，下
一层结点按另一维进行划分，以此类推，各个维循环往复。划分要使得在每个结点，大约一半存
储在子树中的点落入一侧，而另一半落入另一侧。当一个结点中的点数少于给定的最大点数时，划分结束。

在每一层利用某一维的信息进行判断，比如来说，当前的数据是10维的，在第一层的时候用第一层的信息进行二叉排序，第一维数据比较，小于的在左子树，大于的在右子树，在第二层的时候又比较第二维的数据比较了。

那这个是怎么检索最紧邻的数据呢？当然是在lg（n）的复杂度了

 

 对于拥有n个已知点的kD-Tree，其复杂度如下：

1. 构建：O(log²n)
2. 插入：O(log n)
3. 删除：O(log n)
4. 查询：O(n^1-1/k+m) m---每次要搜索的最近点个数

From Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Kd-tree

A 3-dimensional  kd-tree. The first split (red) cuts the root cell (white) into two subcells, each of which is then split (green) into two subcells. Finally, each of those four is split (blue) into two subcells. Since there is no more splitting, the final eight are called leaf cells.

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 neighbor searches). kd-trees are a special case of BSP trees.(二叉搜索树）

 

Informal description

The kd-tree is a binary tree in which every node is a k-dimensional point. Every non-leaf node can be thought of as implicitly generating a splitting hyperplane that divides the space into two parts, known as subspaces. Points to the left of this hyperplane represent the left sub-tree of that node and points right of the hyperplane are represented by the right sub-tree. The hyperplane direction is chosen in the following way: every node in the tree is associated with one of the k-dimensions, with the hyperplane perpendicular to that dimension's axis. So, for example, if for a particular split the "x" axis is chosen, all points in the subtree with a smaller "x" value than the node will appear in the left subtree and all points with larger "x" value will be in the right sub tree. In such a case, the hyperplane would be set by the x-value of the point, and its normal would be the unit x-axis.