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 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, 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.

Technically, the letter k refers to the number of dimensions. A 3-dimensional kd-tree would be called a 3d-tree. However, the phrase "3-dimensional kd-tree" is more commonly used. (It is also more descriptive, since a 3-dimensional tree could be any of a variety of different things, but the term kd-tree refers to a specific type of space-partitioning tree.) The letters k and d are both lowercase, even when the term comes at the beginning of a sentence, and the k is in italics. However, variant spellings are common, such as KD-tree and Kd-tree.

Operations on kd-trees

Constructing a kd-tree

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.

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.

Given a list of n points, the following algorithm will construct a balanced kd-tree containing those points.

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
        sort pointList using predicate: point1[axis] < point2[axis];
        choose 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;
    }
}

This algorithm creates the invariant that for any node, all the nodes in the left subtree are on one side of a splitting plane, and all the nodes in the right subtree are on the other side. The splitting plane of a node goes through the point associated with that node (referred to in the code as node.location).

Adding elements to a kd-tree

One adds a new point to a kd-tree in the same way as one adds an element to any other tree. First, traverse the tree, starting from the root and moving to either the left or the right child depending on whether the point to be inserted is on the "left" or "right" side of the splitting plane. Once you get to a leaf node, add the new point as either the left or right child of the leaf node, again depending on which side of the node's splitting plane contains the new point.

Removing elements from a kd-tree

Remove a point from an existing kd-tree, without breaking the invariant. (Undone)

Balancing a kd-tree

Balancing a kd-tree requires care. Because kd-trees are sorted in multiple dimensions, the tree rotation technique cannot be used to balance them — this may break the invariant.

Orthogonal range search in a kd-tree

Use a kd-tree to find all the points that lie within a given rectangle (or higher-dimensional region analogous to a rectangle). This operation is also called an orthogonal range search . (Undone)

Complexity

• Building a static kd-tree from n points takes O(n log n) time.
• Inserting a new point into a balanced kd-tree takes O(log n) time.
• Removing a point from a balanced kd-tree takes O(log n) time.

External links

• Java applet demonstrating 2-dimensional kd-trees