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## kd-tree

K-D tree implementation.

## Usage

(require-extension kd-tree)

## Documentation

This library implements a k-d tree, a data structure for organizing and searching points in k-dimensional space.

The k-d tree is a binary search tree in which every branching node contains a k-dimensional point, and every leaf node contains a set of points. Every branching node represents a splitting hyperplane that divides the space into two parts, known as half-spaces.

Points to the left of the splitting hyperplane are contained in the left subtree of the node and points right of the hyperplane are contained in the right subtree. The splitting hyperplane is chosen so as to be perpendicular to one of the axes in the k-dimensional space. The axis at each branching level is chosen in a round-robin fashion. For instance, in 3-D space, at level 0, the chosen axis is X, so points are divided according to their X-coordinates; at level 1, the chosen axis is Y, so the points are divided according to their Y-coordinates; at the next branch level the chosen axis is Z, and so on.

### Point

This library currently only supports points in 3D space.

*[procedure]*

`make-point3d:: DOUBLE * DOUBLE * DOUBLE -> POINT3D`

3D point constructor.

*[procedure]*

`point3d? :: POINT3D -> BOOL`

3D point predicate.

*[procedure]*

`point3d-x :: POINT3D -> DOUBLE`

*[procedure]*

`point3d-y :: POINT3D -> DOUBLE`

*[procedure]*

`point3d-z :: POINT3D -> DOUBLE`

Accessors for the x,y,z coordinates of a 3D point.

### KD-tree

A K-D tree (short for k-dimensional tree) is a space-partitioning data structure for organizing points in a k-dimensional space.

#### Constructors

*[procedure]*

`list->kd-tree:: POINT3D LIST -> KD-TREE`

Given a list of points, constructs and returns a K-D tree object.

#### Predicates

*[procedure]*

`kd-tree? :: KD-TREE -> BOOL`

Returns `#t` if the given object is a K-D tree, `#f` otherwise.

*[procedure]*

`kd-tree-empty? :: KD-TREE -> BOOL`

Returns `#t` if the given K-D tree object is empty, `#f` otherwise.

*[procedure]*

`kd-tree-is-valid? :: KD-TREE -> BOOL`

Checks whether the K-D tree property holds for the given tree. Specifically, it tests that all points in the left subtree lie to the left of the plane, p is on the plane, and all points in the right subtree lie to the right.

*[procedure]*

`kd-tree-all-subtrees-are-valid? :: KD-TREE -> BOOL`

Checks whether the K-D tree property holds for the given tree and all subtrees.

#### Accessors

*[procedure]*

`kd-tree->list :: KD-TREE -> POINT3D LIST`

Returns a list with the points contained in the tree.

*[procedure]*

`kd-tree->list* :: KD-TREE -> (INT . POINT3D) LIST`

Returns a list where every element has the form `(i . p)`, where i is the relative index of this point, and `p` is the point.

*[procedure]*

`kd-tree-subtrees :: KD-TREE -> KD-TREE LIST`

*[procedure]*

`kd-tree-point :: KD-TREE -> POINT3D`

#### Combinators

*[procedure]*

`kd-tree-map`

*[procedure]*

`kd-tree-for-each`

*[procedure]*

`kd-tree-for-each*`

*[procedure]*

`kd-tree-fold-right`

*[procedure]*

`kd-tree-fold-right*`

#### Query procedures

*[procedure]*

`kd-tree-nearest-neighbor`

*[procedure]*

`kd-tree-near-neighbors`

*[procedure]*

`kd-tree-near-neighbors*`

*[procedure]*

`kd-tree-k-nearest-neighbors`

*[procedure]*

`kd-tree-slice`

*[procedure]*

`kd-tree-slice*`

#### Modifiers

*[procedure]*

`kd-tree-remove`

## Examples

## About this egg

### Author

### Version history

- 4.0-4.1
- Added with-distance? flag to kd-tree-near-neighbors
- 3.2
- Bug fix in kd-tree-near-neighbors
- 2.0
- Improvements to internal representation
- 1.0
- Initial release

### License

Copyright 2012 Ivan Raikov and the Okinawa Institute of Science and Technology. This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. A full copy of the GPL license can be found at <http://www.gnu.org/licenses/>.