Quadtree

A quadtree is a tree data structure in which each internal node has up to four children. Quadtrees are most often used to partition a two dimensional space by recursively subdividing it into four quadrants or regions. The regions may be square or rectangular, or may have arbitrary shapes. This data structure was named a quadtree by Raphael Finkel and J.L. Bentley in 1974. A similar partitioning is also known as a Q-tree. All forms of Quadtrees share some common features:


 * They decompose space into adaptable cells
 * Each cell (or bucket) has a maximum capacity. When maximum capacity is reached, the bucket splits
 * The tree directory follows the spatial decomposition of the Quadtree

Types
Quadtrees may be classified according to the type of data they represent, including areas, points, lines and curves. Quadtrees may also be classified by whether the shape of the tree is independent of the order data is processed. Some common types of quadtrees are:

The region quadtree
The region quadtree represents a partition of space in two dimensions by decomposing the region into four equal quadrants, subquadrants, and so on with each leaf node containing data corresponding to a specific subregion. Each node in the tree either has exactly four children, or has no children (a leaf node). The region quadtree is not strictly a 'tree' - as the positions of subdivisions are independent of the data. They are more precisely called 'tries'.

A region quadtree with a depth of n may be used to represent an image consisting of 2n × 2n pixels, where each pixel value is 0 or 1. The root node represents the entire image region. If the pixels in any region are not entirely 0s or 1s, it is subdivided. In this application, each leaf node represents a block of pixels that are all 0s or all 1s.

A region quadtree may also be used as a variable resolution representation of a data field. For example, the temperatures in an area may be stored as a quadtree, with each leaf node storing the average temperature over the subregion it represents.

If a region quadtree is used to represent a set of point data (such as the latitude and longitude of a set of cities), regions are subdivided until each leaf contains at most a single point.

Point quadtree
The point quadtree is an adaptation of a binary tree used to represent two dimensional point data. It shares the features of all quadtrees but is a true tree as the center of a subdivision is always on a point. The tree shape depends on the order data is processed. It is often very efficient in comparing two dimensional ordered data points, usually operating in O(log n) time.

Node structure for a point quadtree
A node of a point quadtree is similar to a node of a binary tree, with the major difference being that it has four pointers (one for each quadrant) instead of two ("left" and "right") as in an ordinary binary tree. Also a key is usually decomposed into two parts, referring to x and y coordinates. Therefore a node contains following information:


 * 4 Pointers: quad[‘NW’], quad[‘NE’], quad[‘SW’], and quad[‘SE’]
 * point; which in turn contains:
 * key; usually expressed as x, y coordinates
 * value; for example a name

Edge quadtree
Edge quadtree are specifically used to store lines rather than points. Curves are approximated by subdividing cells to a very fine resolution. This can result in extremely unbalanced trees which may defeat the purpose of indexing.

Some common uses of quadtrees

 * Image representation
 * Spatial indexing
 * Efficient collision detection in two dimensions
 * View frustum culling of terrain data
 * Storing sparse data, such as a formatting information for a spreadsheet or for some matrix calculations
 * Solution of multidimensional fields (computational fluid dynamics, electromagnetism)
 * Conway's Game of Life simulation program.[1]

Quadtrees are the two-dimensional analog of octrees.

Caveats
If geometric subdividing fails to reduce the item count for each quadrant, (e.g., for overlapping data,) QuadTree subpartitioning fails, and the capacity must be breached for the algorithm to continue. For example, if the maximum capacity for a quadrant is 8, and there are 9 points at (0, 0), subpartitioning would produce three empty quadrants, and one containing the original 9 points, and so on. Because the tree must allow more than 8 points in such a quadrant, QuadTrees can approach O(N) complexity for data sets with arbitrary geometry (e.g., maps or graphs).

General references

 * 1) Raphael Finkel and J.L. Bentley (1974). "Quad Trees: A Data Structure for Retrieval on Composite Keys". Acta Informatica 4 (1): 1–9. doi:10.1007/BF00288933.
 * 2) Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf (2000). Computational Geometry (2nd revised ed.). Springer-Verlag. ISBN 3-540-65620-0. Chapter 14: Quadtrees: pp.291–306.