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# 15.2. Spatial Data Structures¶

## 15.2.1. Spatial Data Structures¶

Search trees such as BSTs, AVL trees, splay trees, 2-3 Trees, B-trees, and tries are designed for searching on a one-dimensional key. A typical example is an integer key, whose one-dimensional range can be visualized as a number line. These various tree structures can be viewed as dividing this one-dimensional number line into pieces.

Some databases require support for multiple keys. In other words, records can be searched for using any one of several key fields, such as name or ID number. Typically, each such key has its own one-dimensional index, and any given search query searches one of these independent indices as appropriate.

### 15.2.1.1. Multdimensional Keys¶

A multidimensional search key presents a rather different concept. Imagine that we have a database of city records, where each city has a name and an $xy$ coordinate. A BST or splay tree provides good performance for searches on city name, which is a one-dimensional key. Separate BSTs could be used to index the $x$ and $y$ coordinates. This would allow us to insert and delete cities, and locate them by name or by one coordinate. However, search on one of the two coordinates is not a natural way to view search in a two-dimensional space. Another option is to combine the $xy$ coordinates into a single key, say by concatenating the two coordinates, and index cities by the resulting key in a BST. That would allow search by coordinate, but would not allow for an efficient two-dimensional range query such as searching for all cities within a given distance of a specified point. The problem is that the BST only works well for one-dimensional keys, while a coordinate is a two-dimensional key where neither dimension is more important than the other.

Multidimensional range queries are the defining feature of a spatial application. Because a coordinate gives a position in space, it is called a spatial attribute. To implement spatial applications efficiently requires the use of a spatial data structure. Spatial data structures store data objects organized by position and are an important class of data structures used in geographic information systems, computer graphics, robotics, and many other fields.

A number of spatial data structures are used for storing point data in two or more dimensions. The kd tree is a natural extension of the BST to multiple dimensions. It is a binary tree whose splitting decisions alternate among the key dimensions. Like the BST, the kd tree uses object-space decomposition. The PR quadtree uses key-space decomposition and so is a form of trie. It is a binary tree only for one-dimensional keys (in which case it is a trie with a binary alphabet). For $d$ dimensions it has $2^d$ branches. Thus, in two dimensions, the PR quadtree has four branches (hence the name “quadtree”), splitting space into four equal-sized quadrants at each branch. Two other variations on these data structures are the bintree and the point quadtree. In two dimensions, these four structures cover all four combinations of object- versus key-space decomposition on the one hand, and multi-level binary versus $2^d$-way branching on the other.