Define a range tree and its use in geometric problems.

A range tree is a type of data structure used for efficient range queries and nearest neighbour searches in geometric problems.

A range tree is a balanced binary search tree that is built on a set of one-dimensional points. It is a type of space partitioning data structure, which means it is used to organise points in a multi-dimensional space. The primary use of range trees is in computational geometry, where they are used to solve problems such as range searching and nearest neighbour searching.

In a range tree, each node stores a point, as well as a pointer to a secondary tree that contains all the points. The primary tree is built on the x-coordinates of the points, and each secondary tree is built on the y-coordinates of the points in the corresponding x-range. This structure allows for efficient range queries: to find all points in a given range, you first traverse the primary tree to find the x-range, and then traverse the secondary trees to find the y-range.

The construction of a range tree involves sorting the points according to their x-coordinates and then recursively building a balanced binary search tree. Each node in the tree stores a point, as well as a pointer to a secondary tree that is built in the same way on the y-coordinates of the points. This process can be extended to higher dimensions by building additional trees on the z-coordinates, w-coordinates, and so on.

Range trees are particularly useful in geometric problems because they allow for efficient range queries and nearest neighbour searches. A range query involves finding all points that lie within a given range, and a nearest neighbour search involves finding the point that is closest to a given point. By organising the points in a multi-dimensional space, range trees make these operations much more efficient than they would be in a simple list or array.

In conclusion, a range tree is a powerful tool in computational geometry, providing an efficient way to handle range queries and nearest neighbour searches. Its structure allows for quick access to points in a multi-dimensional space, making it an essential tool for many geometric problems.