Balltree

In computer science, a ball tree, balltree or metric tree, is a space partitioning data structure for organizing points in a multi-dimensional space. The ball tree gets its name from the fact that it partitions data points into a nested set of hyperspheres known as "balls". The resulting data structure has characteristics that make it useful for a number of applications, most notably nearest neighbor search.

A ball tree is a binary tree in which every node defines a D-dimensional hypersphere, or ball, containing a subset of the points to be searched. Each internal node of the tree partitions the data points into two disjoint sets which are associated with different balls. While the balls themselves may intersect, each point is assigned to one or the other ball in the partition according to its distance from the ball's center. Each leaf node in the tree defines a ball and enumerates all data points inside that ball.

Each node in the tree defines the smallest ball that contains all data points in its subtree. This gives rise to the useful property that, for a given test point t, the distance to any point in a ball B in the tree is greater than or equal to the distance from t to the ball. Formally:

Where ${\displaystyle D^{B}(t)}$ is the minimum possible distance from any point in the ball B to some point t.

Ball-trees are related to the M-tree, but only support binary splits, whereas in the M-tree each level splits ${\displaystyle m}$ to ${\displaystyle 2m}$ fold, thus leading to a shallower tree structure. The M-tree also keeps the distances from the parent node precomputed to speed up queries.

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