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Ball-tree

Ball-tree is a binary tree data structure used for organizing points in a multi-dimensional space, particularly useful in vector databases for nearest neighbor search. It partitions data points into hyperspheres (balls), enabling efficient search and scalability in high-dimensional vector spaces.

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Websiteen.wikipedia.org

PublishedMay 13, 2025

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Concepts & Definitions

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#data structure

#nearest neighbor

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Ball-tree

Category: Concepts & Definitions
Tags: data-structure, nearest-neighbor, vector-search, scalability

Description

A Ball-tree is a binary tree data structure used to organize points in multi-dimensional space. It partitions data points into a nested set of hyperspheres (balls), which makes it particularly effective for applications such as nearest neighbor search, especially in high-dimensional vector spaces.

Features

Space Partitioning: Organizes points by recursively splitting them into two disjoint sets, each associated with a D-dimensional ball (hypersphere).
Binary Tree Structure: Each internal node represents the smallest ball containing all data points in its subtree; leaves enumerate all points within their ball.
Efficient Queries: Supports fast nearest neighbor searches by pruning subtrees whose balls cannot contain closer points than already found candidates.
Construction Algorithms: Several algorithms exist for constructing ball trees; the simplest is the k-d construction algorithm, which splits data by the dimension of greatest spread and partitions at the median, resulting in O(n log n) build time.
Volume Minimization: Efficient trees aim to minimize the total volume of their internal balls, balancing construction effort and query efficiency.
High-dimensional Scalability: Performs well for nearest neighbor queries as the number of dimensions increases, compared to some alternative structures.
Comparison to Related Structures:
- M-tree: Supports multi-way splits, potentially shallower trees, and is more suited for disk storage.
- Vantage-point tree: Uses a different binary split strategy (one ball and the remainder).
Distance Properties: For a given test point outside a ball, the minimum distance to any point in the ball is at least the distance from the test point to the surface of the ball.
Customizable Construction: The ideal ball tree structure depends on the desired query type and data distribution; heuristics are often used for practical partitioning.

Applications

Nearest neighbor search in multi-dimensional and high-dimensional spaces
Vector databases and similarity search

References

Wikipedia: Ball-tree

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