KD-Tree

Overview

KD-Trees (k-dimensional trees) are tree-based data structures that partition vectors through recursive axis-aligned splitting. Each split reduces the search space, enabling logarithmic time complexity for balanced data.

How KD-Trees Work

KD-Trees partition data recursively along axis-aligned planes:

1. Select a dimension (axis)
2. Choose a splitting value (often the median)
3. Partition points into left and right subtrees
4. Recursively apply to subtrees

Time Complexity

For balanced data, KD-trees provide:

• Search: O(log n) average case
• Insert: O(log n) average case
• Worst case: O(n) for unbalanced trees

Limitations

Curse of Dimensionality

KD-trees struggle in high-dimensional spaces due to the "curse of dimensionality":

• Search performance degrades as dimensions increase
• Beyond 20-30 dimensions, performance approaches linear search
• Not suitable for typical embedding dimensions (384, 768, 1536, etc.)

Comparison with Ball Trees

Ball Trees organize vectors based on spherical regions instead of axis-aligned splits, making them better suited for high-dimensional data compared to KD-trees.

Use Cases

• Low-dimensional spatial data (2D, 3D)
• Geographic information systems
• Computer graphics
• Not recommended for high-dimensional embeddings

Availability

Implemented in:

• Scikit-learn
• SciPy
• Various spatial libraries

Pricing

Free - algorithmic concept with open-source implementations.