# OpenSource KD tree implementation

This is a post about Open-source KD tree implementation.

# Introduction

This is a post dedicated to give a small comment about implementation of KD-tree way for space partitioning (also known as spatial indexing) available at: https://github.com/burlachenkok/lw_index_datastructs

This algorithm proposed by Jon Bentley when he undergraduate student from Stanford under the supervision of Donald Knuth. In this video Jerome Friedman at 2004 remembers his work with the work of Jon Bentley.

# Thoretical Speed

KD trees allow todo Insert/Search/Delete points in Euclidian space ($$R^d$$) typically in ~lg(N) iterations. At each iteration:

• You compare specified coordinate O(1)
• You evaluate L2 norm or another norm. So if take into account “d” and evaluate L2 norm just usually it will take O(d) single operations
• But both this two numbers in case of $$d < lg(N)$$ can be hidden into O(1)

Range Count / Range search. Find all “R” points from all “N” points that lie in a specific range in which is typical: ~R+lg(N), But the worst case is: ~R+$$N^{0.5}$$

# Applications

Classical applications are everywhere where we use geometric data:

1. Ray-tracing
2. 2d range search
3. Collision detection
4. Nearest neighbors search, etc
5. n-body simulation algorithm
6. Search in databases
7. Computer Graphics

But KD tree generalizes for a higher dimension and this implementation supports it. Based on my knowledge from Datamining lectures CS246 from prof. Jure Leskovec in high dimensional space is almost true that everything is very far from everything so if you choose of using for K Nearest Neighbors https://sites.google.com/site/burlachenkok/k-nearest-neighbors-and-things-around-it and you choose for distance function or dissimilarity metric is based on Eculidan norm in $$R^d$$ then this library will help you for your C++ implementation.

Written on January 14, 2021