How does the KD-tree nearest neighbor search work?

Question

I am looking at the Wikipedia page for KD trees. As an example, I implemented, in python, the algorithm for building a kd tree listed.

The algorithm for doing KNN sea

Answer 1

This book introduction, page 3:

Given a set of n points in a d-dimensional space, the kd-tree is constructed recursively as follows. First, one finds a median of the values of the ith coordinates of the points (initially, i = 1). That is, a value M is computed, so that at least 50% of the points have their ith coordinate greater-or-equal to M, while at least 50% of the points have their ith coordinate smaller than or equal to M. The value of x is stored, and the set P is partitioned into PL and PR , where PL contains only the points with their ith coordinate smaller than or equal to M, and |PR | = |PL |±1. The process is then repeated recursively on both PL and PR , with i replaced by i + 1 (or 1, if i = d). When the set of points at a node has size 1, the recursion stops.

The following paragraphs discuss its use in solving nearest neighbor.

Or, here is the original 1975 paper by Jon Bentley.

EDIT: I should add that SciPy has a kdtree implementation:

• scipy.spatial
• another Stack Overflow question