Random hyperplane search trees in high dimensions

TL;DR

This paper studies the structure of random hyperplane search trees in high-dimensional spaces, showing they become more balanced as dimension increases, with proven bounds on height and depth that are optimal with respect to dimension.

Contribution

It provides the first analysis of the structural properties of random hyperplane search trees in high dimensions, establishing bounds on height and depth that improve with increasing dimension.

Findings

Tree height is at most (1 + O(1/√d)) log n for fixed d.

Average element depth is at most (1 + O(1/d)) log n.

Bounds are asymptotically optimal with respect to dimension d.

Abstract

Given a set S of n \geq d points in general position in R^d, a random hyperplane split is obtained by sampling d points uniformly at random without replacement from S and splitting based on their affine hull. A random hyperplane search tree is a binary space partition tree obtained by recursive application of random hyperplane splits. We investigate the structural distributions of such random trees with a particular focus on the growth with d. A blessing of dimensionality arises--as d increases, random hyperplane splits more closely resemble perfectly balanced splits; in turn, random hyperplane search trees more closely resemble perfectly balanced binary search trees. We prove that, for any fixed dimension d, a random hyperplane search tree storing n points has height at most (1 + O(1/sqrt(d))) log_2 n and average element depth at most (1 + O(1/d)) log_2 n with high probability as n \rightarrow \infty. Further, we show that these bounds are asymptotically optimal with respect to d.