RANDOM HYPERPLANE SEARCH TREES IN HIGH DIMENSIONS

Abstract

Given a set S of n ≥ 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 → ∞. Further, we show that these bounds are asymptotically optimal with respect to d .