RANDOM HYPERPLANE SEARCH TREES

Abstract. A hyperplane search tree is a binary tree used to store a set S of n d-dimensional data points. In a random hyperplane search tree for S, the root represents a hyperplane defined by d data points drawn uniformly at random from S. The remaining data points are split by the hyperplane, and the definition is used recursively on each subset. We assume that the data are points in general position in IR d. We show that uniformly over all such data sets S, the expected height of the hyperplane tree is not worse than that of the k-d tree or the ordinary one-dimensional random binary search tree, and that for any fixed d ≥ 3, the expected height improves over that of the standard random binary search tree by an asymptotic factor strictly greater than one. Key words. Binary search tree, data structures, expected time analysis, height of a tree, hyperplane tree, random tree, large deviation theory, random sampling. AMS subject classifications. 68P05, 68W20, 68W40, 60E15 1. Introduction and Results. Hyperplane Search Trees. A hyperplane search tree is defined as follows. Given is a set S = {x1,..., xn} of points in general position 1 in IR d. The root node is formed by X1,..., Xd, obtained by uniform random sampling without replacement fro

Random hyperplane search trees in high dimensions

Given a set S of n ≥ d points in general position in Rd, 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))) log2 n and average element depth at most (1 + O(1/d)) log2 n with high probability as n → ∞. Further, we show that these bounds are asymptotically optimal with respect to d