Random hyperplane search trees in high dimensions

• Main Authors: Luc Devroye, James King
• Format: Article
• Language: English
• Published: Carleton University 2015-06-01
• Series: Journal of Computational Geometry
• Online Access: http://jocg.org/index.php/jocg/article/view/67

<p>Given a set <em>S</em> of <em>n</em> ≥ <em>d</em> points in general position in <em>R^d</em>, a random hyperplane split is obtained by sampling <em>d</em> points uniformly at random without replacement from <em>S</em> 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 <em>d</em>. A blessing of dimensionality arises—as <em>d</em> increases, random hyperplane splits more closely resemble perfectly balanced splits; in turn, random hyperplane search trees more closely resemble perfectly balanced binary search trees.</p><p>We prove that, for any fixed dimension <em>d</em>, a random hyperplane search tree storing<em> n</em> points has height at most (1 + <em>O</em>(1/sqrt(<em>d</em>))) log₂ <em>n</em> and average element depth at most (1 + <em>O</em>(1/<em>d</em>)) log₂ <em>n</em> with high probability as <em>n</em> → ∞. Further, we show that these bounds are asymptotically optimal with respect to <em>d</em>.</p>