Analysis of range search for random k-d trees

We analyze the expected time complexity of range searching with k-d trees in all dimensions when the data points are uniformly distributed in the unit hypercube. The partial match results of Flajolet and Puech are reproved using elementary probabilistic methods. In addition, we give asymptotic expected time analysis for orthogonal and convex range search, as well as nearest neighbor search. We disprove a conjecture by Bentley that nearest neighbor search for a given random point in the k-d tree can be done in O(1) expected time. Received: 27 July 1999 / 2 June 2000