Updating relaxed K-d trees

  • Authors:

  • Amalia Duch;Conrado Martínez

  • Affiliations:

  • University Politècnica de Catalunya, Barcelona, Spain;University Politècnica de Catalunya, Barcelona, Spain

  • Venue:
  • ACM Transactions on Algorithms (TALG)
  • Year:
  • 2009

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Abstract

In this work we present an in-depth study of randomized relaxed K-d trees. It covers two fundamental aspects: the randomized algorithms that allow to preserve the random properties of relaxed K-d trees and the mathematical analysis of the expected performance of these algorithms. In particular, we describe randomized update algorithms for K-d trees based on the split and join algorithms of Duch et al. [1998]. We carry out an analysis of the expected cost of all these algorithms, using analytic combinatorics techniques. We show that the average cost of split and join is of the form ζ(K) ⋅ nφ(K) + o(nφ(K)), with 1 ≤ φ(K) K) and φ(K). These results on the average performance of split and join imply that the expected cost of an insertion or a deletion is Θ(nφ(K)−1) when K 2 and Θ(log n) for K = 2.