On the Average Cost of Insertions on Random Relaxed K-d Trees

Abstract

In this work we refine the average case analysis of randomized insertions and deletions in random relaxed K-d trees, first given by Broutin et al. in [3]. The analysis is based in the analysis of the split and join algorithms, which recursively call one another and are the basis of the randomized update operations under consideration. For K = 2 the average cost of insertions and deletions is Θ(log n). For K > 2, this average cost is Θ(np(K)-1), for some p(K) > 1. This immediately follows from the analysis of the expected cost sn of splitting a tree of size n, which is the same as the expected cost mn of joining a pair of trees with total size n. These costs are, for K = 2, sn = mn = Θ(n) and, for K > 2, sn = mn = Ω(np(K)). In this abstract we find a closed form for the value of the exponent p(K), as well as the constant factor multiplying the main order term in sn.