Median and Hybrid Median K-Dimensional Trees

Abstract

We consider here two new variants of K-dimensional binary search trees (K-d trees): median K-d trees and hybrid-median K-d trees. These two kinds of trees are designed with the aim to get a tree as balanced as possible. This goal is attained by heuristics that choose for each node of the K-d tree the appropriate coordinate to discriminate. In the case of median K-d trees, the chosen dimension to discriminate at each node is the one whose point value at that node is the most centered one. In hybrid-median K-d trees, the heuristic is similar except that it should be followed in a cyclic way, meaning that, at every path of the tree, no dimension can be re-selected to discriminate unless all the other dimensions have already been selected. We study the expected internal path length (IPL) and the expected cost of random partial match (PM) searches in both variants of K-d trees. For both variants, we prove that the expected IPL is of the form cK· nlog 2n+ lower order terms, and the expected cost of PM is of the form Θ(nα) with α= α(s, K). We give the explicit equations satisfied by the constants cK and the exponents α which we can then numerically solve. Moreover, we prove that as K→ ∞ the trees in both variants tend to be perfectly balanced (cK→ 1 ) and we also show that α→ log 2(2 - s/ K) for median K -d trees when K→ ∞. In the case of hybrid median K -d trees we conjecture that α→ 1 - s/ K, when K→ ∞, which would be optimal.