Partial Match Queries in Relaxed K-dt trees

Abstract

The study of partial match queries on random hierarchical multidimensional data structures dates back to Ph. Flajolet and C. Puech's 1986 seminal paper on partial match retrieval. It was not until recently that fixed (as opposed to random) partial match queries were studied for random relaxed K-d trees, random standard K-d trees, and random 2-dimensional quad trees. Based on those results it seemed natural to classify the general form of the cost of fixed partial match queries into two families: that of either random hierarchical structures or perfectly balanced structures, as conjectured by Duch, Lau and Martínez (On the Cost of Fixed Partial Queries in K-d trees Algorithmica, 75(4): 2016). Here we show that the conjecture just mentioned does not hold by introducing relaxed K-dt trees and providing the average-case analysis for random partial match queries as well as some advances on the average-case analysis for fixed partial match queries on them. In fact this cost -for fixed partial match queries-does not follow the conjectured forms.

Analysis of partial match queries in multidimensional search trees

“…Regarding random PM queries, we report here some further, unpublished, advances that we have achieved after [DL17] for both relaxed and standard K-dt trees. We denote by x k the rising factorial power x(x+1) (x+k −1) [GKP94].…”

“…In Chapter 6, in the article [DL17], we introduce a new K-d tree variant: relaxed K-dt trees, where every subtree of size ≤ 2t+1 is locally rebalanced and the discriminant coordinate at each node is randomly chosen. Standard K-dt trees were defined by Cunto, Lau and Flajolet in the late 1980s [CLF89].…”

“…In [DLM14] we use t as a subindex while in [DLM16a] we use t for s s0, the number of specified coordinates that are not extreme. In later articles we avoided those uses of t because in [DL17] we use t for the local rebalance parameter.…”

“…In the article in this chapter [DL17] we introduce a new K-d tree variant, relaxed K-dt trees, adding local rebalances to relaxed K-d trees, and we analyze both random PM queries and fixed PM queries in them. On Theorem 4.1 of this article (page 128 of this document) we give the expected cost of random PM queries in relaxed K-dt trees.…”

“…The other main result of [DL17] is to prove that the expected cost of fixed PM queries in random relaxed K-d trees is not of the form conjectured in [DLM16a]:…”