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Algorithmica

, Volume 75, Issue 4, pp 684–723 | Cite as

On the Cost of Fixed Partial Match Queries in K-d Trees

  • Amalia Duch
  • Gustavo Lau
  • Conrado Martínez
Article
  • 148 Downloads

Abstract

Partial match queries constitute the most basic type of associative queries in multidimensional data structures such as \(K\)-d trees or quadtrees. Given a query \(\mathbf {q}=(q_0,\ldots ,q_{K-1})\) where s of the coordinates are specified and \(K-s\) are left unspecified (\(q_i=*\)), a partial match search returns the subset of data points \(\mathbf {x}=(x_0,\ldots ,x_{K-1})\) in the data structure that match the given query, that is, the data points such that \(x_i=q_i\) whenever \(q_i\not =*\). There exists a wealth of results about the cost of partial match searches in many different multidimensional data structures, but most of these results deal with random queries. Only recently a few papers have begun to investigate the cost of partial match queries with a fixed query \(\mathbf {q}\). This paper represents a new contribution in this direction, giving a detailed asymptotic estimate of the expected cost \(P_{{n},\mathbf {q}}\) for a given fixed query \(\mathbf {q}\). From previous results on the cost of partial matches with a fixed query and the ones presented here, a deeper understanding is emerging, uncovering the following functional shape for \(P_{{n},\mathbf {q}}\)
$$\begin{aligned} P_{{n},\mathbf {q}} = \nu \cdot \left( \prod _{i:q_i\text { is specified}}\, q_i(1-q_i)\right) ^{\alpha /2}\cdot n^\alpha + \text {l.o.t.} \end{aligned}$$
(l.o.t. lower order terms, throughout this work) in many multidimensional data structures, which differ only in the exponent \(\alpha \) and the constant \(\nu \), both dependent on s and K, and, for some data structures, on the whole pattern of specified and unspecified coordinates in \(\mathbf {q}\) as well. Although it is tempting to conjecture that this functional shape is “universal”, we have shown experimentally that it seems not to be true for a variant of \(K\)-d trees called squarish \(K\)-d trees.

Keywords

Multidimensional search Partial match search K-dimensional search trees Analysis of algorithms Multidimensional data structures 

Notes

Acknowledgments

We are very thankful to the two anonymous reviewers of this manuscript for their detailed reports and useful suggestions.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversitat Politècnica de CatalunyaBarcelonaSpain

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