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Fixed Partial Match Queries in Quadtrees

Authors Amalia Duch , Gustavo Lau , Conrado Martínez

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Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Quadtree
  • Partial match queries
  • Associative queries
  • Multidimensional search
  • Analysis of algorithms

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Abstract

Several recent papers in the literature have addressed the analysis of the cost P_{n,q} of partial match search for a given fixed query q - that has s out of K specified coordinates - in different multidimensional data structures. Indeed, detailed asymptotic estimates for the main term in the expected cost P_{n,q} = E {P_{n,q}} in standard and relaxed K-d trees are known (for any dimension K and any number s of specified coordinates), as well as stronger distributional results on P_{n,q} for standard 2-d trees and 2-dimensional quadtrees. In this work we derive a precise asymptotic estimate for the main order term of P_{n,q} in quadtrees, for any values of K and s, 0 < s < K, under the assumption that the limit of P_{n,q}/n^alpha when n -> infty exists, where alpha is the exponent of n in the expected cost of a random partial match query with s specified coordinates in a random K-dimensional quadtree.

Cite As Get BibTex

Amalia Duch, Gustavo Lau, and Conrado Martínez. Fixed Partial Match Queries in Quadtrees. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.AofA.2018.20

Author Details

Amalia Duch
  • Universitat Politècnica de Catalunya
Gustavo Lau
  • Universitat Politècnica de Catalunya
Conrado Martínez
  • Universitat Politècnica de Catalunya

Funding

This work has been partially supported by funds from the Spanish Ministery of Economy, Industry and Competitiviness (MINECO) and the European Union (FEDER) under grant GRAMM (TIN2017-86727-C2-1-R), and by funds from the Catalan Government (AGAUR) under grant 2017SGR 786.

References

  1. Nicolas Broutin, Ralph Neininger, and Henning Sulzbach. Partial match queries in random quadtrees. In Yuval Rabani, editor, Proc. of the 23^rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1056-1065, 2012. Google Scholar
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  5. Amalia Duch and Gustavo Lau. Partial match queries in relaxed K-dt trees. In Proc. of the Fourteenth ACM-SIAM Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pages 131-138, 2017. URL: http://dx.doi.org/10.1137/1.9781611974775.13.
  6. Amalia Duch, Gustavo Lau, and Conrado Martínez. On the cost of fixed partial match queries in k-d trees. Algorithmica, 75(4):684-723, 2016. URL: http://dx.doi.org/10.1007/s00453-015-0097-4.
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