Fixed Partial Match Queries in Quadtrees

Authors Amalia Duch , Gustavo Lau , Conrado Martínez

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Author Details

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

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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)


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.

Subject Classification

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


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  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
  2. H.-H. Chern and H.-K. Hwang. Partial match queries in random quadtrees. SIAM J. Comput., 32:904-915, 2003. Google Scholar
  3. N. Curien and A. Joseph. Partial match queries in two-dimensional quadtrees: A probabilistic approach. Advances in Applied Probability, 43:178-194, 2011. Google Scholar
  4. A. Duch, R. M. Jiménez, and C. Martínez. Selection by rank in k-dimensional binary search trees. Random Structures and Algorithms, 2012. URL:
  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:
  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:
  7. Philippe Flajolet, Gaston Gonnet, Claude Puech, and John Michael Robson. Analytic variations on quad trees. Algorithmica, 10:473-500, 1993. Google Scholar
  8. Gerald B. Folland. Introduction to Partial Differential Equations. Princeton University Press, 2nd edition, 1995. Google Scholar
  9. Steven G. Krantz and Harold R. Parks. A Primer of Real Analytic Functions. Birkhäuser, 2nd edition, 2002. Google Scholar
  10. S. Ross. A First Course in Probability. Prentice Hall, Upper Saddle River, New Jersey, 8th edition, 2010. Google Scholar
  11. Daniel Zwillinger. Handbook of Differential Equations. Academic Press, 3rd edition, 1997. Google Scholar