The Annals of Applied Probability

A limit process for partial match queries in random quadtrees and $2$-d trees

Nicolas Broutin, Ralph Neininger, and Henning Sulzbach

Full-text: Open access


We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quadtrees and $k$-d trees). We assume the traditional model where the data consist of independent and uniform points in the unit square. For this model, in a structure on $n$ points, it is known that the number of nodes $C_{n}(\xi)$ to visit in order to report the items matching a random query $\xi$, independent and uniformly distributed on $[0,1]$, satisfies $\mathbf{E} [{C_{n}(\xi)}]\sim\kappa n^{\beta}$, where $\kappa$ and $\beta$ are explicit constants. We develop an approach based on the analysis of the cost $C_{n}(s)$ of any fixed query $s\in[0,1]$, and give precise estimates for the variance and limit distribution of the cost $C_{n}(x)$. Our results permit us to describe a limit process for the costs $C_{n}(x)$ as $x$ varies in $[0,1]$; one of the consequences is that $\mathbf{E} [{\max_{x\in[0,1]}C_{n}(x)}]\sim\gamma n^{\beta}$; this settles a question of Devroye [Pers. Comm., 2000].

Article information

Ann. Appl. Probab. Volume 23, Number 6 (2013), 2560-2603.

First available in Project Euclid: 22 October 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60F17: Functional limit theorems; invariance principles 05C05: Trees
Secondary: 11Y16: Algorithms; complexity [See also 68Q25] 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A16: Asymptotic enumeration

Analysis of algorithms quadtree limit distribution contraction method


Broutin, Nicolas; Neininger, Ralph; Sulzbach, Henning. A limit process for partial match queries in random quadtrees and $2$-d trees. Ann. Appl. Probab. 23 (2013), no. 6, 2560--2603. doi:10.1214/12-AAP912.

Export citation


  • [1] Bentley, J. L. (1975). Multidimensional binary search trees used for associative searching. Communication of the ACM 18 509–517.
  • [2] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [3] Broutin, N., Neininger, R. and Sulzbach, H. (2013). Partial match queries in random quadtrees. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (Y. Rabani, ed.) 1056–1065. SIAM, Philadelphia, PA.
  • [4] Chern, H.-H. and Hwang, H.-K. (2003). Partial match queries in random quadtrees. SIAM J. Comput. 32 904–915 (electronic).
  • [5] Chern, H.-H. and Hwang, H.-K. (2006). Partial match queries in random $k$-d trees. SIAM J. Comput. 35 1440–1466 (electronic).
  • [6] Curien, N. and Joseph, A. (2011). Partial match queries in two-dimensional quadtrees: A probabilistic approach. Adv. in Appl. Probab. 43 178–194.
  • [7] Devroye, L. (1987). Branching processes in the analysis of the heights of trees. Acta Inform. 24 277–298.
  • [8] Devroye, L. and Laforest, L. (1990). An analysis of random $d$-dimensional quad trees. SIAM J. Comput. 19 821–832.
  • [9] Drmota, M., Janson, S. and Neininger, R. (2008). A functional limit theorem for the profile of search trees. Ann. Appl. Probab. 18 288–333.
  • [10] Duch, A., Estivill-Castro, V. and Martínez, C. (1998). Randomized $K$-dimensional binary search trees. In Algorithms and Computation (Taejon, 1998) (K.-Y. Chwa and O. Ibarra, eds.). Lecture Notes in Computer Science 1533 199–208. Springer, Berlin.
  • [11] Duch, A., Jiménez, R. and Martínez, C. (2010). Rank selection in multidimensional data. In Proceedings of LATIN (A. López-Ortiz, ed.). Lecture Notes in Computer Science 6034 674–685. Springer, Berlin.
  • [12] Duch, A. and Martínez, C. (2002). On the average performance of orthogonal range search in multidimensional data structures. J. Algorithms 44 226–245.
  • [13] Eickmeyer, K. and Rüschendorf, L. (2007). A limit theorem for recursively defined processes in $L^{p}$. Statist. Decisions 25 217–235.
  • [14] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II. 3rd ed. Wiley, New York.
  • [15] Finkel, R. A. and Bentley, J. L. (1974). Quad trees, a data structure for retrieval on composite keys. Acta Inform. 4 1–19.
  • [16] Flajolet, P., Gonnet, G., Puech, C. and Robson, J. M. (1993). Analytic variations on quadtrees. Algorithmica 10 473–500.
  • [17] Flajolet, P., Labelle, G., Laforest, L. and Salvy, B. (1995). Hypergeometrics and the cost structure of quadtrees. Random Structures Algorithms 7 117–144.
  • [18] Flajolet, P. and Lafforgue, T. (1994). Search costs in quadtrees and singularity perturbation asymptotics. Discrete Comput. Geom. 12 151–175.
  • [19] Flajolet, P. and Puech, C. (1986). Partial match retrieval of multidimensional data. J. Assoc. Comput. Mach. 33 371–407.
  • [20] Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge Univ. Press, Cambridge.
  • [21] Grübel, R. (2009). On the silhouette of binary search trees. Ann. Appl. Probab. 19 1781–1802.
  • [22] Ho-Le, K. (1988). Finite element mesh generation methods: A review and classification. Computer-Aided Design 20 27–38.
  • [23] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • [24] Knuth, D. E. (1975). The Art of Computer Programming, 2nd ed. Addison-Wesley, Reading, MA.
  • [25] Mahmoud, H. M. (1992). Evolution of Random Search Trees. Wiley, New York.
  • [26] Martínez, C., Panholzer, A. and Prodinger, H. (2001). Partial match queries in relaxed multidimensional search trees. Algorithmica 29 181–204.
  • [27] Neininger, R. (2000). Asymptotic distributions for partial match queries in $K$-$\mathrm{d}$ trees. In Proceedings of the Ninth International Conference “Random Structures and Algorithms” (Poznan, 1999) 17 403–427.
  • [28] Neininger, R. (2001). On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Structures Algorithms 19 498–524.
  • [29] Neininger, R. and Rüschendorf, L. (2001). Limit laws for partial match queries in quadtrees. Ann. Appl. Probab. 11 452–469.
  • [30] Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 378–418.
  • [31] Neininger, R. and Rüschendorf, L. (2004). On the contraction method with degenerate limit equation. Ann. Probab. 32 2838–2856.
  • [32] Neininger, R. and Sulzbach, H. (2012). On a functional contraction method. Preprint. Available at arXiv:1202.1370.
  • [33] Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. in Appl. Probab. 27 770–799.
  • [34] Rivest, R. L. (1976). Partial-match retrieval algorithms. SIAM J. Comput. 5 19–50.
  • [35] Rösler, U. (1991). A limit theorem for “Quicksort”. RAIRO Inform. Théor. Appl. 25 85–100.
  • [36] Rösler, U. (1992). A fixed point theorem for distributions. Stochastic Process. Appl. 42 195–214.
  • [37] Rösler, U. (2001). On the analysis of stochastic divide and conquer algorithms. Algorithmica 29 238–261.
  • [38] Samet, H. (1990). The Design and Analysis of Spatial Data Structures. Addison-Wesley, Reading, MA.
  • [39] Samet, H. (1990). Applications of Spatial Data Structures: Computer Graphics, Image Processing, and GIS. Addison-Wesley, Reading, MA.
  • [40] Samet, H. (2006). Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann, San Francisco, CA.
  • [41] Yerry, M. and Shephard, M. (1983). A modified quadtree approach to finite element mesh generation. IEEE Computer Graphics and Applications 3 39–46.