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].
Ann. Appl. Probab.
23(6):
2560-2603
(December 2013).
DOI: 10.1214/12-AAP912
[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. MR3205273 10.1137/1.9781611973099.83[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. MR3205273 10.1137/1.9781611973099.83
[4] Chern, H.-H. and Hwang, H.-K. (2003). Partial match queries in random quadtrees. SIAM J. Comput. 32 904–915 (electronic). MR2001889 10.1137/S0097539702412131[4] Chern, H.-H. and Hwang, H.-K. (2003). Partial match queries in random quadtrees. SIAM J. Comput. 32 904–915 (electronic). MR2001889 10.1137/S0097539702412131
[5] Chern, H.-H. and Hwang, H.-K. (2006). Partial match queries in random $k$-d trees. SIAM J. Comput. 35 1440–1466 (electronic). MR2217152 10.1137/S0097539703437491[5] Chern, H.-H. and Hwang, H.-K. (2006). Partial match queries in random $k$-d trees. SIAM J. Comput. 35 1440–1466 (electronic). MR2217152 10.1137/S0097539703437491
[6] Curien, N. and Joseph, A. (2011). Partial match queries in two-dimensional quadtrees: A probabilistic approach. Adv. in Appl. Probab. 43 178–194. MR2761153 10.1239/aap/1300198518 euclid.aap/1300198518
[6] Curien, N. and Joseph, A. (2011). Partial match queries in two-dimensional quadtrees: A probabilistic approach. Adv. in Appl. Probab. 43 178–194. MR2761153 10.1239/aap/1300198518 euclid.aap/1300198518
[7] Devroye, L. (1987). Branching processes in the analysis of the heights of trees. Acta Inform. 24 277–298. MR894557 10.1007/BF00265991[7] Devroye, L. (1987). Branching processes in the analysis of the heights of trees. Acta Inform. 24 277–298. MR894557 10.1007/BF00265991
[8] Devroye, L. and Laforest, L. (1990). An analysis of random $d$-dimensional quad trees. SIAM J. Comput. 19 821–832. MR1059656 10.1137/0219057[8] Devroye, L. and Laforest, L. (1990). An analysis of random $d$-dimensional quad trees. SIAM J. Comput. 19 821–832. MR1059656 10.1137/0219057
[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. MR2380900 10.1214/07-AAP457 euclid.aoap/1199890024
[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. MR2380900 10.1214/07-AAP457 euclid.aoap/1199890024
[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. MR1733960[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. MR1733960
[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.[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. MR1933200 10.1016/S0196-6774(02)00213-4[12] Duch, A. and Martínez, C. (2002). On the average performance of orthogonal range search in multidimensional data structures. J. Algorithms 44 226–245. MR1933200 10.1016/S0196-6774(02)00213-4
[16] Flajolet, P., Gonnet, G., Puech, C. and Robson, J. M. (1993). Analytic variations on quadtrees. Algorithmica 10 473–500. MR1244619 10.1007/BF01891833[16] Flajolet, P., Gonnet, G., Puech, C. and Robson, J. M. (1993). Analytic variations on quadtrees. Algorithmica 10 473–500. MR1244619 10.1007/BF01891833
[17] Flajolet, P., Labelle, G., Laforest, L. and Salvy, B. (1995). Hypergeometrics and the cost structure of quadtrees. Random Structures Algorithms 7 117–144. MR1369059[17] Flajolet, P., Labelle, G., Laforest, L. and Salvy, B. (1995). Hypergeometrics and the cost structure of quadtrees. Random Structures Algorithms 7 117–144. MR1369059
[18] Flajolet, P. and Lafforgue, T. (1994). Search costs in quadtrees and singularity perturbation asymptotics. Discrete Comput. Geom. 12 151–175. MR1283884[18] Flajolet, P. and Lafforgue, T. (1994). Search costs in quadtrees and singularity perturbation asymptotics. Discrete Comput. Geom. 12 151–175. MR1283884
[19] Flajolet, P. and Puech, C. (1986). Partial match retrieval of multidimensional data. J. Assoc. Comput. Mach. 33 371–407. MR835110 10.1145/5383.5453[19] Flajolet, P. and Puech, C. (1986). Partial match retrieval of multidimensional data. J. Assoc. Comput. Mach. 33 371–407. MR835110 10.1145/5383.5453
[21] Grübel, R. (2009). On the silhouette of binary search trees. Ann. Appl. Probab. 19 1781–1802. MR2569807 10.1214/08-AAP593 euclid.aoap/1255699543
[21] Grübel, R. (2009). On the silhouette of binary search trees. Ann. Appl. Probab. 19 1781–1802. MR2569807 10.1214/08-AAP593 euclid.aoap/1255699543
[23] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30. MR144363 10.1080/01621459.1963.10500830[23] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30. MR144363 10.1080/01621459.1963.10500830
[26] Martínez, C., Panholzer, A. and Prodinger, H. (2001). Partial match queries in relaxed multidimensional search trees. Algorithmica 29 181–204. MR1887303 10.1007/BF02679618[26] Martínez, C., Panholzer, A. and Prodinger, H. (2001). Partial match queries in relaxed multidimensional search trees. Algorithmica 29 181–204. MR1887303 10.1007/BF02679618
[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. MR1801141[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. MR1801141
[28] Neininger, R. (2001). On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Structures Algorithms 19 498–524. MR1871564[28] Neininger, R. (2001). On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Structures Algorithms 19 498–524. MR1871564
[29] Neininger, R. and Rüschendorf, L. (2001). Limit laws for partial match queries in quadtrees. Ann. Appl. Probab. 11 452–469. MR1843054 10.1214/aoap/1015345300 euclid.aoap/1015345300
[29] Neininger, R. and Rüschendorf, L. (2001). Limit laws for partial match queries in quadtrees. Ann. Appl. Probab. 11 452–469. MR1843054 10.1214/aoap/1015345300 euclid.aoap/1015345300
[30] Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 378–418. MR2023025 10.1214/aoap/1075828056 euclid.aoap/1075828056
[30] Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 378–418. MR2023025 10.1214/aoap/1075828056 euclid.aoap/1075828056
[31] Neininger, R. and Rüschendorf, L. (2004). On the contraction method with degenerate limit equation. Ann. Probab. 32 2838–2856. MR2078559 10.1214/009117904000000171 euclid.aop/1091813632
[31] Neininger, R. and Rüschendorf, L. (2004). On the contraction method with degenerate limit equation. Ann. Probab. 32 2838–2856. MR2078559 10.1214/009117904000000171 euclid.aop/1091813632
[32] Neininger, R. and Sulzbach, H. (2012). On a functional contraction method. Preprint. Available at arXiv:1202.1370. 1202.1370[32] Neininger, R. and Sulzbach, H. (2012). On a functional contraction method. Preprint. Available at arXiv:1202.1370. 1202.1370
[33] Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. in Appl. Probab. 27 770–799. MR1341885 10.2307/1428133[33] Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. in Appl. Probab. 27 770–799. MR1341885 10.2307/1428133
[34] Rivest, R. L. (1976). Partial-match retrieval algorithms. SIAM J. Comput. 5 19–50. MR395398 10.1137/0205003[34] Rivest, R. L. (1976). Partial-match retrieval algorithms. SIAM J. Comput. 5 19–50. MR395398 10.1137/0205003
[36] Rösler, U. (1992). A fixed point theorem for distributions. Stochastic Process. Appl. 42 195–214. MR1176497 10.1016/0304-4149(92)90035-O[36] Rösler, U. (1992). A fixed point theorem for distributions. Stochastic Process. Appl. 42 195–214. MR1176497 10.1016/0304-4149(92)90035-O
[37] Rösler, U. (2001). On the analysis of stochastic divide and conquer algorithms. Algorithmica 29 238–261. MR1887306 10.1007/BF02679621[37] Rösler, U. (2001). On the analysis of stochastic divide and conquer algorithms. Algorithmica 29 238–261. MR1887306 10.1007/BF02679621
[41] Yerry, M. and Shephard, M. (1983). A modified quadtree approach to finite element mesh generation. IEEE Computer Graphics and Applications 3 39–46.[41] Yerry, M. and Shephard, M. (1983). A modified quadtree approach to finite element mesh generation. IEEE Computer Graphics and Applications 3 39–46.