Partial match queries in random k-d trees

SIAM Journal on Computing

Volumn 35, Issue 6, 2006, Pages 1440-1466

  Chern, Hua Huai ^a   Hwang, Hsien Kuei ^b  

^a NATIONAL TAIWAN OCEAN UNIVERSITY   (Taiwan)

^b INSTITUTE OF STATISTICAL SCIENCE   (Taiwan)

Author keywords

Asymptotic analysis; Average case analysis of algorithms; Differential equations; K d trees; Method of linear operators; Partial match queries

Indexed keywords

APPROXIMATION THEORY; DIFFERENTIAL EQUATIONS; MATHEMATICAL OPERATORS; POLYNOMIALS; PROBLEM SOLVING; TREES (MATHEMATICS);

ASYMPTOTIC ANALYSIS; AVERAGE-CASE ANALYSIS OF ALGORITHMS; K-D TREES; METHOD OF LINEAR OPERATORS; PARTIAL MATCH QUERIES;

QUERY LANGUAGES;

EID: 33845591240     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539703437491     Document Type: Article

References (37)

1. M. ABRAMOWITZ AND I. A. STEGUN, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972.
2. J. L. BENTLEY, Multidimensional binary search trees used for associative searching, Comm. ACM, 18 (1975), pp. 509-517.
3. J. L. BENTLEY, K-d trees for semidynamic point sets, in Proceedings of the Sixth Annual Symposium on Computational Geometry, ACM, New York, 1990, pp. 187-197.
4. J. L. BENTLEY AND J. H. FRIEDMAN, Data structures for range searching, ACM Comput. Surveys, 11 (1979), pp. 397-409.
5. E. BERTINO, B. CATANIA, B. C. OOI, R. SACKS-DAVIS, K. L. TAN, J. ZOBEL, B. SHIDLOVSKY, AND B. CATANIA, Indexing Techniques for Advanced Database Systems, Kluwer Academic Publishers, Boston, 1997.
6. P. CHANZY, L. DEVROYE, AND C. ZAMORA-CURA, Analysis of range search for random k-d trees, Acta Inform., 37 (2001), pp. 355-383.
7. H.-H. CHERN, M. FUCHS, AND H.-K. HWANG, Phase changes in random point quadtrees, ACM Trans. Algorithms, to appear.
8. H.-H. CHERN AND H.-K. HWANG, Phase changes in random m-ary search trees and generalized quicksort, Random Structures Algorithms, 19 (2001), pp. 316-358.
9. H.-H. CHERN AND H.-K. HWANG, Partial match queries in random quadtrees, SIAM J. Comput., 32 (2003), pp. 904-915.
10. H.-H. CHERN, H.-K. HWANG, AND T.-H. TSAI, An asymptotic theory for Cauchy-Euler differential equations with applications to the analysis of algorithms, J. Algorithms, 44 (2002), pp. 177-225.
11. W. CUNTO, G. LAU, AND P. FLAJOLET, Analysis of kdt-trees: kd-trees improved by local re-ogranisations, in Algorithms and Data Structures, Lecture Notes in Comput. Sci. 382, Springer, Berlin, 1989, pp. 24-38.
12. L. DEVROYE, J. JABBOUR, AND C. ZAMORA-CURA, Squarish k-d trees, SIAM J. Comput., 30 (2000), pp. 1678-1700.
13. A. DUCH AND C. MARTÍNEZ, On the average performance of orthogonal range search in multi-dimensional data structures, J. Algorithms, 44 (2002), pp. 226-245.
14. P. FLAJOLET, G. GONET, C. PUECH, AND J. M. ROBSON, Analytic variations on quadtrees, Algorithmica, 10 (1993), pp. 473-500.
15. P. FLAJOLET, X. GOURDON, AND P. DUMAS, Mellin transforms and asymptotics: Harmonic sums, Theoret. Comput. Sc., 144 (1995), pp. 3-58.
16. P. FLAJOLET AND A. M. ODLYZKO, Singularity analysis of generating functions, SIAM J. Discrete Math., 3 (1990), pp. 216-240.
17. P. FLAJOLET AND P. PUECH, Partial match retrieval of multi-dimensional data, J. ACM, 32 (1986), pp. 371-407.
18. P. FLAJOLET AND R. SEDGEWICK, Analytic Combinatorics, in preparation. Preliminary version available online at http://algo.inria.fr/flajolet/ Publications/publist.html.
19. J. H. FRIEDMAN, J. L. BENTLEY, AND R. A. FINKEL, An algorithm for finding best matches in logarithmic expected time, ACM Trans. Math. Software, 3 (1977), pp. 209-226.
20. G. H. GONNET AND R. BAEZA-YATES, Handbook of Algorithms and Data Structures, Addison-Wesley, Reading, MA, 1991.
21. A. G. GRAY AND A. W. MOORE, "N-body" problems in statistical learning, in Advances in Neural Information Processing Systems, T. K. Leen and T. G. Dietterich, eds., MIT Press, Cambridge, MA, 2001, pp. 521-527.
22. P. J. GROTHER, G. T. CANDELA, AND J. L. BLUE, Fast implementations of nearest neighbor classifiers, Pattern Recognition, 30 (1997), pp. 459-465.
23. T. N. HIBBARD, Some combinatorial properties of certain trees with applications to searching and sorting, J. ACM, 9 (1962), pp. 13-28.
24. E. L. INCE, Ordinary Differential Equations, Dover, New York, 1926.
25. D. T. LEE AND C. K. WONG, Worst-case analysis for region and partial region searches in multidimensional binary search trees and quad trees, Acta Inform., 9 (1977), pp. 23-29.
26. C. MARTÍNEZ, A. PANHOLZER, AND H. PRODINGER, Partial match queries in relaxed multidimensional search trees, Algorithmica, 29 (2001), pp. 181-204.
27. R. NEININGER, Asymptotic distributions for partial match gueries in K-d trees, Random Structures Algorithms, 17 (2000), pp. 403-427.
28. R. C. NICHOL, S. CHONG, A. J. CONNOLLY, S. DAVIES, C. GENOVESE, A. M. HOPKINS, C. J. MILLER, A. W. MOORE, D. PELLEG, G. T. RICHARDS, J. SCHNEIDER, I. SZAPUDI, AND L. WASSERMAN, Computational AstroStatistics: Fast and Efficient Tools for Analysing Huge Astronomical Data Sources, preprint, 2002. Available online at http://arxiv.org/abs/astro-ph/0110230.
29. S. M. OMOHUNDRO, Efficient algorithms with neural network behavior, Complex Systems, 1 (1987), pp. 273-347.
30. P. V. POBLETE AND J. I. MUNRO, The analysis of a fringe heuristic for binary search trees, J. Algorithms, 6 (1985), pp. 336-350.
31. F. P. PREPARATA AND M. I. SHAMOS, Computational Geometry: An Introduction, Springer-Verlag, New York, 1985.
32. J. REISS, J.-J. AUCOUTURIER, AND M. SANDLER, Efficient multidimensional searching routines for music information retrieval: Use k-d trees to retrieve music information, in Proceedings of the Second Annual International Symposium on Music Information Retrieval (Bloomington, IN), J. S. Downie and D. Bainbridge, eds., 2001, pp. 163-171.
33. H. SAMET, The Design and Analysis of Spatial Data Structures, Addison-Wesley, Reading, MA, 1990.
34. H. SAMET, Applications of Spatial Data Structures: Computer Graphics, Image Processing, and GIS, Addison-Wesley, Reading, MA, 1990.
35. F. SCHWARZER AND I. LOTAN, Approximation of protein structure for fast similarity measures, in Proceedings of the Seventh Annual ACM International Conference on Computational Biology (RECOMB), ACM, New York, 2003, pp. 267-276.
36. R. SEDGEWICK, Quicksort, Garland Publishing, New York, 1980.
37. S. S. SKIENA, Who is interested in algorithms and why? Lessons from the Stony Brook algorithms, ACM SIGACT News, 30 (1999), pp. 65-74. Available online at http://www.es. sunysb.edu/~skiena/hits.