Phase changes in random point quadtrees

ACM Transactions on Algorithms

Volumn 3, Issue 2, 2007, Pages

  Chern, Hua Huai ^a   Fuchs, Michael ^b   Hwang, Hsien Kuei ^c,d  

^a NATIONAL TAIWAN OCEAN UNIVERSITY   (Taiwan)

^b NATIONAL CHIAO TUNG UNIVERSITY   (Taiwan)

^c ACADEMIA SINICA   (Taiwan)

^d INSTITUTE OF STATISTICAL SCIENCE   (Taiwan)

Author keywords

Analysis in distribution of algorithms; Asymptotic transfer; Central limit theorems; Depth; Differential equations; Grid trees; Local limit theorems; Mellin transforms; Page usage; Phase transitions; Quadtrees; Total path length

Indexed keywords

ASYMPTOTIC TRANSFER; CENTRAL LIMIT THEOREMS; GRID TREES; LOCAL LIMIT THEOREMS; MELLIN TRANSFORMS; TOTAL PATH LENGTH;

APPROXIMATION THEORY; DIFFERENTIAL EQUATIONS; GRID COMPUTING; NUMERICAL METHODS; THEOREM PROVING;

TREES (MATHEMATICS);

EID: 34250222447     PISSN: 15496325     EISSN: 15496333     Source Type: Journal    
DOI: 10.1145/1240233.1240235     Document Type: Article

References (47)

1. BAI, Z.-D., HWANG, H.-K., AND TSAI, T.-H. 2003. Berry-Esseen bounds for the number of maxima in planar regions. Electron. J. Probab, 8, 9.
2. CHAUVIN, B., AND POUYANNE, N. 2004. w-ary search trees when m ≥ 27: A strong asymptotics for the space requirements. Random Struct. Alg. 24, 2, 133-154.
3. CHERN, H.-H., AND HWANG, H.-K. 2001. Phase changes in random m-ary search trees and generalized quicksort. Random Struct. Alg. 19, 3-4, 316-358.
4. CHERN, H.-H., AND HWANG, H.-K. 2003. Partial match queries in random quadtrees. SIAM J. Comput. 32, 4, 904-915.
5. CHERN, H.-H., AND HWANG, H.-K. 2005a. Limit distribution of the number of consecutive records. Random Struct. Alg. 26, 4, 404-417.
6. CHERN, H.-H., AND HWANG, H.-K. 2005b. Partial match queries in random k-d trees. SIAM J. Comput. 35, 6, 1440-1466.
7. CHERN, H.-H., HWANG, H.-K., AND TSAI, T.-H. 2002. An asymptotic theory for Cauchy-Eulerdifferential equations with applications to the analysis of algorithms. J. Alg. 44, 1, 177-225.
8. DE BERG, M., VAN KREVELD, M., OVERMARS, M., AND SCHWARZKOPF, O. 2000. Computational Geometry, revised ed. Springer Verlag, Berlin.
9. DEAN, D. S., AND MAJUMDAR, S. N. 2002. Phase transition in a random fragmentation problem with applications to computer science. J. Phys. A 35, 32, L501-L507.
10. DEVROYE, L. 1987. Branching processes in the analysis of the heights of trees. Acta Inform. 24, 3, 277-298.
11. DEVROYE, L. 1991. Limit laws for local counters in random binary search trees. Random Struct. Alg. 2, 3, 303-315.
12. DEVROYE, L. 1999. Universal limit laws for depths in random trees. SIAM J. Comput. 28, 2, 409-432.
13. DEVROYE, L., AND LAFOREST, L. 1990. An analysis of random d-dimensional quad trees. SIAM J. Comput. 19, 5, 821-832.
14. ERDÉLYI, A., MAGNUS, W., OBERHETTINGER, F., AND TRICOMI, F. G. 1953. Higher Transcendental Functions., vol. I. McGraw-Hill, New York.
15. FILL, J. A., AND KAPUR, N. 2004. The space requirement of m-ary search trees: Distributional asymptotics for m >=27. eprint arxiv:|math/0405144.
16. FINKEL, R. A., AND BENTLEY, J. L. 1974. Quad trees, A data structure for retrieval on composite keys. Acta Inf. 4, 1-9.
17. FLAJOLET, P., GONNET, G., PUECH, C., AND ROBSON, J. M. 1993. Analytic variations on quadtrees. Algorithmic 10, 6, 473-500.
18. FLAJOLET, P., GOURDON, X., AND MARTINEZ, C. 1997. Patterns in random binary search trees. Random Struct. Alg. 11, 3, 223-244.
19. FLAJOLET, P., LABELLE, G., LAFOREST, L., AND SALVY, B. 1995. Hypergeometrics and the cost structure of quadtrees. Random Struct. Alg. 7, 2, 117-144.
20. FLAJOLET, P., AND LAFFORGUE, T. 1994. Search costs in quadtrees and singularity perturbation asymptotics. Discrete Comput. Geom. 12, 2, 151-175.
21. FLAJOLET, P., AND ODLYZKO, A. 1990. Singularity analysis of generating functions. SIAM J. Discrete Math. 3, 2, 216-240.
22. FLAJOLET, P., AND PUECH, C. 1986. Partial match retrieval of multidimensional data. J. Assoc. Comput. Mach. 33, 2, 371-407.
23. FLAJOLET, P., AND SEDGEWICK, R. 1995. Mellin transforms and asymptotics: Finite differences and Rice's integrals. Theoret. Comput. Sci. 144, 1-2, 101-124.
24. FLAJOLET, P., AND SEDGEWICK, R. 2005. Analytic Combinatorics, in preparation.
25. FOATA, D., HAN, G.-N., AND LASS, B. 2001/02. Les nombres hyperharmoniques et la fratrie du collectionneur de vignettes. Sém. Lothar. Combin. 47, B47a.
26. HOANG, N. M., AND JACOB, G. 2000. Symbolic integration of meromorphic differential systems via Dirichlet functions. Discrete Math. 210, 1-3, 87-116.
27. HOSHI, M., AND FLAJOLET, P. 1992. Page usage in a quadtree index. BIT 32, 3, 384-402.
28. HWANG, H.-K. 1998. On convergence rates in the central limit theorems for combinatorial structures. European J. Combin. 19, 3, 329-343.
29. HWANG, H.-K. 2003. Second phase changes in random m-ary search trees and generalized quicksort: Convergence rates. Ann. Probab. 31, 2, 609-629.
30. HWANG, H.-K. 2004. Phase changes in random recursive structures and algorithms (a brief survey). In Proceedings of the Workshop on Probability with Applications to Finance and Insurance (Singapore), T. L. Lai et al., eds. World Scientific, River Edge, NJ. 82-97.
31. HWANG, H.-K., AND NEININGER, R. 2002. Phase change of limit laws in the quicksort recurrence under varying toll functions. SIAM J. Comput. 31, 6, 1687-1722.
32. HWANG, H.-K., AND TSAI, T.-H. 2003. An asymptotic theory for recurrence relations based on minimization and maximization. Theoret. Comput. Sci. 290, 3, 1475-1501.
33. INCE, E. L. 1944. Ordinary Differential Equations. Dover, New York.
34. JANSON, S. 2004. Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110, 2, 177-245.
35. JANSON, S. 2005. Congruence properties of depths in some random trees, preprint arxiv:|math/050947|v|.
36. LABELLE, G., AND LAFOREST, L. 1995a. Combinatorial variations on multidimensional quadtrees. J. Combin. Theory Ser. A 69, 1, 1-16.
37. LABELLE, G., AND LAFOREST, L. 1995b. Sur la distribution de l'arité de la racine d'une arborescence hyperquaternaire à d dimensions. Discrete Math. 139, 1-3, 287-302.
38. LABELLE, G., AND LAFOREST, L. 1996. Étude de constantes universelles pour les arborescences hyperquaternaires de recherche. Discrete Math. 153, 1-3, 199-211.
39. LOÈVE, M. 1977. Probability Theory. I, 4th ed. Springer Verlag, New York.
40. MAHMOUD, H. M. 1992. Evolution of Random Search Trees. John Wiley, New York.
41. MARTÍNEZ, C., PANHOLZER, A., AND PRODINGER, H. 2001. Partial match queries in relaxed multidimensional search trees. Algorithmica 29, 1-2, 181-204.
42. NEININGER, R., AND RÜSCHENDORF, L. 1999. On the internal path length of d-dimensional quad trees. Random Struct. Alg. 15, 1, 25-41.
43. NEININGER, R., AND RÜSCHENDORF, L. 2001. Limit laws for partial match queries in quadtrees. Ann. Appl. Probab. 11, 2, 452-469.
44. PETROV, V. V. 1975. Sums of Independent Random Variables. Springer Verlag, New York.
45. SAMET, H. 1990a. Applications of Spatial Data Structures: Computer Graphics, Image Processing, and GIS. Series in Computer Science. Addison-Wesley, Reading, MA.
46. SAMET, H. 1990b. The Design and Analysis of Spatial Data Structures. Addison-Wesley, Reading, MA.
47. SEDGEWICK, R. 1980. Quicksort. Garland, New York. Reprint of Ph.D. thesis, Stanford University, 1975.