Balanced Aspect Ratio Trees: Combining the Advantages of k-d Trees and Octrees

Journal of Algorithms

Volumn 38, Issue 1, 2001, Pages 303-333

  Duncan, Christian A ^a   Goodrich, Michael T ^b   Kobourov, Stephen ^c  

^a Department of Computer Science and Center for Computational Science   (United States)

^b JOHNS HOPKINS UNIVERSITY   (United States)

^c University of Arizona   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001470860     PISSN: 01966774     EISSN: None     Source Type: Journal    
DOI: 10.1006/jagm.2000.1135     Document Type: Article

References (39)

1. P. K. Agarwal and J. Erickson, Geometric range searching and its relatives, in "Advances in Discrete and Comput. Geom." (B. Chazelle, E. Goodman, and R. Pollack, Eds.), Amer. Math. Soc., Providence, RI, 1998.
2. P. K. Agarwal, E. F. Grove, T. M. Murali, and J. S. Vitter, Binary space partitions for fat rectangles, in "Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., Oct. 1996," pp. 482-491.
3. P. K. Agarwal, M. Katz, and M. Sharir, Computing depth orders for fat objects and related problems, Comput. Geom. Theory Appl. 5 (1995), 187-206.
4. W. G. Aref and H. Samet, Perspective viewing of objects represented by octrees, Comput. Graph Forum 14 (1995), 59-66. [Also University of Maryland Computer Science TR-2757.]
5. S. Arya and D. M. Mount, Approximate range searching, in "Proc. 11th Annu. ACM Sympos. Comput. Geom., 1995," pp. 172-181.
6. S. Arya, D. M. Mount, N. S. Netanyahu, R. Silverman, and A. Wu, An optimal algorithm for approximate nearest neighbor searching, in "Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, 1994," pp. 573-582.
7. F. Aurenhammer, "Voronoi Diagrams: A Survey of a Fundamental Geometric Data Structure," Report B-90-09, Fachber. Math., Free Univ. Berlin, Berlin, West Germany, 1990.
8. J. L. Bentley, Multidimensional binary search trees used for associative searching, Commun. Assoc. Comput. Mach. 18 (1975), 509-517.
9. J. L. Bentley, K-d trees for semidynamic point sets, in "Proc. 6th Annu. ACM Sympos. Comput. Geom., 1990," pp. 187-197.
10. J. L. Bentley and J. H. Friedman, Data structures for range searching, ACM Comput. Surv. 11 (1979), 397-409.
11. J. L. Bentley and H. A. Maurer, Efficient worst-case data structures for range searching, Acta. Inform. 13 (1980), 155-168.
12. P. B. Callahan and S. R. Kosaraju, Algorithms for dynamic closest-pair and n-body potential fields, in "Proc. 6th ACM-SIAM Sympos. Discrete Algorithms, 1995," pp. 263-272.
13. P. B. Callahan and S. R. Kosaraju, A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields, J. Assoc. Comput. Mach. 42 (1995), 67-90.
14. B. Chazelle, Lower bounds for orthogonal range searching, I: The reporting case, J. Assoc. Comput. Mach. 37 (1990), 200-212.
15. H. H. Chen and T. S. Huang, A survey of construction and manipulation of octrees, Comput. Vision Graphics Image Process 43 (1988), 409-431.
16. N. Chin and S. Feiner, Near real-time shadow generation using BSP trees, in "Proc. SIG-GRAPH '89," Comput. Graph. 23 (1989), 99-106.
17. M. de Berg, Linear size binary space partitions for fat objects, in "Proc. 3rd Annu. European Sympos. Algorithms," Lecture Notes in Computer Science, Vol. 979, pp. 252-263, Springer-Verlag, Berlin/New York, 1995.
18. D. P. Dobkin and H. Edelsbrunner, Space searching for intersecting objects, J. Algorithms 8 (1987), 348-361.
19. C. A. Duncan, M. T. Goodrich, and S. G. Kobourov, Balanced aspect ratio trees and their use for drawing very large graphs, in "Sixth Symposium on Graph Drawing, 1998."
20. C. A. Duncan, M. T. Goodrich, and S. G. Kobourov, Balanced aspect ratio trees: Combining the advantages of k-d trees and octrees, in "Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, Jan. 1999."
21. A. Efrat and M. Sharir, On the complexity of the union of fat objects in the plane, in "Proc. 13th Annu. ACM Sympos. Comput. Geom., 1997," pp. 104-112.
22. R. A. Finkel and J. L. Bentley, Quad trees: a data structure for retrieval on composite keys, Acta Inform. 4 (1974), 1-9.
23. S. Fortune, Voronoi diagrams and Delaunay triangulations, in "Handbook of Discrete and Computational Geometry" (J. E. Goodman and J. O'Rourke, Eds.), chap. 20, pp. 377-388, CRC Press, Boca Raton, FL, 1997.
24. H. Fuchs, Z. M. Kedem, and B. Naylor, On visible surface generation by a priori tree structures, in "Proc. SIGGRAPH '80," Comput. Graph 14 (1980), 124-133.
25. A. Henrich, Improving the performance of multi-dimensional access structures based on kd-trees, in "Proc. 12th IEEE Intl. Conf. on Data Engineering, 1996," pp. 68-74.
26. C. Jackins and S. L. Tanimoto, Oct-trees and their use in representing 3-d objects, Comput. Graph. Image Process. 14 (1980), 249-270.
27. M. J. Katz, 3-D vertical ray shooting and 2-D point enclosure, range searching, and arc shooting amidst convex fat objects, Comput. Geom. Theory Appl. 8 (1997), 299-316.
28. J. Matoušek, "Geometric Range Searching," Tech. Report B-93-09, Fachbereich Mathematik und Informatik, Free Univ. Berlin, 1993.
29. J. Matoušek, Geometric range searching, ACM Comput. Surv. 26 (1994), 421-461.
30. J. Matoušek, J. Pach, M. Sharir, S. Sifrony, and E. Welzl, Fat triangles determine linearly many holes, SIAM J. Comput. 23 (1994), 154-169.
31. M. H. Overmars, Efficient data structures for range searching on a grid, J. Algorithms 9 (1988), 254-275.
32. M. H. Overmars and A. F. van der Stappen, Range searching and point location among fat objects, J. Algorithms 21 (1996), 629-656.
33. M. H. Overmars and J. van Leeuwen, Dynamic multi-dimensional data structures based on quad- and k-d trees, Acta Inform 17 (1982), 267-285.
34. M. S. Paterson and F. F. Yao, Efficient binary space partitions for hidden-surface removal and solid modeling, Discrete Comput. Geom. 5 (1990), 485-503.
35. M. S. Paterson and F. F. Yao, Optimal binary space partitions for orthogonal objects, J. Algorithms 13 (1992), 99-113.
36. H. Samet, The quadtree and related hierarchical data structures, ACM Comput. Surv. 16 (1984), 187-260.
37. H. Samet, An overview of quadtrees, octrees, and related hierarchical data structures, in "Theoretical Foundations of Computer Graphics and CAD" (R. A. Earnshaw, Ed.), NATO ASI Series F, Vol. 40, pp. 51-68, Springer-Verlag, Berlin, 1988.
38. H. Samet, "The Design and Analysis of Spatial Data Structures," Addison-Wesley, Reading, MA, 1990.
39. E. Torres, Optimization of the binary space partition algorithm (BSP) for the visualization of dynamic scenes, in "Eurographics '90," pp. 507-518, North-Holland, Amsterdam, 1990.