Fast construction of nets in low-dimensional metrics and their applications

SIAM Journal on Computing

Volumn 35, Issue 5, 2006, Pages 1148-1184

  Har Peled, Sariel ^a   Mendel, Manor ^a,b  

^a University of Illinois at Urbana Champaign   (United States)

^b OPEN UNIVERSITY OF ISRAEL   (Israel)

Author keywords

Approximate distance oracle; Distance labeling; Doubling dimension; Doubling measure; Metric nets; Nearest neighbor search; Quadtree

Indexed keywords

DOUBLING MEASURE; FINITE METRIC SPACES; LINEAR TIME ALGORITHM; NEAREST NEIGHBOR;

ALGORITHMS; APPROXIMATION THEORY; COMPUTATION THEORY; DATA STRUCTURES; HIERARCHICAL SYSTEMS; PROBLEM SOLVING;

LINEAR ALGEBRA;

EID: 33749260659     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539704446281     Document Type: Article

References (48)

1. S. ARYA, D. M. MOUNT, N. S. NETANYAHU, R. SILVERMAN, AND A. Y. WU, An optimal algorithm for approximate nearest neighbor searching in fixed dimensions, J. ACM, 45 (1998), pp. 891-923.
2. n, Bull. Soc. Math. France, 111 (1983), pp. 429-448.
3. A. BARVINOK, A Course in Convexity, Grad. Stud. Math. 54, AMS, Providence, RI, 2002.
4. M. A. BENDER AND M. FARACH-COLTON, The lea problem revisited, in Proceedings of the 4th Latin American Symposium on Theoretical Informatics, Springer-Verlag, Berlin, 2000, pp. 88-94.
5. M. A. BENDER AND M. FARACH-COLTON, The level ancestor problem simplified, Theoret. Comput. Sci., 321 (2004), pp. 5-12.
6. J. L. BENTLEY AND J. B. SAXE, Decomposable searching problems i: Static-to-dynamic transformation, J. Algorithms, 1 (1980), pp. 301-358.
7. Y. BENYAMINI AND J. LINDENSTRAUSS, Geometric Nonlinear Functional Analysis, AMS, Providence, RI, 2000.
8. A. BEYGELZIMER, S. KAKADE, AND J. LANGFORD, Cover Trees for Nearest Neighbor, http://ttic.uchicago.edu/~sham/papers/ml/cover.tree.pdf (2004).
9. M. BǍDOIU, K. DHAMDHERE, A. GUPTA, Y. RABINOVICH, H. RÄCKE, R. RAVI, AND A. SIDIROPOULOS, Approximation algorithms for low-distortion embedding into low-dimensional spaces, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, ACM, New York, 2005, pp. 119-128.
10. P. B. CALLAHAN, The Well Separated Pair Decomposition and Its Application, Tech. report, Johns Hopkins University, Baltimore, MD, 1995.
11. 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), pp. 67-90.
12. H. T.-H. CHAN, A. GUPTA, B. M. MAGGS, AND S. ZHOU, On hierarchical routing in doubling metrics, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, ACM, New York, 2005, pp. 762-771.
13. K. L. CLARKSON, Fast algorithms for the all nearest neighbors problem, in Proceedings of the 24th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 1983, pp. 226-232.
14. K. L. CLARKSON, Nearest neighbor queries in metric spaces, Discrete Comput. Geom., 22 (1999), pp. 63-93.
15. K. L. CLARKSON, Nearest Neighbor Searching in Metric Spaces: Experimental Results for sb(s), http://cm.bell-labs.com/who/clarkson/Msb/white.paper.pdf (2002).
16. K. L. CLARKSON AND P. W. SHOR, Applications of random sampling in computational geometry, II, Discrete Comput. Geom., 4 (1989), pp. 387-421.
17. T. FEDER AND D. H. GREENE, Optimal algorithms for approximate clustering, in Proceedings of the 20th Annual ACM Symposium on Theory of Computing, ACM, New York, 1988, pp. 434-444.
18. C. GAVOILLE, M. KATZ, N. A. KATZ, C. PAUL, AND D. PELEG, Approximate distance labeling schemes, in Proceedings of the 9th Annual European Symposium on Algorithms, Lecture Notes in Comput. Sci. 2161, Springer, Berlin, 2001, pp. 476-487.
19. C. GAVOILLE, D. PELEG, S. PERENNES, AND R. RAZ, Distance labeling in graphs, J. Algorithms, 53 (2004), pp. 85-112.
20. T. GONZALEZ, Clustering to minimize the maximum intercluster distance, Theoret. Comput. Sci., 38 (1985), pp. 293-306.
21. J. GUDMUNDSSON, C. LEVCOPOULOS, G. NARASIMHAN, AND M. SMID, Approximate distance oracles for geometric graphs, in Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, ACM, New York, 2002, pp. 828-837.
22. J. GUDMUNDSSON, C. LEVCOPOULOS, G. NARASIMHAN, AND M. H. M. SMID, Approximate distance oracles revisited, in Proceedings of the 13th Annual International Symposium on Algorithms and Computation, Springer, Berlin, 2002, pp. 357-368.
23. A. GUPTA, R. KRAUTHGAMER, AND J. R. LEE, Bounded geometries, fractals, and low-distortion embeddings, in Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 2003, pp. 534-543.
24. S. HAR-PELED, A replacement for Voronoi diagrams of near linear size, in Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 2001, pp. 94-103.
25. S. HAR-PELED, Clustering motion, Discrete Comput. Geom., 31 (2004), pp. 545-565.
26. S. HAR-PELED AND S. MAZUMDAR, Coresets for k-means and k-median clustering and their applications, in Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 291-300.
27. J. HEINONEN, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001.
28. K. HILDRUM, J. KUBIATOWICZ, S. MA, AND S. RAO, A note on the nearest neighbor in growth-restricted metrics, in Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, ACM, New York, 2004, pp. 553-554.
29. P. INDYK AND R. MOTWANI, Approximate nearest neighbors: Towards removing the curse of dimensionality, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, ACM, New York, 1998, pp. 604-613.
30. D. R. KARGER AND M. RUHL, Finding nearest neighbors in growth-restricted metrics, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing, ACM, New York, 2002. pp. 741-750.
31. J. M. KEIL, Approximating the complete Euclidean graph, in Proceedings of the 1st Scandinavian Workshop on Algorithm Theory, Lecture Notes in Comput. Sci. 318, Springer, Berlin, 1988, pp. 208-213.
32. D. E. KNUTH, The Art of Computer Programming: Seminumerical Algorithms, 3rd ed., Addison-Wesley, New York, 1997.
33. R. KRAUTHGAMER AND J. R. LEE, The black-box complexity of nearest neighbor search, in Proceedings of the 31st International Colloquium on Automata, Languages and Programming, Springer, Berlin, 2004, pp. 858-869.
34. R. KRAUTHGAMER AND J. R. LEE, Navigating nets: Simple algorithms for proximity search, in Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, ACM, New York, 2004, pp. 791-800.
35. N, Bull. London Math. Soc., 34 (2002), pp. 667-676.
36. J. LUUKKAINEN AND E. SAKSMAN, Every complete doubling metric space carries a doubling measure, Proc. Amer. Math. Soc., 126 (1998), pp. 531-534.
37. J. MATOUŠEK, Using the Borsuk-Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry, Universitext, Springer-Verlag, Berlin, 2003.
38. M. H. OVERMARS AND J. VAN LEEUWEN, Maintenance of configurations in the plane, J. Comput. System Sci., 23 (1981), pp. 166-204.
39. D. PELEG, Informative labeling schemes for graphs, Theoret. Comput. Sci., 340 (2005), pp. 577-593.
40. D. PELEG AND A. SCHÄFFER, Graph spanners, J. Graph Theory, 13 (1989), pp. 99-116.
41. ∞-weights, Rev. Mat. Iberoamericana, 12 (1996), pp. 337-410.
42. A. SLIVKINS, Distance Estimation and Object Location via Rings of Neighbors, Tech. Report TR2005-1977, Cornell CIS, Cornell University, Ithaca, NY, 2005.
43. A. SLIVKINS, Distributed approaches to triangulation and embedding, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, ACM, New York, 2005, pp. 640-649.
44. K. TALWAR, Bypassing the embedding: Algorithms for low dimensional metrics, in Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 281-290.
45. M. THORUP AND U. ZWICK, Approximate distance oracles, in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 183-192.
46. P. M. VAIDYA, An o(n log n) algorithm for the all-nearest-neighbors problem, Discrete Comput. Geom., 4 (1989), pp. 101-115.
47. A. L. VOL'BERG AND S. V. KONYAGIN, On measures with the doubling condition, Izv. Akad. Nauk SSSR Ser. Mat, 51 (1987), pp. 666-675.
48. J.-M. Wu, Hausdorff dimension and doubling measures on metric spaces, Proc. Amer. Math. Soc., 126 (1998), pp. 1453-1459.