Algorithms on negatively curved spaces

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

Volumn , Issue , 2006, Pages 119-128

  Krauthgamer, Robert ^a   Lee, Jmes R ^b  

^a IBM ALMADEN RESEARCH CENTER   (United States)

^b INSTITUTE FOR ADVANCED STUDY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

GROMOV'S (COMBINATORIAL) HYPERBOLIC CONDITION; NEGATIVELY CURVED SPACES; REAL-HYPERBOLIC SPACE;

BANACH SPACES; COMPUTER SCIENCE; COMPUTER VISION; TRAVELING SALESMAN PROBLEM;

APPROXIMATION ALGORITHMS;

EID: 38049022753     PISSN: 02725428     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/FOCS.2006.9     Document Type: Conference Paper

References (27)

1. J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short. Notes on word hyperbolic groups. In Group theory from a geometrical viewpoint (Trieste, 1990), pages 3-63. World Sci. Publishing, River Edge, NJ, 1991. Edited by H. Short.
2. S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM, 45(5):753-782, 1998.
3. 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(6):891-923, 1998.
4. n. Bull. Soc. Math. France, 111(4):429-448, 1983.
5. Y. Bartal. Probabilistic approximations of metric space and its algorithmic application. In 37th Annual Symposium on Foundations of Computer Science, pages 183-193, Oct. 1996.
6. E. Begelfor and M. Werman. The world is not always flat or learning curved manifolds. Technical Report 65, Leibniz Center, The Hebrew University of Jerusalem, 2005.
7. A. Beygelzimer, S. Kakade, and J. Langford. Cover trees for nearest neighbor. To appear, ICML 2006. Source code at http://hunch.net/-jl/projects/ cover_tree/cover_tree.html.
8. M. Bonk and O. Schramm. Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal., 10(2):266-306, 2000.
9. S. Buyalo and V. Schroeder. Embedding of hyperbolic spaces in the product of trees. Geom. Dedicata, 113:75-93, 2005.
10. S. Buyalo and V. Schroeder. A product of trees as universal space for hyperbolic groups. Preprint, arxiv:math.GR/0509355, 2005.
11. H. T.-H. Chan, A. Gupta, B. Maggs, and S. Zhou. On hierarchical routing in doubling metrics. In Proceedings of the 16th Annual Symposium on Discrete Algorithms, 2005.
12. R. Cole and L. Gottlieb. Searching dynamic point sets in spaces with bounded doubling dimension. In Proceedings of the 38th Annual Symposium on the Theory of Computing, 2006.
13. M. Elkin and D. Peleg. (1 + ∈, β )-spanner constructions for general graphs. SIAM J. Comput., 33(3):608-631, 2004.
14. É. Ghys and P. de la Harpe, editors. Sur les groupes hyperboliques d'après Mikhael Gromov, volume 83 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1990. Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988.
15. M. Gromov. Hyperbolic groups. In Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ., pages 75-263. Springer, New York, 1987.
16. A. Gupta, R. Krauthgamer, and J. R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In 44th Annual IEEE Symposium on Foundations of Computer Science, pages 534-543, Oct. 2003.
17. S. Har-Peled and M. Mendel. Fast construction of nets in low dimensional metrics, and their applications. In Proceedings of the 21st Annual Symposium on Computational Geometry, 2005.
18. D. Karger and M. Ruhl. Finding nearest neighbors in growth-restricted metrics. In Proceedings of the 34th Annual ACM Symposium on the Theory of Computing, pages 63-66, 2002.
19. R. Krauthgamer and J. R. Lee. Navigating nets: Simple algorithms for proximity search. In 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 791-801, Jan. 2004.
20. R. Krauthgamer and J. R. Lee. The black-box complexity of nearest-neighbor search. Theoret. Comput. Sci., 348(2-3):262-276, 2005.
21. R. Krauthgamer, J. R. Lee, M. Mendel, and A. Naor. Measured descent: A new embedding method for finite metrics. Geom. Funct. Anal., 15(4):839-858, 2005.
22. J. M. Lee. Riemannian manifolds, volume 176 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. An introduction to curvature.
23. J. R. Lee and A. Naor. Extending Lipschitz functions via random metric partitions. Invent. Math., 160(1):59-95, 2005.
24. S. Rao. Small distortion and volume preserving embeddings for planar and Euclidean metrics. In Proceedings of the 15 th Annual Symposium on Computational Geometry, pages 300-306. ACM, 1999.
25. E. Sharon and D. Mumford. 2d-shape analysis using conformal mapping. In CVPR (2), pages 350-357, 2004.
26. Y. Shavitt and T. Tankel. On the curvature of the internet and its usage for overlay construction and distance estimation. In Proceedings of the 23th Annual INFOCOM, 2004.
27. K. Talwar. Bypassing the embedding: algorithms for low dimensional metrics. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 281-290, New York, 2004. ACM.