Triangulation and embedding using small sets of beacons

Journal of the ACM

Volumn 56, Issue 6, 2009, Pages

  Kleinberg, Jon ^a   Slivkins, Aleksandrs ^b   Wexler, Tom ^c  

^a Department of Obstetrics Gynecology   (United States)

^b MICROSOFT RESEARCH   (United States)

^c DENISON UNIVERSITY   (United States)

Author keywords

Distributed algorithms; Doubling dimension; Embeddings; Metric spaces; Triangulation

Indexed keywords

EID: 70349637877     PISSN: 00045411     EISSN: 1557735X     Source Type: Journal    
DOI: 10.1145/1568318.1568322     Document Type: Article

References (59)

1. Abraham, I., Bartal, Y., Chan, H. T.-H., Dhamdhere, K., Gupta, A., Kleinberg, J., Neiman, O., and Slivkins, A. 2005. Metric embeddings with relaxed guarantees. In Proceedings of the 46th IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society Press, Los Alamitos, CA, 83-100.
2. Abraham, I., Bartal, Y., and Neiman, O. 2006. Advances in metric embedding theory. In Proceedings of the 38th ACM Symposium on Theory of Computing (STOC). ACM, New York.
3. Abraham, I., and Malkhi, D. 2005. Name independent routing for growth bounded networks. In Proceedings of the 17th ACM Symposium on Parallel Algorithms and Architectures (SPAA). ACM, New York, 49-55.
4. Alon, N., and Spencer, J. 2000. The Probabilistic Method, 2nd ed. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York.
5. n. Bull. Soc. Math. France 111, 4, 429-448.
6. Bartal, Y., Linial, N., Mendel, M., and Naor, A. 2005. On metric ramsey-type phenomena. Ann. Math. 162, 2, 643-709. (Preliminary version in 35th ACM STOC, 2003.)
7. Bartal, Y., and Mendel, M. 2004. Dimension reduction for ultrametrics. In Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA). ACM, New York, 664-665.
8. Bourgain, J. 1985. On Lipschitz embeddings of finite metric spaces in Hilbert space. Israel J. Math. 52, 1-2, 46-52.
9. Chan, H. T.-H., Dhamdhere, K., Gupta, A., Kleinberg, J., and Slivkins, A. 2009. Metric embeddings with relaxed guarantees. SIAM J. Comput. 38, 6 (Mar.), 2303-2329. (Preliminary version (merged with an independent effort by another research group) has appeared in 46th IEEE FOCS, 2005.)
10. Crippen, G., and Havel, T. 1988. Distance Geometry and Molecular Conformation. Wiley, New York.
11. Dabek, F., Cox, R., Kaashoek, F., and Morris, R. 2004. Vivaldi: A decentralized network coordinate system. In Proceedings of the ACM SIGCOMM. ACM, New York.
12. p-spaces. Ark. Mat. 8, 103-105.
13. Figiel, T., Lindenstrauss, J., and Milman, V. 1977. The dimension of almost spherical sections of convex bodies. Acta Math. 139, 53-94.
14. Fomenkov, M., Claffy, K., Huffaker, B., and Moore, D. 2001. Macroscopic Internet topology and performance measurements from the DNS root name servers. In Usenix LISA.
15. Francis, P., Jamin, S., Jin, C., Jin, Y., Raz, D., Shavitt, Y., and Zhang, L. 2001. IDMaps: A global Internet host distance estimation service. IEEE/ACM Trans. Netw. 9, 525-540. (Preliminary version in IEEE INFOCOM 1999.)
16. Gavoille, C., Peleg, D., Perennes, S., and Raz, R. 2004. Distance labeling in graphs. J. Algorithms 53, 1, 85-112. (Preliminary version in 12th ACM-SIAM SODA, 2001).
17. Goldreich, O., Goldwasser, S., and Ron:, D. 1998. Property testing and its connection to learning and approximation. J. ACM 45, 4, 653-750. (Preliminary version in 37th IEEE FOCS, 2006.)
18. Gummadi, K. P., Saroiu, S., and Gribble, S. D. 2002. King: Estimating latency between arbitrary internet end hosts. In Proceedings of the 2nd ACM SIGCOMM Internet Measurement Workshop (IMW). ACM, New York, 5-18.
19. Gupta, A., Krauthgamer, R., and Lee, J. R. 2003. Bounded geometries, fractals, and low-distortion embeddings. In Proceedings of the 44th IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society Press, Los Alamitos, CA, 534-543.
20. Guyton, J., and Schwartz, M. 1995. Locating nearby copies of replicated Internet servers. In Proceedings of the ACM SIGCOMM. ACM, New York.
21. Hildrum, K., Kubiatowicz, J., Ma, S., and Rao, S. 2004. A note on the nearest neighbor in growth-restricted metrics. In Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA). ACM, New York, 560-561.
22. Huffaker, B., Fomenkov, M., Plummer, D., Moore, D., and Claffy, K. 2002. Distance metrics in the internet. In Proceedings of the IEEE International Telecommunications Symposium (ITS). IEEE Computer Society Press, Los Alamitos, CA.
23. Indyk, P. 1999. Sublinear time algorithms for metric space problems. In Proceedings of the 31st ACM Symposium on Theory of Computing (STOC). ACM, New York, 428-434.
24. Indyk, P. 2001. Algorithmic applications of low-distortion geometric embeddings. In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society Press, Los Alamitos, CA. 10-33. (Invited survey).
25. Indyk, P., and Matouek, J. 2004. Low-distortion embeddings of finite metric spaces. In Handbook of Discrete and Computational Geometry, Second ed., J. E. Goodman and J. O'Rourke, Eds. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, Chapter 8, xviii+1539.
26. Johnson, W. B., and Lindenstrauss, J. 1984. Extensions of Lipschitz maps into a Hilbert space. Contemp. Math. 26, 189-206.
27. Karger, D., and Ruhl, M. 2002. Finding nearest neighbors in growth-restricted metrics. In Proceedings of the 34th ACM Symposium on Theory of Computing (STOC). ACM, New York, 63-66.
28. Kempe, D., Kleinberg, J., and Demers, A. 2004. Spatial gossip and resource location protocols. J. ACM 51, 6, 943-967. (Preliminary version in 33rd ACM STOC, 2001.)
29. Kleinberg, J., Slivkins, A., and Wexler, T. 2004. Triangulation and embedding using small sets of beacons. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society Press, Los Alamitos, CA, 444-453.
30. Kommareddy, C., Shankar, N., and Bhattacharjee, B. 2001. Finding close friends on the Internet. In Proceedings of the 12th IEEE International Conference on Network Protocols (ICNP). IEEE Computer Society Press, Los Alamitos, CA.
31. Krauthgamer, R., Lee, J., Mendel, M., and Naor, A. 2005. Measured descent: A new embedding method for finite metrics. Geomet. Funct. Anal. (GAFA) 15, 4, 839-858. (Preliminary version in 45th IEEE FOCS, 2004.)
32. Krauthgamer, R., and Lee, J. R. 2003. The intrinsic dimensionality of graphs. In Proceedings of the 35th ACM Symposium on Theory of Computing (STOC). ACM, New York, 438-447.
33. Krauthgamer, R., and Lee, J. R. 2004. Navigating nets: Simple algorithms for proximity search. In Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA). ACM, New York, 798-807.
34. Krauthgamer, R., and Sasson, O. 2003. Property testing of data dimensionality. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms. ACM, New York, 18-27.
35. N. Bull. London Math. Soc. 34, 6, 667-676.
36. Lang, U., and Plaut, C. 2001. Bilipschitz embeddings of metric spaces into space forms. Geom. Dedicata 87, 1-3, 285-307.
37. Linial, N. 2002. Finite metric spacescombinatorics, geometry and algorithms. In Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002). Beijing, China, 573-586.
38. Linial, N., London, E., and Rabinovich, Y. 1995. The geometry of graphs and some of its algorithmic applications. Combinatorica 15, 2, 215-245. (Preliminary version in 35th IEEE FOCS, 1994.)
39. Linial, N., and Wigderson, A. 2002. Course notes: Expander graphs and their applications. http://www.math.ias.edu/boaz/ExpanderCourse/.
40. p spaces. Israel J. Math. 102, 189-197.
41. Matouek, J. 2002. Lectures on discrete geometry. Graduate Texts in Mathematics, vol. 212. Springer-Verlag, New York, Chapter 15 (Embedding Finite Metric Spaces into Normed Spaces), 323-362.
42. Mendel, M., and Har-Peled, S. 2005. Fast construction of nets in low dimensional metrics, and their applications. In Proceedings of the 21st ACM Symposium on Computational Geometry (SoCG). ACM, New York, 150-158.
43. Motwani, R., and Raghavan, P. 1995. Randomized algorithms. Cambridge University Press, Cambridge, MA.
44. Ng, T. E., and Zhang, H. 2002. Predicting Internet network distance with coordinates-based approaches. In Proceedings of the IEEE INFOCOM. IEEE Computer Society Press, Los Alamitos, CA.
45. Parnas, M., and Ron, D. 2003. Testing metric properties. Info. Comput. 187, 2, 155-195. (Preliminary version in 33th ACM STOC, 2001.)
46. Percacci, R., and Vespignani, A. 2003. Scale-free behavior of the Internet global performance. Europ. Phys. J. B 32, 4, 411-414. (Also appeared as arXiv e-print cond-mat/0209619, September 2002.)
47. Pias, M., Crowcroft, J., Wilbur, S., Harris, T., and Bhatti, S. 2003. Lighthouses for scalable distributed location. In Proceedings of the 2nd International Workshop on Peer-to-Peer Systems (IPTPS).
48. Pittel, B. 1987. On spreading a rumor. SIAM J. Appl. Math. 47, 1, 213-223.
49. Plaxton, C. G., Rajaraman, R., and Richa, A. W. 1999. Accessing nearby copies of replicated objects in a distributed environment. Theory Comput. Syst. 32, 3, 241-280. (Preliminary version in 9th ACM SPAA, 1997.)
50. Semmes, S. 1996. On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A1-weights. Rev. Mat. Iberoamericana 12, 2, 337-410.
51. Shavitt, Y., and Tankel, T. 2003. Big-bang simulation for embedding network distances in euclidean space. In Proceedings of the IEEE INFOCOM. IEEE Computer Society Press, Los Alamitos, CA.
52. Slivkins, A. 2005. Distributed approaches to triangulation and embedding. In Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA). ACM, New York, 640-649.
53. Slivkins, A. 2006. Embedding, distance estimation and object location in networks. Ph.D. dissertation, Cornell University, Ithaca, NY 14850. (Available online at http://research.microsoft.com/en-us/people/slivkins.)
54. Slivkins, A. 2007a. Distance estimation and object location via rings of neighbors. Distrib. Comput. 19, 4 (Mar.), 313-333. (Special issue for 24th ACM PODC, 2005.)
55. Slivkins, A. 2007b. Towards fast decentralized construction of locality-aware overlay networks. In Proceedings of the 26th Annual ACM Symposium on Principles of Distributed Computing (PODC). ACM, New York, 89-98. (Full version available online at http://research.microsoft.com/en-us/people/slivkins. )
56. Talwar, K. 2004. Bypassing the embedding: approximation schemes and compact representations for growth restricted metrics. In Proceedings of the 36th ACM Symposium on Theory of Computing (STOC). ACM, New York, 281-290.
57. Vazquez, A., Pastor-Satorras, R., and Vespignani, A. 2002. Large-scale topological and dynamical properties of Internet. Phys. Rev. E 65, 066130.
58. Xiao, D. Y. 2003. The evolution of expander graphs. BA dissertation, Harvard University. (Available from http://www.cs.princeton.edu/dxiao.)
59. Zhao, B., Huang, L., Rhea, S., Stribling, J., Joseph, A., and Kubiatowicz, J. 2004. Tapestry: A resilient global-scale overlay for service deployment. IEEE J. Select. Areas Commun. 22, 1, 41-53.