Metric embeddings with relaxed guarantees

SIAM Journal on Computing

Volumn 38, Issue 6, 2009, Pages 2303-2329

  Chan, T H Hubert ^a   Dhamdhere, Kedar ^b   Gupta, Anupam ^c   Kleinberg, Jon ^d   Slivkins, Aleksandrs ^e  

^a MAX PLANCK INSTITUTE FOR INFORMATICS   (Germany)

^b GOOGLE INC   (United States)

^c Carnegie Mellon University   (United States)

^d Department of Computer Science and School of Operations Research and Information Engineering   (United States)

^e MICROSOFT RESEARCH   (United States)

Author keywords

Low distortion embeddings; Metric decompositions; Metric embeddings; Metric spaces; Randomized algorithms

Indexed keywords

LOW-DISTORTION EMBEDDINGS; METRIC DECOMPOSITIONS; METRIC EMBEDDINGS; METRIC SPACES; RANDOMIZED ALGORITHMS;

ALGORITHMS; CONTROL THEORY; SET THEORY; TOPOLOGY;

METRIC SYSTEM;

EID: 65949122721     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/060670511     Document Type: Article

References (44)

1. I. ABRAHAM, Y. BARTAL, T.-H. H. CHAN, K. DHAMDHERE, A. GUPTA, J. KLEINBERG, O. NEIMAN, AND A. SLIVKINS, Metric embeddings with relaxed guarantees, in Proceedings ofthe 46th IEEE Symposium on Foundations of Computer Science (FOCS), Pittsburgh, PA, 2005, pp. 83-100.
2. I. ABRAHAM, Y. BARTAL, AND O. NEIMAN, Advances in metric embedding theory, in Proceedings of the 38th ACM Symposium on Theory of Computing (STOC), Seattle, WA, 2006.
3. N. ALON, R. M. KARP, D. PELEG, AND D. WEST, A graph-theoretic game and its application to the k-server problem, SIAM J. Comput., 24 (1995), pp. 78-100.
4. n, Bull. Soc. Math. France, 111 (1983), pp. 429-448.
5. Y. BARTAL, Probabilistic approximations of metric spaces and its algorithmic applications, in Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS), Burlington, VT, 1996, pp. 184-193.
6. Y. BARTAL, On approximating arbitrary metrics by tree metrics, in Proceedings of the 30th ACM Symposium on Theory of Computing (STOC), Dallas, TX, 1998, pp. 161-168.
7. Y. BARTAL, Graph decomposition lemmas and their role in metric embedding methods, in Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA), Bergen, Norway, 2004, pp. 89-97.
8. Y. BARTAL, B. BOLLOBÁS, AND M. MENDEL, Ramsey-type theorems for metric spaces with applications to online problems, J. Comput. System Sci., 72 (2002), pp. 890-921. Preliminary version in 42nd IEEE FOCS, 2001.
9. Y. BARTAL, N. LINIAL, M. MENDEL, AND A. NAOR, On metric Ramsey-type phenomena, Ann. of Math., 162 (2005), pp. 643-709. Preliminary version in 35th ACM STOC, 2003.
10. J. BOURGAIN, On Lipschitz embeddings of finite metric spaces in Hilbert space, Israel J. Math., 52 (1985), pp. 46-52.
11. J. BOURGAIN, T. FIGIEL, AND V. MILMAN, On Hilbertian subsets of finite metric spaces, Israel J. Math., 55 (1986), pp. 147-152.
12. 1, J. ACM, 52 (2005), pp. 766-788. Preliminary version in 44th IEEE FOCS, 2003
13. F. DABEK, R. COX, F. KAASHOEK, AND R. MORRIS, Vivaldi: A decentralized network coordinate system, in Proceedings of the ACM Special Interest Group on Data Communications (SIGCOMM), Portland, OR, 2004.
14. J. FAKCHAROENPHOL, S. RAO, AND K. TALWAR, A tight bound on approximating arbitrary metrics by tree metrics, J. Comput. System Sci., 69 (2004), pp. 485-497. Preliminary version in 35th ACM STOC, 2003.
15. J. FAKCHAROENPHOL AND K. TALWAR, An improved decomposition theorem for graphs excluding a fixed minor, in Proceedings of the RANDOM-APPROX, Princeton, NJ, Springer-Verlag, 2003, pp. 36-46.
16. M. FOMENKOV, L. CLAFFY, B. HUFFAKER, AND D. MOORE, Macroscopic internet topology and performance measurements from the DNS root name servers, in Proceedings of the Usenix Large Installation System Administration (LISA) Conference, San Diego, CA, 2001.
17. P. FRANCIS, S. JAMIN, C. JIN, Y. JIN, D. RAZ, Y. SHAVITT, AND L. ZHANG, IDMaps: A global Internet host distance estimation service, IEEE/ACM Trans. Networking, 9 (2001), pp. 525-540. Preliminary version in IEEE INFOCOM 1999.
18. A. GUPTA, Steiner points in tree metrics don't (really) help, in Proceedings of the 12th ACMSIAM Symposium on Discrete Algorithms (SODA), Philadelphia, 2001, pp. 220-227.
19. A. GUPTA, R. KRAUTHGAMER, AND J. R. LEE, Bounded geometries, fractals, and low-distortion embeddings, in Proceedings of the 44th IEEE Symposium on Foundations of Computer Science (FOCS), Cambridge, MA, 2003, pp. 534-543.
20. 1-embeddings of graphs, Combinatorica, 24 (2004), pp. 233-269. Preliminary version in 40th FOCS, 1999.
21. J. D. GUYTON AND M, F. SCHWARTZ, Locating nearby copies of replicated Internet servers, in Proceedings of the ACM Special Interest Group on Data Communications (SIGCOMM), Cambridge, MA, 1995.
22. S. HAR-PELED AND M. MENDEL, Fast construction of nets in low dimensional metrics and their applications, SIAM J. Comput., 35 (2006), pp. 1148-1184. Preliminary version in 21st ACM SoCG, 2005.
23. J. HEINONEN, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001.
24. P. INDYK, Algorithmic applications of low-distortion geometric embeddings, in Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science (FOCS), Los Vegas, NV, 2001, pp. 10-33.
25. P. INDYK AND J. MATOUŠEK, Low-distortion embeddings of finite metric spaces, in Handbook of Discrete and Computational Geometry, 2nd ed., J. E. Goodman and J. O'Rourke, eds., Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2004, pp. 177-196.
26. D. R. KARGER AND M. RUHL, Finding nearest neighbors in growth-restricted metrics, in Proceedings of the 34th ACM Symposium on Theory of Computing (STOC), Montreal, Canada, 2002, pp. 63-66.
27. P. KLEIN, S. A. PLOTKIN, AND S. B. RAO, Excluded minors, network decomposition, and multicommodity flow, in Proceedings of the 25th ACM Symposium on Theory of Computing (STOC), 1993, San Diego, CA, pp. 682-690.
28. J. KLEINBERG, A. SLIVKINS, AND T. WEXLER, Triangulation and embedding using small sets of beacons, in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science (FOCS), Rome, Italy, 2004, pp. 444-453.
29. C. KOMMAREDDY, N. SHANKAR, AND B. BHATTACHARJEE, Finding close friends on the Internet, in Proceedings of the 12th IEEE International Conference on Network Protocols (ICNP), Berlin, Germany, 2001.
30. R. KRAUTHGAMER, J. LEE, M. MENDEL, AND A. NAOR, Measured descent: A new embedding method for finite metrics, Geom. Funct. Anal., 15 (2005), pp. 839-858. Preliminary version in 45th IEEE FOCS, 2004.
31. J. R. LEE, On distance scales, embeddings, and efficient relaxations of the cut cone, in Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA), Vancouver, Canada, 2005, pp. 92-101.
32. J. R. LEE, M. MENDEL, AND A. NAOR, Metric structures in L1: Dimension, snowflakes, and average distortion, European J. Combin., 6 (2005), pp. 1180-1190. Preliminary version in LATIN, 2004.
33. N. LINIAL, E. LONDON, AND Y. RABINOVICH, The geometry of graphs and some of its algorithmic applications, Combinatorica, 15 (1995), pp. 215-245. Preliminary version in 35th IEEE FOCS, 1994.
34. p spaces, Israel J. Math., 102 (1997), pp. 189-197.
35. J. MATOUŠEK, Lectures on Discrete Geometry, Grad. Texts in Math. 212, Springer-Verlag, New York, 2002.
36. J. MATOUŠEK, Bi-Lipschitz embeddings into low dimensional Euclidean spaces, Comment. Math. Univ. Carolin., 31 (1990), pp. 589-600.
37. T. S. E. NG AND H. ZHANG, Predicting Internet network distance with coordinates-based approaches, in Proceedings of the 21st Joint Conference of the IEEE Computer and Communication (INFOCOM), New York, 2002.
38. T. S. E. NG AND H. ZHANG, A network positioning system for the Internet, in Proceedings of the USENIX, Boston, 2004.
39. C. G. PLAXTON, R. RAJARAMAN, AND A. W. RICHA, Accessing nearby copies of replicated objects in a distributed environment, Theory Comput. Syst., 32 (1999), pp. 241-280. Preliminary version in 9th ACM SPAA, 1997.
40. Y. RABINOVICH AND R. RAZ, Lower bounds on the distortion of embedding finite metric spaces in graphs, Discrete Comput. Geom., 19 (1998), pp. 79-94.
41. S. B. RAO, Small distortion and volume preserving embeddings for planar and Euclidean metrics, in Proceedings of the 15th ACM Symposium on Computational Geometry (SoCG), Miami, FL, 1999, pp. 300-306.
42. Y. SHAVITT AND T. TANKEL, Big-bang simulation for embedding network distances in Euclidean space, IEEE/ACM Trans. Netw., 12 (2004), pp. 993-1006. Preliminary version in 22nd IEEE INFOCOM, 2003.
43. A. SLIVKINS, Distributed approaches to triangulation and embedding, in Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA), Vancouver, Canada, 2005, pp. 640-649.
44. K. TALWAR, Bypassing the embedding: Algorithms for low-dimensional metrics, in Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago, IL, 2004, pp. 281-290.