skip to main content
research-article

Metric 1-Median Selection: Query Complexity vs. Approximation Ratio

Authors Info & Claims
Published:14 December 2017Publication History
Skip Abstract Section

Abstract

Consider the problem of finding a point in a metric space ({ 1,2,…, n}, d) with the minimum average distance to other points. We show that this problem has no deterministic o(n1+1/(h-1)/h)-query 2h· (1-ϵ))-approximation algorithms for any constant ϵ >0 and any h=h(n)∈ Z+ \ {1} satisfying h=o(n1/(h-1)). Combining our result with existing ones, we determine the best approximation ratio achievable by deterministic O(n1+ϵ)-query (respectively, O(n1+ϵ)-time) algorithms to be 2⌈ 1/ϵ ⌉, for all constants ϵ ∈ (0,1).

References

  1. Sanjeev Arora and Boaz Barak. 2009. Computational Complexity: A Modern Approach. Cambridge University Press. Google ScholarGoogle ScholarCross RefCross Ref
  2. David A. Bader and Kamesh Madduri. 2006. Parallel algorithms for evaluating centrality indices in real-world networks. In Proceedings of the 2006 International Conference on Parallel Processing (ICPP’06). Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Maria-Florina Balcan, Avrim Blum, and Anupam Gupta. 2013. Clustering under approximation stability. J. ACM 60, 2, Article 8 (2013), 34 pages. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Alex Bavelas. 1950. Communication patterns in task-oriented groups. J. Acoust. Soc. Amer. 22, 6 (1950), 725--730.Google ScholarGoogle ScholarCross RefCross Ref
  5. Prosenjit Bose, Anil Maheshwari, and Pat Morin. 2003. Fast approximations for sums of distances, clustering and the Fermat--Weber problem. Comput. Geom. 24, 3 (2003), 135--146. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Ulrik Brandes. 2001. A faster algorithm for betweenness centrality. J. Math. Sociol. 25, 2 (2001), 163--177.Google ScholarGoogle ScholarCross RefCross Ref
  7. Ulrik Brandes. 2008. On variants of shortest-path betweenness centrality and their generic computation. Social Netw. 30, 2 (2008), 136--145.Google ScholarGoogle ScholarCross RefCross Ref
  8. Ulrik Brandes and Christian Pich. 2007. Centrality estimation in large networks. Int. J. Bifurcat. Chaos 17, 7 (2007), 2303--2318.Google ScholarGoogle ScholarCross RefCross Ref
  9. Jack Brimberg. 1995. The fermat--weber location problem revisited. Math. Program. 71, 1 (1995), 71--76. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Domenico Cantone, Gianluca Cincotti, Alfredo Ferro, and Alfredo Pulvirenti. 2005. An efficient approximate algorithm for the -median problem in metric spaces. SIAM J. Optim. 16, 2 (2005), 434--451. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Domenico Cantone, Alfredo Ferro, Rosalba Giugno, Giuseppe L. Presti, and Alfredo Pulvirenti. 2005. Multiple-winners randomized tournaments with consensus for optimization problems in generic metric spaces. In Proceedings of the 4th International Workshop on Experimental and Efficient Algorithms (WEA’05). 265--276. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Shu Yan Chan, Ian X. Y. Leung, and Pietro Liò. 2009. Fast centrality approximation in modular networks. In Proceedings of the 18th ACM Conference on Information and Knowledge Management (CIKM’09). 31--38. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Ching-Lueh Chang. 2012. Some results on approximate -median selection in metric spaces. Theoret. Comput. Sci. 426 (2012), 1--12. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Ching-Lueh Chang. 2013. Deterministic sublinear-time approximations for metric 1-median selection. Inform. Process. Lett. 113, 8 (2013), 288--292. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Ching-Lueh Chang. 2015. A deterministic sublinear-time nonadaptive algorithm for metric -median selection. Theoret. Comput. Sci. 602 (2015), 149--157. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Ching-Lueh Chang. 2017. A lower bound for metric -median selection. J. Comput. System Sci. 84, C (2017), 44--51. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Ching-Lueh Chang. 2017. Metric 1-median selection with fewer queries. In Proceedings of the 2017 IEEE International Conference on Applied System Innovation. 1056--1059.Google ScholarGoogle ScholarCross RefCross Ref
  18. Moses Charikar, Sudipto Guha, Éva Tardos, and David B. Shmoys. 2002. A constant-factor approximation algorithm for the k-median problem. J. Comput. System Sci. 65, 1 (2002), 129--149. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Moses Charikar, Liadan O’Callaghan, and Rina Panigrahy. 2003. Better streaming algorithms for clustering problems. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC’03). 30--39. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Shiri Chechik. 2015. Approximate distance oracles with improved bounds. In Proceedings of the 47th Annual Symposium on the Theory of Computing (STOC’15). 1--10. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Shiri Chechik, Edith Cohen, and Haim Kaplan. 2015. Average distance queries through weighted samples in graphs and metric spaces: High scalability with tight statistical guarantees. In Proceedings of the 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and the 19th International Workshop on Randomization and Computation (APPROX-RANDOM’15). 659--679.Google ScholarGoogle Scholar
  22. Ke Chen. 2009. On coresets for k-median and k-means clustering in metric and Euclidean spaces and their applications. SIAM J. Comput. 39, 3 (2009), 923--947. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Flavio Chierichetti, Ravi Kumar, Sandeep Pandey, and Sergei Vassilvitskii. 2010. Finding the jaccard median. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’10). 293--311. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. 2009. Introduction to Algorithms (3rd ed.). The MIT Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Artur Czumaj and Christian Sohler. 2007. Sublinear-time approximation algorithms for clustering via random sampling. Random Structures 8 Algorithms 30, 1--2 (2007), 226--256. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Artur Czumaj and Christian Sohler. 2010. Sublinear-time algorithms. In Property Testing, O. Goldreich (Ed.). Springer-Verlag, Berlin, Heidelberg, 41--64. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. Shlomi Dolev, Yuval Elovici, and Rami Puzis. 2010. Routing betweenness centrality. J. ACM 57, 4, Article 25 (2010), 27 pages. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. David Eppstein and Joseph Wang. 2004. Fast approximation of centrality. Journal of Graph Algorithms and Applications 8, 1 (2004), 39--45.Google ScholarGoogle ScholarCross RefCross Ref
  29. Eldar Fischer. 2001. The art of uninformed decisions: A primer to property testing. Bull. Eur. Assoc. Theoret. Comput. Sci. 75 (2001), 97--126.Google ScholarGoogle Scholar
  30. Linton C. Freeman. 1978. Centrality in social networks: Conceptual clarification. Social Netw. 1, 3 (1978), 215--239.Google ScholarGoogle ScholarCross RefCross Ref
  31. Oded Goldreich and Dana Ron. 2008. Approximating average parameters of graphs. Random Struct. Algor. 32, 4 (2008), 473--493. Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Sudipto Guha, Adam Meyerson, Nina Mishra, Rajeev Motwani, and Liadan O’Callaghan. 2003. Clustering data streams: Theory and practice. IEEE Trans. Knowl. Data Eng. 15, 3 (2003), 515--528. Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. Sudipto Guha and Nina Mishra. 2016. Clustering data streams. In Data Stream Management: Processing High-Speed Data Streams, M. Garofalakis, J. Gehrke, and R. Rastogi (Eds.). Springer-Verlag, Berlin, 169--187.Google ScholarGoogle Scholar
  34. Piotr Indyk. 1999. Sublinear time algorithms for metric space problems. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC’99). 428--434. Google ScholarGoogle ScholarDigital LibraryDigital Library
  35. Piotr Indyk. 2000. High-Dimensional Computational Geometry. Ph.D. Dissertation. Stanford University.Google ScholarGoogle Scholar
  36. Kamal Jain, Mohammad Mahdian, and Amin Saberi. 2002. A new greedy approach for facility location problems. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC’02). 731--740. Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. Kamal Jain and Vijay V. Vazirani. 2001. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM 48, 2 (2001), 274--296. Google ScholarGoogle ScholarDigital LibraryDigital Library
  38. Valentine Kabanets and Russell Impagliazzo. 2004. Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complex. 13, 1--2 (2004), 1--46. Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. Christos Kaklamanis, Danny Krizanc, and Thanasis Tsantilas. 1991. Tight bounds for oblivious routing in the hypercube. Math. Syst. Theory 24, 1 (1991), 223--232.Google ScholarGoogle ScholarCross RefCross Ref
  40. Amit Kumar, Yogish Sabharwal, and Sandeep Sen. 2010. Linear-time approximation schemes for clustering problems in any dimensions. J. ACM 57, 2, Article 5 (2010), 32 pages. Google ScholarGoogle ScholarDigital LibraryDigital Library
  41. Ramgopal R. Mettu and Charles Greg Plaxton. 2004. Optimal time bounds for approximate clustering. Mach. Learn. 56, 1--3 (2004), 35--60. Google ScholarGoogle ScholarDigital LibraryDigital Library
  42. François Nicolas and Eric Rivals. 2005. Hardness results for the center and median string problems under the weighted and unweighted edit distances. J. Discrete Algor. 3, 2--4 (2005), 390--415.Google ScholarGoogle Scholar
  43. Ronitt Rubinfeld and Asaf Shapira. 2011. Sublinear time algorithms. SIAM J. Discrete Math. 25, 4 (2011), 1562--1588. Google ScholarGoogle ScholarDigital LibraryDigital Library
  44. Gert Sabidussi. 1966. The centrality index of a graph. Psychometrika 31, 4 (1966), 581--603.Google ScholarGoogle ScholarCross RefCross Ref
  45. Robert Scheidweiler and Eberhard Triesch. 2013. A lower bound for the complexity of monotone graph properties. SIAM J. Discrete Math. 27, 1 (2013), 257--265.Google ScholarGoogle ScholarCross RefCross Ref
  46. Leslie G. Valiant. 1982. A scheme for fast parallel communication. SIAM J. Comput. 11, 2 (1982), 350--361.Google ScholarGoogle ScholarDigital LibraryDigital Library
  47. Stanley Wasserman and Katherine Faust. 1994. Social Network Analysis: Methods and Applications. Cambridge University Press.Google ScholarGoogle Scholar
  48. Bang Ye Wu. 2014. On approximating metric -median in sublinear time. Inform. Process. Lett. 114, 4 (2014), 163--166. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Metric 1-Median Selection: Query Complexity vs. Approximation Ratio

    Recommendations

    Comments

    Login options

    Check if you have access through your login credentials or your institution to get full access on this article.

    Sign in

    Full Access

    PDF Format

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader
    About Cookies On This Site

    We use cookies to ensure that we give you the best experience on our website.

    Learn more

    Got it!