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).
- Sanjeev Arora and Boaz Barak. 2009. Computational Complexity: A Modern Approach. Cambridge University Press. Google Scholar
Cross Ref
- 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 Scholar
Digital Library
- Maria-Florina Balcan, Avrim Blum, and Anupam Gupta. 2013. Clustering under approximation stability. J. ACM 60, 2, Article 8 (2013), 34 pages. Google Scholar
Digital Library
- Alex Bavelas. 1950. Communication patterns in task-oriented groups. J. Acoust. Soc. Amer. 22, 6 (1950), 725--730.Google Scholar
Cross Ref
- 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 Scholar
Digital Library
- Ulrik Brandes. 2001. A faster algorithm for betweenness centrality. J. Math. Sociol. 25, 2 (2001), 163--177.Google Scholar
Cross Ref
- Ulrik Brandes. 2008. On variants of shortest-path betweenness centrality and their generic computation. Social Netw. 30, 2 (2008), 136--145.Google Scholar
Cross Ref
- Ulrik Brandes and Christian Pich. 2007. Centrality estimation in large networks. Int. J. Bifurcat. Chaos 17, 7 (2007), 2303--2318.Google Scholar
Cross Ref
- Jack Brimberg. 1995. The fermat--weber location problem revisited. Math. Program. 71, 1 (1995), 71--76. Google Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- Ching-Lueh Chang. 2012. Some results on approximate -median selection in metric spaces. Theoret. Comput. Sci. 426 (2012), 1--12. Google Scholar
Digital Library
- Ching-Lueh Chang. 2013. Deterministic sublinear-time approximations for metric 1-median selection. Inform. Process. Lett. 113, 8 (2013), 288--292. Google Scholar
Digital Library
- Ching-Lueh Chang. 2015. A deterministic sublinear-time nonadaptive algorithm for metric -median selection. Theoret. Comput. Sci. 602 (2015), 149--157. Google Scholar
Digital Library
- Ching-Lueh Chang. 2017. A lower bound for metric -median selection. J. Comput. System Sci. 84, C (2017), 44--51. Google Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. 2009. Introduction to Algorithms (3rd ed.). The MIT Press. Google Scholar
Digital Library
- 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 Scholar
Digital Library
- Artur Czumaj and Christian Sohler. 2010. Sublinear-time algorithms. In Property Testing, O. Goldreich (Ed.). Springer-Verlag, Berlin, Heidelberg, 41--64. Google Scholar
Digital Library
- Shlomi Dolev, Yuval Elovici, and Rami Puzis. 2010. Routing betweenness centrality. J. ACM 57, 4, Article 25 (2010), 27 pages. Google Scholar
Digital Library
- David Eppstein and Joseph Wang. 2004. Fast approximation of centrality. Journal of Graph Algorithms and Applications 8, 1 (2004), 39--45.Google Scholar
Cross Ref
- Eldar Fischer. 2001. The art of uninformed decisions: A primer to property testing. Bull. Eur. Assoc. Theoret. Comput. Sci. 75 (2001), 97--126.Google Scholar
- Linton C. Freeman. 1978. Centrality in social networks: Conceptual clarification. Social Netw. 1, 3 (1978), 215--239.Google Scholar
Cross Ref
- Oded Goldreich and Dana Ron. 2008. Approximating average parameters of graphs. Random Struct. Algor. 32, 4 (2008), 473--493. Google Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
- 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 Scholar
Digital Library
- Piotr Indyk. 2000. High-Dimensional Computational Geometry. Ph.D. Dissertation. Stanford University.Google Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- Valentine Kabanets and Russell Impagliazzo. 2004. Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complex. 13, 1--2 (2004), 1--46. Google Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
Digital Library
- Ramgopal R. Mettu and Charles Greg Plaxton. 2004. Optimal time bounds for approximate clustering. Mach. Learn. 56, 1--3 (2004), 35--60. Google Scholar
Digital Library
- 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 Scholar
- Ronitt Rubinfeld and Asaf Shapira. 2011. Sublinear time algorithms. SIAM J. Discrete Math. 25, 4 (2011), 1562--1588. Google Scholar
Digital Library
- Gert Sabidussi. 1966. The centrality index of a graph. Psychometrika 31, 4 (1966), 581--603.Google Scholar
Cross Ref
- 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 Scholar
Cross Ref
- Leslie G. Valiant. 1982. A scheme for fast parallel communication. SIAM J. Comput. 11, 2 (1982), 350--361.Google Scholar
Digital Library
- Stanley Wasserman and Katherine Faust. 1994. Social Network Analysis: Methods and Applications. Cambridge University Press.Google Scholar
- Bang Ye Wu. 2014. On approximating metric -median in sublinear time. Inform. Process. Lett. 114, 4 (2014), 163--166. Google Scholar
Digital Library
Index Terms
Metric 1-Median Selection: Query Complexity vs. Approximation Ratio
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