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Sub-logarithmic distributed algorithms for metric facility location

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Abstract

The facility location problem consists of a set of facilities \({\mathcal {F}}\), a set of clients \({\mathcal {C}}\), an opening cost \(f_i\) associated with each facility \(x_i\), and a connection cost \(D(x_i,y_j)\) between each facility \(x_i\) and client \(y_j\). The goal is to find a subset of facilities to open, and to connect each client to an open facility, so as to minimize the total facility opening costs plus connection costs. We consider distributed versions of facility location in which the underlying network is either a clique, in which each node is both a client and a facility (and \({\mathcal {F}} = {\mathcal {C}}\)), or a complete bipartite network, with \(({\mathcal {F}}, {\mathcal {C}})\) forming the bipartition. This paper presents the first expected-sub-logarithmic-round distributed \(O(1)\)-approximation algorithms in the \({\mathcal {CONGEST}}\) model for the metric facility location problem. We present our approach first for a clique network, and then extend it to a bipartite network. Our algorithms have expected running times of \(O((\log \log n)^c)\) rounds, where \(n = |{\mathcal {F}}| + |{\mathcal {C}}|\), and where \(c = 1\) for a clique network and \(c = 3\) for a bipartite network (These results were first published in ICALP 2012 and DISC 2013). In order to obtain these results, our paper makes four main technical contributions. First, we show a new lower bound for metric facility location, extending the lower bound of Bădoiu et al. (ICALP 2005) that applies only to the special case of uniform facility opening costs. Next, we demonstrate a reduction of the distributed metric facility location problem to the problem of computing an \(O(1)\)-ruling set of an appropriate spanning subgraph. Third, we provide an expected-sub-logarithmic-round algorithm for computing a \(2\)-ruling set in a spanning subgraph of a clique. Finally, we present a new technique based on probabilistic hashing to solve a problem of information dissemination on bipartite networks.

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References

  1. Balinski, M.L.: On finding integer solutions to linear programs. In: Proceedings of IBM Scientific Computing Symposium on Combinatorial Problems, pp. 225–248 (1966)

  2. Berns, A., Hegeman, J., Pemmaraju, S.V.: Super-fast distributed algorithms for metric facility location. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) Automata, Languages, and Programming, Lecture Notes in Computer Science, vol. 7392, pp. 428–439. Springer, Berlin Heidelberg (2012)

    Chapter  Google Scholar 

  3. Berns, A., Hegeman, J., Pemmaraju, S.V.: Super-fast distributed algorithms for metric facility location. CoRR (2013). arXiv:1308.2473

  4. Bădoiu, M., Czumaj, A., Indyk, P., Sohler, C.: Facility location in sublinear time. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 3580, pp. 866–877. Springer, Berlin Heidelberg (2005)

    Chapter  Google Scholar 

  5. Cornuejols, G., Nemhouser, G., Wolsey, L.: Discrete Location Theory. Wiley, New York (1990)

    Google Scholar 

  6. Drucker, A., Kuhn, F., Oshman, R.: On the power of the congested clique model. In: Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing, PODC ’14, pp. 367–376. ACM, New York, NY, USA (2014)

  7. Frank, C.: Algorithms for Sensor and Ad Hoc Networks. Springer, Berlin (2007)

    Google Scholar 

  8. Gehweiler, J., Lammersen, C., Sohler, C.: A distributed O(1)-approximation algorithm for the uniform facility location problem. Algorithmica 68(3), 643–670 (2014)

    Article  MathSciNet  Google Scholar 

  9. Guha, S., Khuller, S.: Greedy strikes back: improved facility location algorithms. In: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’98, pp. 649–657. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1998)

  10. Hamburger, M.J., Kuehn, A.A.: A heuristic program for locating warehouses. Manag. Sci. 9(4), 643–666 (1963)

    Article  Google Scholar 

  11. Hegeman, J., Pemmaraju, S.V.: A super-fast distributed algorithm for bipartite metric facility location. In: Afek, Y. (ed.) Distributed Computing, Lecture Notes in Computer Science, vol. 8205, pp. 522–536. Springer, Berlin Heidelberg (2013)

    Google Scholar 

  12. Hegeman, J., Pemmaraju, S.V.: A super-fast distributed algorithm for bipartite metric facility location. CoRR (2013). arXiv:1308.2694

  13. Hegeman, J.W., Pemmaraju, S.V.: Lessons from the congested clique applied to MapReduce. In: Halldórsson, M.M. (ed.) Structural Information and Communication Complexity, Lecture Notes in Computer Science, vol. 8576, pp. 149–164. Springer, Berlin (2014)

    Google Scholar 

  14. Hegeman, J.W., Pemmaraju, S.V., Sardeshmukh, V.B.: Near-constant-time distributed algorithms on a congested clique. In: Kuhn, F. (ed.) Distributed Computing, Lecture Notes in Computer Science, vol. 8784, pp. 514–530. Springer, Berlin Heidelberg (2014)

    Google Scholar 

  15. Karloff, H., Suri, S., Vassilvitskii, S.: A model of computation for mapreduce. In: Proceedings of the Twenty-first Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’10, pp. 938–948. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2010)

  16. Kaufman, L., Eede, M.V., Hansen, P.: A plant and warehouse location problem. Oper. Res. Q. (1970–1977) 28(3), 547–554 (1977)

    Article  MATH  Google Scholar 

  17. Klauck, H., Nanongkai, D., Pandurangan, G., Robinson, P.: Distributed Computation of Large-Scale Graph Problems, chap. 28, pp. 391–410. SODA ’15. Society for Industrial and Applied Mathematics (2015)

  18. Lenzen, C.: Optimal deterministic routing and sorting on the congested clique. In: Proceedings of the 2013 ACM Symposium on Principles of Distributed Computing, PODC ’13, pp. 42–50. ACM, New York, NY, USA (2013)

  19. Lenzen, C., Wattenhofer, R.: Brief announcement: exponential speed-up of local algorithms using non-local communication. In: Proceedings of the 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, PODC ’10, pp. 295–296. ACM, New York, NY, USA (2010)

  20. Lenzen, C., Wattenhofer, R.: Tight bounds for parallel randomized load balancing: extended abstract. In: Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC ’11, pp. 11–20. ACM, New York, NY, USA (2011)

  21. Li, S.: A 1.488 approximation algorithm for the uncapacitated facility location problem. In: Proceedings of the 38th International Conference on Automata, Languages and Programming, Volume Part II, ICALP ’11, pp. 77–88. Springer-Verlag, Berlin, Heidelberg (2011)

  22. Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15(4), 1036–1053 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lotker, Z., Patt-Shamir, B., Peleg, D.: Distributed MST for constant diameter graphs. Distrib. Comput. 18(6), 453–460 (2006)

    Article  MATH  Google Scholar 

  24. Lotker, Z., Patt-shamir, B., Pavlov, E., Peleg, D.: Minimum-weight spanning tree construction in O(log log n) communication rounds. SIAM J. Comput. 35, 120–131 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Malewicz, G., Austern, M.H., Bik, A.J., Dehnert, J.C., Horn, I., Leiser, N., Czajkowski, G.: Pregel: a system for large-scale graph processing. In: Proceedings of the 2010 ACM SIGMOD International Conference on Management of Data, SIGMOD ’10, pp. 135–146. ACM, New York, NY, USA (2010)

  26. Mettu, R.R., Plaxton, C.G.: The online median problem. SIAM J. Comput. 32(3), 816–832 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Moscibroda, T., Wattenhofer, R.: Facility location: distributed approximation. In: Proceedings of the Twenty-fourth Annual ACM Symposium on Principles of Distributed Computing, PODC ’05, pp. 108–117. ACM, New York, NY, USA (2005)

  28. Pandit, S., Pemmaraju, S.: Return of the primal-dual: distributed metric facility location. In: Proceedings of the 28th ACM Symposium on Principles of Distributed Computing, PODC ’09, pp. 180–189. ACM, New York, NY, USA (2009)

  29. Pandit, S., Pemmaraju, S.V.: Finding facilities fast. In: Garg, V., Wattenhofer, R., Kothapalli, K. (eds.) Distributed Computing and Networking, Lecture Notes in Computer Science, vol. 5408, pp. 11–24. Springer, Berlin Heidelberg (2009)

    Chapter  Google Scholar 

  30. Pandit, S., Pemmaraju, S.V.: Rapid randomized pruning for fast greedy distributed algorithms. In: Proceedings of the 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, PODC ’10, pp. 325–334. ACM, New York, NY, USA (2010)

  31. Peleg, D.: Distributed computing: a locality-sensitive approach, vol. 5. Society for Industrial and Applied Mathematics (2000)

  32. Patt-Shamir, B., Teplitsky, M.: The round complexity of distributed sorting: extended abstract. In: Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, PODC ’11, pp. 249–256. ACM, New York, NY, USA (2011)

  33. Stollsteimer, J.F.: A working model for plant numbers and locations. J. Farm Econ. 45(3), 631–645 (1963)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, The University of Iowa, Iowa City, Iowa, 52242-1419, USA

    James W. Hegeman & Sriram V. Pemmaraju

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  1. James W. Hegeman
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  2. Sriram V. Pemmaraju
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Correspondence to Sriram V. Pemmaraju.

Additional information

This work is supported in part by National Science Foundation grants CCF 0915543 and CCF 1318166. This paper combines and extends work that has appeared in ICALP 2012 and DISC 2013.

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Hegeman, J.W., Pemmaraju, S.V. Sub-logarithmic distributed algorithms for metric facility location. Distrib. Comput. 28, 351–374 (2015). https://doi.org/10.1007/s00446-015-0243-x

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