Skip to main content
SpringerLink
Log in

Faster Algorithms for k-Medians in Trees

  • Conference paper
Mathematical Foundations of Computer Science 2003 (MFCS 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2747))

Abstract

In the k-median problem we are given a connected graph with non-negative weights associated with the nodes and lengths associated with the edges. The task is to compute locations of k facilities in order to minimize the sum of the weighted distances between each node and its closest facility. In this paper we consider the case when the graph is a tree. We show that this problem can be solved in time \(O(n {\mbox{\rm polylog}} (n))\) for the following cases: (i) directed trees (and any fixed k), (ii) balanced undirected trees, and (iii) undirected trees with k=3.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arora, S., Raghavan, P., Rao, S.: Approximation schemes for euclidean k-medians and related problems. In: Proc. 30th Annual ACM Symposium on Theory of Computing (STOC 1998), pp. 106–113 (1998)

    Google Scholar 

  2. Arya, V., Garg, N., Khandekar, R., Meyerson, A., Mungala, K., Pandit, V.: Local seach heuristic for k-median and facility location problems. In: Proc. 16th Annual ACOM Symposium on Computing, pp. 21–29 (2001)

    Google Scholar 

  3. Auletta, V., Parente, D., Persiano, G.: Dynamic and static algorithms for optimal placement of resources in a tree. Theoretical Computer Science 165, 441–461 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Auletta, V., Parente, D., Persiano, G.: Placing resources on a growing line. Journal of Algorithms 26, 87–100 (1998)

    MATH  MathSciNet  Google Scholar 

  5. Benkoczi, R.R., Bhattacharya, B.K.: Spine tree decomposition. Technical Report CMPT1999-09, School of Computing Science, Simon Fraser University, Canada (1999)

    Google Scholar 

  6. Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location and k-median problems. In: Proc. 40th Symposium on Foundations of Computer Science (FOCS 1999), pp. 378–388 (1999)

    Google Scholar 

  7. Charikar, M., Guha, S., Tardos, E., Shmoys, D.: A constant-factor approximation algorithm for the k-median problem. In: Proc. 31st Annual ACM Symposium on Theory of Computing (STOC 1999), pp. 1–10 (1999)

    Google Scholar 

  8. Chrobak, M., Larmore, L., Rytter, W.: The k-median problem for directed trees. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 136, pp. 260–271. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  9. Garey, M.R., Johnson, D.S.: Computers and Intractability: a Guide to the Theory of NP-completeness. W.H. Freeman and Co., New York (1979)

    MATH  Google Scholar 

  10. Gavish, R., Sridhar, S.: Computing the 2-median on tree networks in \(O(n \ {\log} \ n)\) time. Networks 26, 305–317 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hassin, R., Tamir, A.: Improved complexity bounds for location problems on the real line. Operation Research Letters 10, 395–402 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hsu, W.L.: The distance-domination numbers of trees. Operation Research Letters 1, 96–100 (1982)

    Article  MATH  Google Scholar 

  13. Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems II: The p-medians. SIAM Journal on Applied Mathematics 37, 539–560 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  14. Korupolu, M.R., Plaxton, C.G., Rajaraman, R.: Analysis of a local search heuristic for facility location problems. Journal of Algorithms 37, 146–188 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Li, B., Deng, X., Golin, M., Sohraby, K.: On the optimal placement of web proxies on the internet: linear topology. In: Proc. 8th IFIP Conference on High Peformance Netwworking (HPN 1998), pp. 485–495 (1998)

    Google Scholar 

  16. Li, B., Golin, M.J., Italiano, G.F., Deng, X., Sohraby, K.: On the optimal placement of web proxies in the internet. In: IEEE InfoComm 1999, pp. 1282–1290 (1999)

    Google Scholar 

  17. Shah, R., Farach-Colton, M.: Undiscretized dynamic programming: faster algorithms for facility location and related problems on trees. In: Proc. 13th Annual Symposium on Discrete Algorithms (SODA), pp. 108–115 (2002)

    Google Scholar 

  18. Shah, R., Langerman, S., Lodha, S.: Algorithms for efficient filtering in contentbased multicast. In: Proc. 9th Annual European Symposium on Algorithms (ESA), pp. 428–439 (2001)

    Google Scholar 

  19. Tamir, A.: An O(pn 2) algorithm for the p-median and related problems on tree graphs. Operations Research Letters 19, 59–64 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Vigneron, A., Gao, L., Golin, M., Italiano, G., Li, B.: An algorithm for finding a k-median in a directed tree. Information Processing Letters 74, 81–88 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Woeginger, G.: Monge strikes again: optimal placement of web proxies in the internet. Operations Research Letters 27, 93–96 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computing Science, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Robert Benkoczi & Binay Bhattacharya

  2. Department of Computer Science, University of California, Riverside, CA, 92521, USA

    Marek Chrobak

  3. Department of Computer Science, University of Nevada, Las Vegas, NV, 89154-4019, USA

    Lawrence L. Larmore

  4. Instytut Informatyki, Uniwersytet Warszawski, Banacha 2, 02–097, Warszawa, Poland

    Wojciech Rytter

  5. Department of Computer Science, and New Jersey Institute of Technology, Newark, New Jersey, 07102-1982, USA

    Wojciech Rytter

Authors
  1. Robert Benkoczi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Binay Bhattacharya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marek Chrobak
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lawrence L. Larmore
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Wojciech Rytter
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia

    Branislav Rovan

  2. Charles University, Prague, Czech Republic

    Peter Vojtáš

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Benkoczi, R., Bhattacharya, B., Chrobak, M., Larmore, L.L., Rytter, W. (2003). Faster Algorithms for k-Medians in Trees. In: Rovan, B., Vojtáš, P. (eds) Mathematical Foundations of Computer Science 2003. MFCS 2003. Lecture Notes in Computer Science, vol 2747. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45138-9_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-45138-9_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40671-6

  • Online ISBN: 978-3-540-45138-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation