Advertisement

Algorithmica

, Volume 81, Issue 7, pp 2917–2933 | Cite as

A \({o}\mathopen {}\left( n\right) \mathclose {}\)-Competitive Deterministic Algorithm for Online Matching on a Line

  • Antonios Antoniadis
  • Neal Barcelo
  • Michael Nugent
  • Kirk Pruhs
  • Michele ScquizzatoEmail author
Article

Abstract

Online matching on a line involves matching an online stream of requests to a given set of servers, all in the real line, with the objective of minimizing the sum of the distances between matched server-request pairs. The best previously known upper and lower bounds on the optimal deterministic competitive ratio are linear in the number of requests, and constant, respectively. We show that online matching on a line is essentially equivalent to a particular search problem, which we call k-lost-cows. We then obtain the first deterministic sub-linearly competitive algorithm for online matching on a line by giving such an algorithm for the k-lost-cows problem.

Keywords

Online algorithms Competitive analysis Metric matching Search problems 

Notes

Supplementary material

References

  1. 1.
    Antoniadis, A., Barcelo, N., Nugent, M., Pruhs, K., Scquizzato, M.: A \(o(n)\)-competitive deterministic algorithm for online matching on a line. In: Proceedings of the 12th International Workshop on Approximation and Online Algorithms (WAOA), pp. 11–22 (2014)Google Scholar
  2. 2.
    Antoniadis, A., Fischer, C., Tönnis, A.: A collection of lower bounds for online matching on the line. In: Proceedings of the 13th Latin American Theoretical Informatics Symposium (LATIN), pp. 52–65 (2018)Google Scholar
  3. 3.
    Baeza-Yates, R.A., Culberson, J.C., Rawlins, G.J.E.: Searching in the plane. Inf. Comput. 106(2), 234–252 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bansal, N., Buchbinder, N., Gupta, A., Naor, J.: A randomized \({O}(\log ^2 k)\)-competitive algorithm for metric bipartite matching. Algorithmica 68(2), 390–403 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chung, C., Pruhs, K., Uthaisombut, P.: The online transportation problem: on the exponential boost of one extra server. In: Proceedings of the 8th Latin American Theoretical Informatics Symposium (LATIN), pp. 228–239 (2008)Google Scholar
  6. 6.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci. 69(3), 485–497 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fuchs, B., Hochstättler, W., Kern, W.: Online matching on a line. Theor. Comput. Sci. 332(1–3), 251–264 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gupta, A., Lewi, K.: The online metric matching problem for doubling metrics. In: Proceedings of the 39th International Colloquium on Automata, Languages, and Programming (ICALP), pp. 424–435 (2012)Google Scholar
  9. 9.
    Kalyanasundaram, B., Pruhs, K.: Online weighted matching. J. Algorithms 14(3), 478–488 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kalyanasundaram, B., Pruhs, K.: Online network optimization problems. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms: The State of the Art, pp. 268–280. Springer, Berlin Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Kalyanasundaram, B., Pruhs, K.: The online transportation problem. SIAM J. Discrete Math. 13(3), 370–383 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kao, M.-Y., Reif, J.H., Tate, S.R.: Searching in an unknown environment: an optimal randomized algorithm for the cow-path problem. Inf. Comput. 131(1), 63–79 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Khuller, S., Mitchell, S.G., Vazirani, V.V.: On-line algorithms for weighted bipartite matching and stable marriages. Theor. Comput. Sci. 127(2), 255–267 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Koutsoupias, E., Nanavati, A.: The online matching problem on a line. In: Proceedings of the 1st International Workshop on Approximation and Online Algorithms (WAOA), pp. 179–191 (2003)Google Scholar
  15. 15.
    Koutsoupias, E., Papadimitriou, C.H.: On the \(k\)-server conjecture. J. ACM 42(5), 971–983 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    López-Ortiz, A.: On-Line Target Searching in Bounded and Unbounded Domains. Ph.D. thesis, University of Waterloo (1996)Google Scholar
  17. 17.
    Meyerson, A., Nanavati, A., Poplawski, L.J.: Randomized online algorithms for minimum metric bipartite matching. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 954–959 (2006)Google Scholar
  18. 18.
    Nayyar, K., Raghvendra, S.: An input sensitive online algorithm for the metric bipartite matching problem. In: Proceedings of the 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pp. 505–515 (2017)Google Scholar
  19. 19.
    Raghvendra, S.: A robust and optimal online algorithm for minimum metric bipartite matching. In: Proceedings of the 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pp. 18:1–18:16 (2016)Google Scholar
  20. 20.
    Raghvendra, S.: Optimal analysis of an online algorithm for the bipartite matching problem on a line. In: Proceedings of the 34th International Symposium on Computational Geometry (SoCG), pp. 67:1–67:14 (2018)Google Scholar
  21. 21.
    Reingold, E.M., Tarjan, R.E.: On a greedy heuristic for complete matching. SIAM J. Comput. 10(4), 676–681 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    van Stee, R.: SIGACT news online algorithms column 27: Online matching on the line, part 1. SIGACT News 47(1), 99–110 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    van Stee, R.: SIGACT news online algorithms column 28: Online matching on the line, part 2. SIGACT News 47(2), 40–51 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Max-Planck Institut für InformatikSaarbrückenGermany
  2. 2.University of PittsburghPittsburghUSA
  3. 3.University of PadovaPaduaItaly

Personalised recommendations