Advertisement

On the Tree Search Problem with Non-uniform Costs

  • Ferdinando CicaleseEmail author
  • Balázs Keszegh
  • Bernard Lidický
  • Dömötör Pálvölgyi
  • Tomáš Valla
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

Searching in partially ordered structures has been considered in the context of information retrieval and efficient tree-like indices, as well as in hierarchy based knowledge representation. In this paper we focus on tree-like partial orders and consider the problem of identifying an initially unknown vertex in a tree by asking edge queries: an edge query e returns the component of \(T-e\) containing the vertex sought for, while incurring some known cost c(e).

The Tree Search Problem with Non-Uniform Cost is the following: given a tree T on n vertices, each edge having an associated cost, construct a strategy that minimizes the total cost of the identification in the worst case.

Finding the strategy guaranteeing the minimum possible cost is an NP-complete problem already for input trees of degree 3 or diameter 6. The best known approximation guarantee was an \(O(\log n/\log \log \log n)\)-approximation algorithm of [Cicalese et al. TCS 2012].

We improve upon the above results both from the algorithmic and the computational complexity point of view: We provide a novel algorithm that provides an \(O(\frac{\log n}{\log \log n})\)-approximation of the cost of the optimal strategy. In addition, we show that finding an optimal strategy is NP-hard even when the input tree is a spider of diameter 6, i.e., at most one vertex has degree larger than 2.

Notes

Acknowledgment

We are very grateful to Balázs Patkós for organizing \(5^{\mathrm {th}}\) Emléktábla Workshop where we collaborated on this paper.

References

  1. 1.
    Ahlswede, R., Wegener, I.: Search Problems. Wiley, Chichester-New York (1987)zbMATHGoogle Scholar
  2. 2.
    Aigner, M.: Combinatorial Search. Wiley-Teubner, New York-Stuttgart (1988)zbMATHGoogle Scholar
  3. 3.
    Ben-Asher, Y., Farchi, E., Newman, I.: Optimal search in trees. SIAM J. Comput. 28(6), 2090–2102 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cicalese, F., Jacobs, T., Laber, E., Valentim, C.: The binary identification problem for weighted trees. Theor. Comput. Sci. 459, 100–112 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de la Torre, P., Greenlaw, R., Schäffer, A.: Optimal edge ranking of trees in polynomial time. Algorithmica 13(6), 592–618 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dereniowski, D.: Edge ranking of weighted trees. Discrete Appl. Math. 154, 1198–1209 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dereniowski, D.: Edge ranking and searching in partial orders. Discrete Appl. Math. 156(13), 2493–2500 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computer and Intractability. W.H. Freeman & Co., New York (1979)Google Scholar
  9. 9.
    Iyer, A.V., Ratliff, H.D., Vijayan, G.: On an edge ranking problem of trees and graphs. Discrete Appl. Math. 30(1), 43–52 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Knuth, D.: Searching and Sorting. The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1998)Google Scholar
  11. 11.
    Lam, T.W., Yue, F.L.: Optimal edge ranking of trees in linear time. In: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1998, pp. 436–445, Philadelphia, PA, USA, Society for Industrial and Applied Mathematics (1998)Google Scholar
  12. 12.
    Linial, N., Saks, M.: Searching order structures. J. Algorithms 6, 86–103 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Makino, K., Uno, Y., Ibaraki, T.: On minimum edge ranking spanning trees. J. Algorithms 38, 411–437 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mozes, S., Onak, K., Weimann, O.: Finding an optimal tree searching strategy in linear time. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 1096–1105 (2008)Google Scholar
  15. 15.
    Wermelinger, M.: Searching Efficiently in Posets. New University of Lisbon, Topics in Programming Technology (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Ferdinando Cicalese
    • 1
    Email author
  • Balázs Keszegh
    • 2
  • Bernard Lidický
    • 3
  • Dömötör Pálvölgyi
    • 4
  • Tomáš Valla
    • 5
  1. 1.Department of Computer ScienceUniversity of VeronaVeronaItaly
  2. 2.Rényi InstituteBudapestHungary
  3. 3.Department of MathematicsIowa State UniversityAmesUSA
  4. 4.Eötvös UniversityBudapestHungary
  5. 5.Faculty of Information TechnologyCzech Technical UniversityPragueCzech Republic

Personalised recommendations