Advertisement

Querying with Uncertainty

  • Huda Chuangpishit
  • Kostantinos Georgiou
  • Evangelos Kranakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10718)

Abstract

We introduce and study a new optimization problem on querying with uncertainty. k robots are required to locate a hidden item that is placed uniformly at random in one of n different locations, each associated with a probability \(p_i\), \(i=1,\ldots ,n\). If the item is placed in location i, a query trial by any of the robots reveals the item with probability \(p_i\). Each robot j is assigned a subset \(A_j\) of the locations, and is allowed to perform a random walk among them, each time step querying the current location (being visited) for the item. We are interested in determining sets \(\{A_j\}_{j=1,\ldots ,k}\) so as to minimize the expected discovery time of the item. We measure the cost by the number of queries, while there is no cost for hopping from node to node.

Our first contribution is to prove a closed formula for the expected number of steps until the treasure is found when the robots execute unanimous queries. Then we focus on querying problems where the sets \(A_j\) are restricted to be either pairwise disjoint or identical. Our findings allow us to obtain optimal solutions, when sets \(A_j\) are exclusively pairwise disjoint, requiring time \(n^{O(k)}\). In our second contribution, we devise an optimal polynomial time algorithm for querying with \(k=2\) robots even when the sets \(A_1,A_2\) are allowed to overlap. All our algorithms are based on special concavity-type properties of the expected termination time when the robots execute unanimous queries, thus inducing special structural properties of optimal solutions for the general problem.

Keywords

Searching Querying Random walk Partition Assignment Optimization 

References

  1. 1.
    Agarwal, P.K., Aronov, B., Har-Peled, S., Phillips, J.M., Yi, K., Zhang, W.: Nearest neighbor searching under uncertainty ii. In: Proceedings of the 32nd ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pp. 115–126. ACM (2013)Google Scholar
  2. 2.
    Ahlswede, R., Wegener, I.: Search Problems. Wiley-Interscience, Hoboken (1987)zbMATHGoogle Scholar
  3. 3.
    Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous, vol. 55. Kluwer Academic Publishers, Dordrecht (2002)zbMATHGoogle Scholar
  4. 4.
    Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. Springer, Heidelberg (2003).  https://doi.org/10.1007/b100809 zbMATHGoogle Scholar
  5. 5.
    Baeza Yates, R., Culberson, J., Rawlins, G.: Searching in the plane. Inf. Comput. 106(2), 234–252 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beck, A.: On the linear search problem. Isr. J. Math. 2(4), 221–228 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Beck, A., Warren, P.: The return of the linear search problem. Isr. J. Math. 14(2), 169–183 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bellman, R.: An optimal search. SIAM Rev. 5(3), 274 (1963)CrossRefGoogle Scholar
  9. 9.
    Chakravarty, A.K., Orlin, J.B., Rothblum, U.G.: Technical note–a partitioning problem with additive objective with an application to optimal inventory groupings for joint replenishment. Oper. Res. 30(5), 1018–1022 (1982)CrossRefzbMATHGoogle Scholar
  10. 10.
    Chrobak, M., Gąsieniec, L., Gorry, T., Martin, R.: Group search on the line. In: Italiano, G.F., Margaria-Steffen, T., Pokorný, J., Quisquater, J.-J., Wattenhofer, R. (eds.) SOFSEM 2015. LNCS, vol. 8939, pp. 164–176. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46078-8_14 Google Scholar
  11. 11.
    Czyzowicz, J., Georgiou, K., Kranakis, E., Krizanc, D., Narayanan, L., Opatrny, J., Shende, S.: Search on a line with byzantine robots. In: ISAAC. LIPCS (2016)Google Scholar
  12. 12.
    Czyzowicz, J., Kranakis, E., Krizanc, D., Narayanan, L., Opatrny, J.: Search on a line with faulty robots. In: Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, pp. 405–414. ACM (2016)Google Scholar
  13. 13.
    Deng, X., Kameda, T., Papadimitriou, C.: How to learn an unknown environment. In: FOCS, pp. 298–303. IEEE (1991)Google Scholar
  14. 14.
    Feinerman, O., Korman, A.: Memory lower bounds for randomized collaborative search and implications for biology. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 61–75. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-33651-5_5 CrossRefGoogle Scholar
  15. 15.
    Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theoret. Comput. Sci. 399(3), 236–245 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hanusse, N., Ilcinkas, D., Kosowski, A., Nisse, N.: Locating a target with an agent guided by unreliable local advice: how to beat the random walk when you have a clock? In: Proceedings of the 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, pp. 355–364. ACM (2010)Google Scholar
  17. 17.
    Hanusse, N., Kavvadias, D.J., Kranakis, E., Krizanc, D.: Memoryless search algorithms in a network with faulty advice. TCS 402(2–3), 190–198 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kao, M.-Y., Ma, Y., Sipser, M., Yin, Y.: Optimal constructions of hybrid algorithms. J. Algorithms 29(1), 142–164 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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
  20. 20.
    Kranakis, E., Krizanc, D.: Searching with uncertainty. In: 6th International Colloquium on Structural Information & Communication Complexity, SIROCCO 1999, Lacanau-Ocean, France, 1–3 July 1999, pp. 194–203 (1999)Google Scholar
  21. 21.
    Stone, L.: Theory of Optimal Search. Academic Press, New York (1975)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Huda Chuangpishit
    • 1
  • Kostantinos Georgiou
    • 1
  • Evangelos Kranakis
    • 2
  1. 1.Department of MathematicsRyerson UniversityTorontoCanada
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations