Skip to main content

Tradeoffs between Cost and Information for Rendezvous and Treasure Hunt

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8878)

Abstract

Rendezvous and treasure hunt are two basic tasks performed by mobile agents in networks. In rendezvous, two agents, initially located at distinct nodes of the network, traverse edges in synchronous rounds and have to meet at some node. In treasure hunt, a single agent has to find a stationary target (treasure) situated at an unknown node. The network is modeled as an undirected connected graph whose nodes have distinct identities. The cost of a rendezvous algorithm is the worst-case total number of edge traversals performed by both agents until meeting. The cost of a treasure hunt algorithm is the worst-case number of edge traversals performed by the agent until the treasure is found. If the agents have no information about the network, the cost of both rendezvous and treasure hunt can be as large as Θ(e) for networks with e edges.

We study tradeoffs between the amount of information available a priori to the agents and the cost of rendezvous and treasure hunt. Following the paradigm of algorithms with advice, this information is provided to the agents at the start of their navigation by an oracle knowing the network, the starting positions of the agents, and, in the case of treasure hunt, the node where the treasure is hidden. The oracle assists the agents by providing them with a binary string called advice, which can be used by each agent during the algorithm execution. In the case of rendezvous, the advice given to each agent can be different. The length of the string given to the agent in treasure hunt and the sum of the lengths of strings given to the agents in rendezvous is called the size of advice.

Our goal is to find the smallest size of advice which enables the agents to solve rendezvous and treasure hunt at a given cost C in a network with e edges. This size turns out to depend on the initial distance D and on the ratio g = e/C, which is the relative cost gain due to advice. For arbitrary graphs, we give upper and lower bounds of O(Dlog(Dg)) and Ω(Dlogg), respectively, on the optimal size of advice. Hence, our bounds leave only a logarithmic gap in the general case. For the class of trees we give tight upper and lower bounds of Θ(Dlogg).

Keywords

  • rendezvous
  • treasure hunt
  • advice
  • deterministic algorithm
  • mobile agent
  • cost

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-14472-6_18
  • Chapter length: 14 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   69.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-14472-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   89.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abiteboul, S., Kaplan, H., Milo, T.: Compact labeling schemes for ancestor queries. In: Proc. 12th ACM-SIAM Symp. on Discrete Algorithms (SODA ), pp. 547–556 (2001)

    Google Scholar 

  2. Alpern, S.: The rendezvous search problem. SIAM J. on Control and Optimization 33, 673–683 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Alpern, S.: Rendezvous search on labelled networks. Naval Reaserch Logistics 49, 256–274 (2002)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Alpern, S., Gal, S.: The theory of search games and rendezvous. Int. Series in Operations research and Management Science. Kluwer Academic Publisher (2002)

    Google Scholar 

  5. Anderson, E., Weber, R.: The rendezvous problem on discrete locations. Journal of Applied Probability 28, 839–851 (1990)

    CrossRef  MathSciNet  Google Scholar 

  6. Anderson, E., Fekete, S.: Two-dimensional rendezvous search. Operations Research 49, 107–118 (2001)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Baeza-Yates, R.A., Culberson, J.C., Rawlins, G.J.E.: Searching in the plane. Information and Computation 106, 234–252 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Bampas, E., Czyzowicz, J., Gąsieniec, L., Ilcinkas, D., Labourel, A.: Almost optimal asynchronous rendezvous in infinite multidimensional grids. In: Lynch, N.A., Shvartsman, A.A. (eds.) DISC 2010. LNCS, vol. 6343, pp. 297–311. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  9. Baston, V., Gal, S.: Rendezvous search when marks are left at the starting points. Naval Reaserch Logistics 48, 722–731 (2001)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. Bose, P., De Carufel, J.-L., Durocher, S.: Revisiting the Problem of Searching on a Line. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 205–216. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  11. Caminiti, S., Finocchi, I., Petreschi, R.: Engineering tree labeling schemes: A case study on least common ancestor. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 234–245. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  12. Cieliebak, M., Flocchini, P., Prencipe, G., Santoro, N.: Distributed computing by mobile robots: Gathering. SIAM J. Comput. 41, 829–879 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Label-guided graph exploration by a finite automaton. ACM Transactions on Algorithms 4 (2008)

    Google Scholar 

  14. Czyzowicz, J., Kosowski, A., Pelc, A.: How to meet when you forget: Log-space rendezvous in arbitrary graphs. Distributed Computing 25, 165–178 (2012)

    CrossRef  MATH  Google Scholar 

  15. Czyzowicz, J., Labourel, A., Pelc, A.: How to meet asynchronously (almost) everywhere. ACM Transactions on Algorithms 8, article 37 (2012)

    Google Scholar 

  16. Dessmark, A., Fraigniaud, P., Kowalski, D., Pelc, A.: Deterministic rendezvous in graphs. Algorithmica 46, 69–96 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. Dieudonné, Y., Pelc, A., Villain, V.: How to meet asynchronously at polynomial cost. In: Proc. 32nd ACM Symp. on Principles of Distributed Comp. (PODC), pp. 92–99 (2013)

    Google Scholar 

  18. Ellis, R.: Volume of an N-Simplex by Multiple Integration. Elemente der Mathematik 31, 57–59 (1976)

    MATH  Google Scholar 

  19. Emek, Y., Fraigniaud, P., Korman, A., Rosen, A.: Online computation with advice. Theoretical Computer Science 412, 2642–2656 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Fraigniaud, P., Ilcinkas, D., Pelc, A.: Communication algorithms with advice. Journal of Computer and System Sciences 76, 222–232 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  21. Fraigniaud, P., Ilcinkas, D., Pelc, A.: Tree exploration with advice. Information and Computation 206, 1276–1287 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. Fraigniaud, P., Korman, A., Lebhar, E.: Local MST computation with short advice. Theory of Computing Systems 47, 920–933 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. Fusco, E., Pelc, A.: Trade-offs between the size of advice and broadcasting time in trees. Algorithmica 60, 719–734 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  24. Fusco, E.G., Pelc, A., Petreschi, R.: Use knowledge to learn faster: Topology recognition with advice. In: Afek, Y. (ed.) DISC 2013. LNCS, vol. 8205, pp. 31–45. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  25. Gavoille, C., Peleg, D., Pérennes, S., Raz, R.: Distance labeling in graphs. Journal of Algorithms 53, 85–112 (2004)

    Google Scholar 

  26. Hipke, C.A., Icking, C., Klein, R., Langetepe, E.: How to find a point on a line within a fixed distance. Disc. App. Math. 93, 67–73 (1999)

    CrossRef  MATH  MathSciNet  Google Scholar 

  27. Katz, M., Katz, N., Korman, A., Peleg, D.: Labeling schemes for flow and connectivity. SIAM Journal of Computing 34, 23–40 (2004)

    CrossRef  MATH  MathSciNet  Google Scholar 

  28. Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. Distributed Computing 22, 215–233 (2010)

    CrossRef  MATH  Google Scholar 

  29. Kowalski, D.R., Malinowski, A.: How to meet in anonymous network. In: Flocchini, P., Gąsieniec, L. (eds.) SIROCCO 2006. LNCS, vol. 4056, pp. 44–58. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  30. Kranakis, E., Krizanc, D., Morin, P.: Randomized Rendez-Vous with Limited Memory. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 605–616. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  31. Lopez-Ortiz, A., Schuierer, S.: The ultimate strategy to search on m rays? Theoretical Computer Science 261, 267–295 (2001)

    CrossRef  MATH  MathSciNet  Google Scholar 

  32. Miller, A., Pelc, A.: Fast rendezvous with advice. In: Proc. 10th Int. Symp. on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics (ALGOSENSORS 2014) (2014), Full version at arxiv:1407.1428v1 [cs.DS]

    Google Scholar 

  33. Miller, A., Pelc, A.: Time versus cost tradeoffs for deterministic rendezvous in networks. In: Proc. 33rd Annual ACM Symposium on Principles of Distributed Computing (PODC 2014 ), pp. 282–290 (2014)

    Google Scholar 

  34. Nisse, N., Soguet, D.: Graph searching with advice. Theoretical Computer Science 410, 1307–1318 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  35. Panaite, P., Pelc, A.: Exploring unknown undirected graphs. Journal of Algorithms 33, 281–295 (1999)

    CrossRef  MATH  MathSciNet  Google Scholar 

  36. Pelc, A.: Deterministic rendezvous in networks: A comprehensive survey. Networks 59, 331–347 (2012)

    CrossRef  MathSciNet  Google Scholar 

  37. Reingold, O.: Undirected connectivity in log-space. Journal of the ACM 55 (2008)

    Google Scholar 

  38. Ta-Shma, A., Zwick, U.: Deterministic rendezvous, treasure hunts and strongly universal exploration sequences. In: Proc. 18th ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 599–608 (2007)

    Google Scholar 

  39. Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52, 1–24 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  40. Xin, Q.: Faster treasure hunt and better strongly universal exploration sequences. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 549–560. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Miller, A., Pelc, A. (2014). Tradeoffs between Cost and Information for Rendezvous and Treasure Hunt. In: Aguilera, M.K., Querzoni, L., Shapiro, M. (eds) Principles of Distributed Systems. OPODIS 2014. Lecture Notes in Computer Science, vol 8878. Springer, Cham. https://doi.org/10.1007/978-3-319-14472-6_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-14472-6_18

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-14471-9

  • Online ISBN: 978-3-319-14472-6

  • eBook Packages: Computer ScienceComputer Science (R0)