Skip to main content

Fast Rendezvous with Advice

Part of the Lecture Notes in Computer Science book series (LNCCN,volume 8847)

Abstract

Two mobile agents, starting from different nodes of an \(n\)-node network at possibly different times, have to meet at the same node. This problem is known as rendezvous. Agents move in synchronous rounds using a deterministic algorithm. In each round, an agent decides to either remain idle or to move to one of the adjacent nodes. Each agent has a distinct integer label from the set \(\{1,\ldots ,L\}\), which it can use in the execution of the algorithm, but it does not know the label of the other agent.

The main efficiency measure of a rendezvous algorithm’s performance is its time, i.e., the number of rounds from the start of the later agent until the meeting. If \(D\) is the distance between the initial positions of the agents, then \(\varOmega (D)\) is an obvious lower bound on the time of rendezvous. However, if each agent has no initial knowledge other than its label, time \(O(D)\) is usually impossible to achieve. We study the minimum amount of information that has to be available a priori to the agents to achieve rendezvous in optimal time \(\varTheta (D)\). Following the standard paradigm of algorithms with advice, this information is provided to the agents at the start by an oracle knowing the entire instance of the problem, i.e., the network, the starting positions of the agents, their wake-up rounds, and both of their labels. The oracle helps the agents by providing them with the same binary string called advice, which can be used by the agents during their navigation. The length of this string is called the size of advice. Our goal is to find the smallest size of advice which enables the agents to meet in time \(\varTheta (D)\). We show that this optimal size of advice is \(\varTheta (D\log (n/D)+\log \log L)\). The upper bound is proved by constructing an advice string of this size, and providing a natural rendezvous algorithm using this advice that works in time \(\varTheta (D)\) for all networks. The matching lower bound, which is the main contribution of this paper, is proved by exhibiting classes of networks for which it is impossible to achieve rendezvous in time \(\varTheta (D)\) with smaller advice.

Keywords

  • Rendezvous
  • Advice
  • Deterministic distributed algorithm
  • Mobile agent
  • Time

A. Pelc—Partially supported by NSERC discovery grant and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-662-46018-4_5
  • Chapter length: 13 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   39.99
Price excludes VAT (USA)
  • ISBN: 978-3-662-46018-4
  • 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   49.99
Price excludes VAT (USA)
Fig. 1.

References

  1. Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. International Series in Operations research and Management Science. Springer, New York (2002)

    Google Scholar 

  2. Anderson, E., Fekete, S.: Asymmetric rendezvous on the plane. In: Proceedings of the 14th Annual ACM Symposium on Computational Geometry, pp. 365–373 (1998)

    Google Scholar 

  3. 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 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Label-guided graph exploration by a finite automaton. ACM Trans. Algorithms 4, 1–18 (2008)

    CrossRef  MathSciNet  Google Scholar 

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

    Google Scholar 

  7. Dereniowski, D., Pelc, A.: Drawing maps with advice. J. Parallel Distrib. Comput. 72, 132–143 (2012)

    CrossRef  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Dieudonné, Y., Pelc, A., Villain, V.: How to meet asynchronously at polynomial cost. In: Proceedings of the 32nd ACM Symposium on Principles of Distributed Computing (PODC 2013), pp. 92–99 (2013)

    Google Scholar 

  10. Ellis, R.: Volume of an N-simplex by multiple integration. Elem. Math. 31, 57–59 (1976)

    MathSciNet  MATH  Google Scholar 

  11. Emek, Y., Fraigniaud, P., Korman, A., Rosen, A.: Online computation with advice. Theor. Comput. Sci. 412, 2642–2656 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Fraigniaud, P., Ilcinkas, D., Pelc, A.: Communication algorithms with advice. J. Comput. Syst. Sci. 76, 222–232 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Fraigniaud, P., Ilcinkas, D., Pelc, A.: Tree exploration with advice. Inf. Comput. 206, 1276–1287 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Fraigniaud, P., Korman, A., Lebhar, E.: Local MST computation with short advice. Theor. Comput. Syst. 47, 920–933 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Gibbons, A.: Algorithmic Graph Theory. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  17. Katz, M., Katz, N., Korman, A., Peleg, D.: Labeling schemes for flow and connectivity. SIAM J. Comput. 34, 23–40 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. Distrib. Comput. 22, 215–233 (2010)

    CrossRef  MATH  Google Scholar 

  19. Nisse, N., Soguet, D.: Graph searching with advice. Theor. Comput. Sci. 410, 1307–1318 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Avery Miller .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Miller, A., Pelc, A. (2015). Fast Rendezvous with Advice. In: Gao, J., Efrat, A., Fekete, S., Zhang, Y. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2014. Lecture Notes in Computer Science(), vol 8847. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46018-4_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-46018-4_5

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-46017-7

  • Online ISBN: 978-3-662-46018-4

  • eBook Packages: Computer ScienceComputer Science (R0)