# Fast Rendezvous with Advice

## 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## 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)CrossRefGoogle Scholar
- 4.Cieliebak, M., Flocchini, P., Prencipe, G., Santoro, N.: Distributed computing by mobile robots: gathering. SIAM J. Comput.
**41**, 829–879 (2012)CrossRefMathSciNetzbMATHGoogle 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)CrossRefMathSciNetGoogle Scholar - 6.Czyzowicz, J., Labourel, A., Pelc, A.: How to meet asynchronously (almost) everywhere. ACM Trans. Algorithms
**8**(2012). article 37Google Scholar - 7.Dereniowski, D., Pelc, A.: Drawing maps with advice. J. Parallel Distrib. Comput.
**72**, 132–143 (2012)CrossRefzbMATHGoogle Scholar - 8.Dessmark, A., Fraigniaud, P., Kowalski, D., Pelc, A.: Deterministic rendezvous in graphs. Algorithmica
**46**, 69–96 (2006)CrossRefMathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar - 11.Emek, Y., Fraigniaud, P., Korman, A., Rosen, A.: Online computation with advice. Theor. Comput. Sci.
**412**, 2642–2656 (2011)CrossRefMathSciNetzbMATHGoogle Scholar - 12.Fraigniaud, P., Ilcinkas, D., Pelc, A.: Communication algorithms with advice. J. Comput. Syst. Sci.
**76**, 222–232 (2010)CrossRefMathSciNetzbMATHGoogle Scholar - 13.Fraigniaud, P., Ilcinkas, D., Pelc, A.: Tree exploration with advice. Inf. Comput.
**206**, 1276–1287 (2008)CrossRefMathSciNetzbMATHGoogle Scholar - 14.Fraigniaud, P., Korman, A., Lebhar, E.: Local MST computation with short advice. Theor. Comput. Syst.
**47**, 920–933 (2010)CrossRefMathSciNetzbMATHGoogle Scholar - 15.Fusco, E., Pelc, A.: Trade-offs between the size of advice and broadcasting time in trees. Algorithmica
**60**, 719–734 (2011)CrossRefMathSciNetzbMATHGoogle Scholar - 16.Gibbons, A.: Algorithmic Graph Theory. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar
- 17.Katz, M., Katz, N., Korman, A., Peleg, D.: Labeling schemes for flow and connectivity. SIAM J. Comput.
**34**, 23–40 (2004)CrossRefMathSciNetzbMATHGoogle Scholar - 18.Korman, A., Kutten, S., Peleg, D.: Proof labeling schemes. Distrib. Comput.
**22**, 215–233 (2010)CrossRefzbMATHGoogle Scholar - 19.Nisse, N., Soguet, D.: Graph searching with advice. Theor. Comput. Sci.
**410**, 1307–1318 (2009)CrossRefMathSciNetzbMATHGoogle Scholar - 20.Pelc, A.: Deterministic rendezvous in networks: a comprehensive survey. Networks
**59**, 331–347 (2012)CrossRefMathSciNetGoogle 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 2007Google Scholar
- 22.Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM
**52**, 1–24 (2005)CrossRefMathSciNetzbMATHGoogle Scholar