Two mobile agents, starting from different nodes of an unknown network, have to meet at a node. Agents move in synchronous rounds using a deterministic algorithm. Each agent has a different label, which it can use in the execution of the algorithm, but it does not know the label of the other agent. Agents do not know any bound on the size of the network. In each round an agent decides if it remains idle or if it wants to move to one of the adjacent nodes. Agents are subject to delay faults: if an agent incurs a fault in a given round, it remains in the current node, regardless of its decision. If it planned to move and the fault happened, the agent is aware of it. We consider three scenarios of fault distribution: random (independently in each round and for each agent with constant probability \(0<p<1\)), unbounded adversarial (the adversary can delay an agent for an arbitrary finite number of consecutive rounds) and bounded adversarial (the adversary can delay an agent for at most c consecutive rounds, where c is unknown to the agents). The quality measure of a rendezvous algorithm is its cost, which is the total number of edge traversals. For random faults, we show an algorithm with cost polynomial in the size n of the network and polylogarithmic in the larger label L, which achieves rendezvous with very high probability in arbitrary networks. By contrast, for unbounded adversarial faults we show that rendezvous is not possible, even in the class of rings. Under this scenario we give a rendezvous algorithm with cost \(O(n\ell )\), where \(\ell \) is the smaller label, working in arbitrary trees, and we show that \(\varOmega (\ell )\) is the lower bound on rendezvous cost, even for the two-node tree. For bounded adversarial faults, we give a rendezvous algorithm working for arbitrary networks, with cost polynomial in n, and logarithmic in the bound c and in the larger label L.
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Aldous, D.J.: Meeting times for independent Markov chains. Stoch. Process. Appl. 38, 185–193 (1991)
Alpern, S.: The rendezvous search problem. SIAM J. Control Optim. 33, 673–683 (1995)
Alpern, S.: Rendezvous search on labelled networks. Nav. Res. Logist. 49, 256–274 (2002)
Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous. International Series in Operations Research and Management Science. Kluwer, Dordrecht (2002)
Anderson, E., Weber, R.: The rendezvous problem on discrete locations. J. Appl. Probab. 28, 839–851 (1990)
Anderson, E., Fekete, S.: Asymmetric rendezvous on the plane. In: Proceedings of 14th Annual ACM Symposium on Computational Geometry, pp. 365–373 (1998)
Anderson, E., Fekete, S.: Two-dimensional rendezvous search. Oper. Res. 49, 107–118 (2001)
Bampas, E., Czyzowicz, J., Gasieniec, L., Ilcinkas, D., Labourel, A.: Almost optimal asynchronous rendezvous in infinite multidimensional grids. In: Proceedings of 24th International Symposium on Distributed Computing (DISC 2010), pp. 297–311
Baston, V., Gal, S.: Rendezvous on the line when the players’ initial distance is given by an unknown probability distribution. SIAM J. Control Optim. 36, 1880–1889 (1998)
Baston, V., Gal, S.: Rendezvous search when marks are left at the starting points. Nav. Res. Logist. 48, 722–731 (2001)
Chalopin, J., Das, S., Widmayer, P.: Deterministic symmetric rendezvous in arbitrary graphs: overcoming anonymity, failures and uncertainty. In: Alpern, S., et al. (eds.) Search Theory: A Game Theoretic Perspective, pp. 175–195. Springer, Berlin (2013)
Cieliebak, M., Flocchini, P., Prencipe, G., Santoro, N.: Distributed computing by mobile robots: gathering. SIAM J. Comput. 41, 829–879 (2012)
Czyzowicz, J., Kosowski, A., Pelc, A.: How to meet when you forget: log-space rendezvous in arbitrary graphs. Distrib. Comput. 25, 165–178 (2012)
Czyzowicz, J., Labourel, A., Pelc, A.: How to meet asynchronously (almost) everywhere. ACM Trans. Algorithms 8(4), 37 (2012)
Das, S.: Mobile agent rendezvous in a ring using faulty tokens. In: Proceedings of 9th International Conference on Distributed Computing and Networking (ICDCN 2008), pp. 292–297
Das, S., Mihalak, M., Sramek, R., Vicari, E., Widmayer, P.: Rendezvous of mobile agents when tokens fail anytime. In: Proceedings of 12th International Conference on Principles of Distributed Systems (OPODIS 2008), pp. 463–480
De Marco, G., Gargano, L., Kranakis, E., Krizanc, D., Pelc, A., Vaccaro, U.: Asynchronous deterministic rendezvous in graphs. Theor. Comput. Sci. 355, 315–326 (2006)
Dessmark, A., Fraigniaud, P., Kowalski, D., Pelc, A.: Deterministic rendezvous in graphs. Algorithmica 46, 69–96 (2006)
Dieudonné, Y., Pelc, A.: Deterministic network exploration by a single agent with Byzantine tokens. Inf. Process. Lett. 112, 467–470 (2012)
Dieudonné, Y., Pelc, A., Peleg, D.: Gathering despite mischief. ACM Trans. Algorithms 11(1), 1 (2014)
Dieudonné, Y., Pelc, A., Villain, V.: How to meet asynchronously at polynomial cost. SIAM J. Comput. 44, 844–867 (2015)
Flocchini, P., Kranakis, E., Krizanc, D., Luccio, F.L., Santoro, N., Sawchuk, C.: Mobile agents rendezvous when tokens fail. In: Proceedings of 11th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2004), pp. 161–172
Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of asynchronous robots with limited visibility. Theor. Comput. Sci. 337, 147–168 (2005)
Fraigniaud, P., Pelc, A.: Delays induce an exponential memory gap for rendezvous in trees. ACM Trans. Algorithms 9(2), 17 (2013)
Israeli, A., Jalfon, M.: Token management schemes and random walks yield self stabilizing mutual exclusion. In: Proceedings of 9th Annual ACM Symposium on Principles of Distributed Computing (PODC 1990), pp. 119–131
Koucký, M.: Universal traversal sequences with backtracking. J. Comput. Syst. Sci. 65, 717–726 (2002)
Kowalski, D., Malinowski, A.: How to meet in anonymous network. In: Proceedings of 13th International Colloquium on Structural Information and Communication Complexity, (SIROCCO 2006), pp. 44–58
Kranakis, E., Krizanc, D., Morin, P.: Randomized rendez-vous with limited memory. In: Proceedings of 8th Latin American Theoretical Informatics (LATIN 2008), pp. 605–616
Kranakis, E., Krizanc, D., Santoro, N., Sawchuk, C.: Mobile agent rendezvous in a ring. In: Proceedings of 23rd International Conference on Distributed Computing Systems (ICDCS 2003), pp. 592–599
Lim, W., Alpern, S.: Minimax rendezvous on the line. SIAM J. Control Optim. 34, 1650–1665 (1996)
Lynch, N.L.: Distributed Algorithms. Morgan Kaufmann Publ. Inc., San Francisco (1996)
Pelc, A.: Deterministic rendezvous in networks: a comprehensive survey. Networks 59, 331–347 (2012)
Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 17 (2008)
Ta-Shma, A., Zwick, U.: Deterministic rendezvous, treasure hunts and strongly universal exploration sequences. In: Proceedings of 18th ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 599–608
Thomas, L.: Finding your kids when they are lost. J. Oper. Res. Soc. 43, 637–639 (1992)
A preliminary version of this paper, entitled “Fault-Tolerant Rendezvous in Networks”, appeared in Proceedings of 41st International Colloquium on Automata, Languages and Programming (ICALP 2014), July 2014, Copenhagen, Denmark, 411–422. Partially supported by NSERC discovery Grant 8136—2013, by the Research Chair in Distributed Computing at the Université du Québec en Outaouais and by the French ANR Project MACARON (anr-13-js02-0002).
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Chalopin, J., Dieudonné, Y., Labourel, A. et al. Rendezvous in networks in spite of delay faults. Distrib. Comput. 29, 187–205 (2016). https://doi.org/10.1007/s00446-015-0259-2
- Deterministic algorithm
- Mobile agent
- Delay fault