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Time versus cost tradeoffs for deterministic rendezvous in networks

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Abstract

Two mobile agents, starting from different nodes of a network at possibly different times, have to meet at the same node. This problem is known as rendezvous. Agents move in synchronous rounds. Each agent has a distinct integer label from the set \(\{1,\ldots ,L\}\). Two main efficiency measures of rendezvous are its time (the number of rounds until the meeting) and its cost (the total number of edge traversals). We investigate tradeoffs between these two measures. A natural benchmark for both time and cost of rendezvous in a network is the number of edge traversals needed for visiting all nodes of the network, called the exploration time. Hence we express the time and cost of rendezvous as functions of an upper bound E on the time of exploration (where E and a corresponding exploration procedure are known to both agents) and of the size L of the label space. We present two natural rendezvous algorithms. Algorithm Cheap has cost O(E) (and, in fact, a version of this algorithm for the model where the agents start simultaneously has cost exactly E) and time O(EL). Algorithm Fast has both time and cost \(O(E\log L)\). Our main contributions are lower bounds showing that, perhaps surprisingly, these two algorithms capture the tradeoffs between time and cost of rendezvous almost tightly. We show that any deterministic rendezvous algorithm of cost asymptotically E (i.e., of cost \(E+o(E)\)) must have time \(\varOmega (EL)\). On the other hand, we show that any deterministic rendezvous algorithm with time complexity \(O(E\log L)\) must have cost \(\varOmega (E\log L)\).

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Notes

  1. A different way of counting time and cost (under which our results still hold) is discussed in Sect. 4.

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Acknowledgments

We thank the anonymous referee for suggesting Algorithm FastWithRelabeling, which replaces a less efficient algorithm that we had in an earlier version. This work was partially supported by NSERC discovery Grant 8136-2013 and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais.

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  1. Université du Québec en Outaouais, Gatineau, Canada

    Avery Miller & Andrzej Pelc

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Correspondence to Avery Miller.

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Miller, A., Pelc, A. Time versus cost tradeoffs for deterministic rendezvous in networks. Distrib. Comput. 29, 51–64 (2016). https://doi.org/10.1007/s00446-015-0253-8

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