Learning a ring cheaply and fast

  • Authors:

  • Emanuele G. Fusco;Andrzej Pelc;Rossella Petreschi

  • Affiliations:

  • Computer Science Department, Sapienza, University of Rome, Rome, Italy;Département d'informatique, Université du Québec en Outaouais, Gatineau, Québec, Canada;Computer Science Department, Sapienza, University of Rome, Rome, Italy

  • Venue:
  • ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part II
  • Year:
  • 2013

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Abstract

We consider the task of learning a ring in a distributed way: each node of an unknown ring has to construct a labeled map of it. Nodes are equipped with unique labels. Communication proceeds in synchronous rounds. In every round every node can send arbitrary messages to its neighbors and perform arbitrary local computations. We study tradeoffs between the time (number of rounds) and the cost (number of messages) of completing this task in a deterministic way: for a given time T we seek bounds on the smallest number of messages needed for learning the ring in time T. Our bounds depend on the diameter D of the ring and on the delayθ=T−D above the least possible time D in which this task can be performed. We prove a lower bound Ω(D2/θ) on the number of messages used by any algorithm with delay θ, and we design a class of algorithms that give an almost matching upper bound: for any positive constant 0εθ≤D and using O(D2 (log*D)/θ1−ε) messages.