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Investigating the Cost of Anonymity on Dynamic Networks

13 May 2015
Giuseppe Antonio Di Luna
R. Baldoni
ArXiv (abs)PDFHTML
Abstract

In this paper we study the difficulty of counting nodes in a synchronous dynamic network where nodes share the same identifier, they communicate by using a broadcast with unlimited bandwidth and, at each synchronous round, network topology may change. To count in such setting, it has been shown that the presence of a leader is necessary. We focus on a particularly interesting subset of dynamic networks, namely \textit{Persistent Distance} - G({\cal G}(G(PD)h)_{h})h​, in which each node has a fixed distance from the leader across rounds and such distance is at most hhh. In these networks the dynamic diameter DDD is at most 2h2h2h. We prove the number of rounds for counting in G({\cal G}(G(PD)2)_{2})2​ is at least logarithmic with respect to the network size ∣V∣|V|∣V∣. Thanks to this result, we show that counting on any dynamic anonymous network with DDD constant w.r.t. ∣V∣|V|∣V∣ takes at least D+Ω(log ∣V∣)D+ \Omega(\text{log}\, |V| )D+Ω(log∣V∣) rounds where Ω(log ∣V∣)\Omega(\text{log}\, |V|)Ω(log∣V∣) represents the additional cost to be payed for handling anonymity. At the best of our knowledge this is the fist non trivial, i.e. different from Ω(D)\Omega(D)Ω(D), lower bounds on counting in anonymous interval connected networks with broadcast and unlimited bandwith.

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