Naming and Counting in Anonymous Unknown Dynamic Networks

TL;DR

This paper investigates the fundamental problems of naming and counting in anonymous, unknown, and dynamic networks, establishing impossibility results and proposing solutions under certain conditions, including the use of a leader and degree bounds.

Contribution

It provides new impossibility results for static and dynamic networks and introduces conditions under which counting and naming become feasible, including the use of a leader and degree knowledge.

Findings

Counting is impossible without a leader in static networks.

Naming is impossible even with a leader in static networks.

A leader enables linear-time counting in static networks.

Abstract

In this work, we study the fundamental naming and counting problems (and some variations) in networks that are anonymous, unknown, and possibly dynamic. In counting, nodes must determine the size of the network n and in naming they must end up with unique identities. By anonymous we mean that all nodes begin from identical states apart possibly from a unique leader node and by unknown that nodes have no a priori knowledge of the network (apart from some minimal knowledge when necessary) including ignorance of n. Network dynamicity is modeled by the 1-interval connectivity model, in which communication is synchronous and a worst-case adversary chooses the edges of every round subject to the condition that each instance is connected. We first focus on static networks with broadcast where we prove that, without a leader, counting is impossible to solve and that naming is impossible to solve even with a leader and even if nodes know n. These impossibilities carry over to dynamic networks as well. We also show that a unique leader suffices in order to solve counting in linear time. Then we focus on dynamic networks with broadcast. We conjecture that dynamicity renders nontrivial computation impossible. In view of this, we let the nodes know an upper bound on the maximum degree that will ever appear and show that in this case the nodes can obtain an upper bound on n. Finally, we replace broadcast with one-to-each, in which a node may send a different message to each of its neighbors. Interestingly, this natural variation is proved to be computationally equivalent to a full-knowledge model, in which unique names exist and the size of the network is known.