Short Labeling Schemes for Topology Recognition in Wireless Tree Networks [article]

Barun Gorain, Andrzej Pelc
2017 arXiv   pre-print
We consider the problem of topology recognition in wireless (radio) networks modeled as undirected graphs. Topology recognition is a fundamental task in which every node of the network has to output a map of the underlying graph i.e., an isomorphic copy of it, and situate itself in this map. In wireless networks, nodes communicate in synchronous rounds. In each round a node can either transmit a message to all its neighbors, or stay silent and listen. At the receiving end, a node v hears a
more » ... ge from a neighbor w in a given round, if v listens in this round, and if w is its only neighbor that transmits in this round. Nodes have labels which are (not necessarily different) binary strings. The length of a labeling scheme is the largest length of a label. We concentrate on wireless networks modeled by trees, and we investigate two problems. * What is the shortest labeling scheme that permits topology recognition in all wireless tree networks of diameter D and maximum degree Δ? * What is the fastest topology recognition algorithm working for all wireless tree networks of diameter D and maximum degree Δ, using such a short labeling scheme? We are interested in deterministic topology recognition algorithms. For the first problem, we show that the minimum length of a labeling scheme allowing topology recognition in all trees of maximum degree Δ≥ 3 is Θ(Δ). For such short schemes, used by an algorithm working for the class of trees of diameter D≥ 4 and maximum degree Δ≥ 3, we show almost matching bounds on the time of topology recognition: an upper bound O(DΔ), and a lower bound Ω(DΔ^ϵ), for any constant ϵ<1.
arXiv:1704.01927v1 fatcat:oderucn2p5gsrj6gegm66xqy7a