Short Labeling Schemes for Topology Recognition in Wireless Tree Networks

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 message 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 ∆? • What is the fastest topology recognition algorithm working for all wireless tree networks of diameter D and maximum degree ∆, 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 ∆ ≥ 3 is Θ(log log ∆). For such short schemes, used by an algorithm working for the class of trees of diameter D ≥ 4 and maximum degree ∆ ≥ 3, we show almost matching bounds on the time of topology recognition: an upper bound O(D∆), and a lower bound Ω(D∆ ), for any constant < 1. Our upper bounds are proven by constructing a topology recognition algorithm using a labeling scheme of length O(log log ∆) and using time O(D∆). Our lower bounds are proven by constructing a class of trees for which any topology recognition algorithm must use a labeling scheme of length at least Ω(log log ∆), and a class of trees for which any topology recognition algorithm using a labeling scheme of length O(log log ∆) must use time at least Ω(D∆ ), on some tree of this class. keywords: topology recognition, wireless network, labeling scheme, feasibility, tree, time ∗Département d’informatique, Université du Québec en Outaouais, Gatineau, Québec J8X 3X7, Canada. [email protected] †Département d’informatique, Université du Québec en Outaouais, Gatineau, Québec J8X 3X7, Canada. [email protected]. Partially supported by NSERC discovery grant 8136–2013 and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais. ar X iv :1 70 4. 01 92 7v 1 [ cs .D C ] 6 A pr 2 01 7