Short Labeling Schemes for Topology Recognition in Wireless Tree Networks

Abstract

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 $\varDelta$?

• What is the fastest topology recognition algorithm working for all wireless tree networks of diameter D and maximum degree $\varDelta$, 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 $\varDelta \ge 3$ is $\varTheta (\log \log \varDelta )$. For such short schemes, used by an algorithm working for the class of trees of diameter $D\ge 4$ and maximum degree $\varDelta \ge 3$, we show almost matching bounds on the time of topology recognition: an upper bound $O(D\varDelta )$, and a lower bound $\varOmega (D\varDelta ^{\epsilon })$, for any constant $\epsilon <1$.

Our upper bounds are proven by constructing a topology recognition algorithm using a labeling scheme of length $O(\log \log \varDelta )$ and using time $O(D\varDelta )$. 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 $\varOmega (\log \log \varDelta )$, and a class of trees for which any topology recognition algorithm using a labeling scheme of length $O(\log \log \varDelta )$ must use time at least $\varOmega (D\varDelta ^{\epsilon })$, on some tree of this class.

A. Pelc—Partially supported by NSERC discovery grant 8136–2013 and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais.