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. 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.