Deterministic Size Discovery and Topology Recognition in Radio Networks with Short Labels

Abstract

We consider the fundamental problems of size discovery and topology recognition in radio networks modeled by simple undirected graphs. Size discovery calls for all nodes to output the number of nodes in the graph, called its size, and in the task of topology recognition each node has to learn the topology of the graph and its position in it. In radio networks, nodes communicate in synchronous rounds and start in the same round. In each round a node can either transmit the same 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. If more than one neighbor of a node v transmits in a given round, there is a collision at v. We do not assume collision detection: in case of a collision, node v does not hear anything (except the background noise that it also hears when no neighbor transmits). The time of a deterministic algorithm for each of the above problems is the worst-case number of rounds it takes to solve it. Nodes have labels which are (not necessarily different) binary strings. Each node knows its own label and can use it when executing the algorithm. The length of a labeling scheme is the largest length of a label. Our goal is to construct short labeling schemes for size discovery and topology recognition in arbitrary radio networks, and to design efficient deterministic algorithms for each of these tasks, using these short schemes. It turns out that the optimal length of labeling schemes for both problems depends on the maximum degree Δ of the graph. For size discovery, we construct a labeling scheme of length O(log log Δ) (which is known to be optimal, even if collision detection is available) and we design an algorithm for this problem using this scheme and working in time O(log2n), where n is the size of the graph. We also show that time complexity O(log2n) is optimal for the problem of size discovery, whenever the labeling scheme is of optimal length O(log log Δ). For topology recognition, we construct a labeling scheme of length O(log Δ), and we design an algorithm for this problem using this scheme and working in time O(DΔ+min(Δ2,n), where D is the diameter of the graph. We also show that the length of our labeling scheme is asymptotically optimal, by proving that topology recognition in the class of arbitrary radio networks requires labeling schemes of length Ω(logΔ).