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

We consider the fundamental problems of size discovery and topology recognition in radio networks modeled by simple undirected connected 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. The time of a deterministic algorithm for each of the above problems is the worst-case number of rounds it takes to solve it. Our goal is to construct short labeling schemes for size discovery and topology recognition in arbitrary radio networks, and to design efficient deterministic algorithms using these schemes. For size discovery, we construct a labeling scheme of length $O(\log\log\Delta)$ and we design an algorithm for this problem using this scheme and working in time $O(\log^2 n)$, where $n$ is the size of the graph. We also show that time complexity $O(\log^2 n)$ is optimal for the problem of size discovery, whenever the labeling scheme is of optimal length. For topology recognition, we construct a labeling scheme of length $O(\log\Delta)$, and we design an algorithm for this problem using this scheme working in time $O\left(D\Delta+\min(\Delta^2,n)\right)$. We also show that the length of our labeling scheme is asymptotically optimal.