Optimal-Length Labeling Schemes for Fast Deterministic Communication in Radio Networks

We consider two fundamental communication tasks in arbitrary radio networks: broadcasting (information from one source has to reach all nodes) and gossiping (every node has a message and all messages have to reach all nodes). Nodes are assigned labels that are (not necessarily different) binary strings. Each node knows its own label and can use it as a parameter in the same deterministic algorithm. The length of a labeling scheme is the largest length of a label. The goal is to find labeling schemes of asymptotically optimal length for the above tasks, and to design fast deterministic distributed algorithms for each of them, using labels of optimal length. Our main result concerns broadcasting. We show the existence of a labeling scheme of constant length that supports broadcasting in time $O(D+\log^2 n)$, where $D$ is the diameter of the network and $n$ is the number of nodes. This broadcasting time is an improvement over the best currently known $O(D\log n + \log^2 n)$ time of broadcasting with constant-length labels, due to Ellen and Gilbert (SPAA 2020). It also matches the optimal broadcasting time in radio networks of known topology. Hence, we show that appropriately chosen node labels of constant length permit to achieve, in a distributed way, the optimal centralized broadcasting time. This is, perhaps, the most surprising finding of this paper. We are able to obtain our result thanks to a novel methodological tool of propagating information in radio networks, that we call a 2-height respecting tree. Next, we apply our broadcasting algorithm to solve the gossiping problem. We get a gossiping algorithm working in time $O(D + \Delta\log n + \log^2 n)$, using a labeling scheme of optimal length $O(\log \Delta)$, where $\Delta$ is the maximum degree. Our time is the same as the best known gossiping time in radio networks of known topology.