Constant-Length Labelling Schemes for Faster Deterministic Radio Broadcast

In this paper, we consider the problem of broadcast from a specified source node in a known synchronous radio network. In 2019, Ellen, Gorain, Miller and Plc showed that this is possible if each node in the network only stores 2 (carefully chosen) bits of information. They proved that in an n-node network, their algorithm ensures that the broadcast completes within 2n-3 rounds. We show that storing only a small constant number of additional bits, it is possible to broadcast significantly faster when the source eccentricity, D, of the network is o(n). We begin by presenting a modification of Ellen, Gorain, Miller and Pelc's algorithm that stores 4 bits per node and completes within O(√Dn) rounds. Then we define a class of broadcast algorithms that includes both these algorithms and prove that any algorithm in this class requires Ω(√nD) rounds. Next, we show (non-constructively) that there exists a broadcast algorithm which stores only 3 bits of information per node, but completes within O(D log n+log2 n) rounds. Finally, using ideas from Chlamtac and Weinstein (1991), we show how to construct the relevant information and do almost as well, giving an algorithm using 3 bits per node that runs in O(D log^2 n) rounds.