Average-time complexity of gossiping in radio networks

Radio networks model wireless synchronous communication with only one wave frequency used for transmissions. In the problem of many-to-all (M2A) communication, some nodes hold input rumors, and the goal is to have all nodes learn all the rumors. We study the average time complexity of distributed many-to-all communication by deterministic protocols in directed networks under two scenarios: of combined messages, in which all input rumors can be sent in one packet, and of separate messages, in which every rumor requires a separate packet to be transmitted. Let n denote the size of a network and k be the number of nodes activated with rumors; the case when k = n is called gossiping. We give & gossiping protocol for combined messages that works in the average time script O sign(n/log n), which is shown to be optimal. For the general M2A communication problem, we show that it can be performed in the average time script O sign(min{k log(n/k),n/log n}) with combined messages, and that Ω(k/log n + log n) is a lower bound. We give a gossiping protocol for separate messages that works in the average time script O sign(n log n), which is shown to be optimal. For the general M2A communication problem, we develop a protocol for separate messages with the average time script O sign(k log(n/k) log n), and show that Ω(k log n) is a lower bound.