On many-to-many communication in packet radio networks

Radio networks model wireless data communication when bandwidth is limited to one wave frequency. The key restriction of such networks is mutual interference of packets arriving simultaneously to a node. The many-to-many (m2m) communication primitive involves p participant nodes of a distance at most d between any pair of them, from among n nodes in the network, and the task is to have all participants get to know all input messages. We consider three cases of the m2m communication problem. In the ad-hoc case, each participant knows only its name and the values of n, p and d. In the partially centralized case, each participant knows the topology of the network and the values of p and d, but does not know the names of other participants. In the centralized case each participant knows the topology of the network and the names of all the participants. For the centralized m2m problem, we give deterministic protocols, for both undirected and directed networks, working in O(d + p) time, which is provably optimal. For the partially centralized m2m problem, we give a randomized protocol for undirected networks working in O((d + p+log² n) log p) time with high probability (whp), and we show that any deterministic protocol requires Ω(plog_n/p n+d) time. For the ad-hoc m2m problem, we develop a randomized protocol for undirected networks that works in O((d +log p) log² n+ plog p) time whp. We show two lower bounds for the ad-hoc m2m problem. One states that any m2m deterministic protocol requires Ω(nlog_n/d+1 n) time when n-p=Ω(n) and d >1;Ω(n) time when n-p=o(n); and Ω(plog_n/p n) time when d = 1. The other lower bound states that any m2m randomized protocol requires Ω(p+d log(n/d +1)+log² n) expected time.