Centralized Deterministic Broadcasting in Undirected Multi-hop Radio Networks

We consider centralized deterministic broadcasting in radio networks. The aim is to design a polynomial algorithm, which, given a graph G, produces a fast broadcasting scheme in the radio network represented by G. The problem of finding an optimal broadcasting scheme for a given graph is NP-hard, hence we can only hope for a good approximation algorithm. We give a deterministic polynomial algorithm which produces a broadcasting scheme working in time \({\cal O}(D \log n + \log ^2 n)\), for every n-node graph of diameter D. It has been proved recently [15,16] that a better order of magnitude of broadcasting time is impossible unless \(NP \subseteq BPTIME(n^{{\cal O}(\log \log n)})\). In terms of approximation ratio, we have a \({\cal O}(\log (n/D))\)-approximation algorithm for the radio broadcast problem, whenever D=Ω (log n).