Improved schedule for radio broadcast

We show that for every radio network G = (V, E) and source s ε V, there exists a radio broadcast schedule for G of length Rad(G, s) + O(√Rad(G, s) · log² n) = O(Rad(G, s) + log⁴ n), where Rad(G, s) is the radius of the radio network G with respect to the source s. This result improves the previously best-known upper bound of O(Rad(G, s) + log⁵ n) due to Gaber and Mansour. For graphs with small genus, particularly for planar graphs, we provide an even better upper bound of Rad(G, S) + O(√Rad(G, s) · log n + log³ n) = O(Rad(G, s)+ log³n).