On adaptive deterministic gossiping in ad hoc radio networks

We study deterministic algorithms for gossiping problem in ad hoc radio networks. The efficiency of communication algorithms in radio networks is very often expressed in terms of: maximum eccentricity D, maximum in-degree Δ, and size (number of nodes) n of underlying graph of connections. The maximum eccentricity D of a network is the maximum of the lengths of shortest directed paths from a node u to a node v, taken over all ordered pairs (u,v) of nodes in the network. The maximum in-degree Δ of a network is the maximum of in-degrees of its nodes. We propose a new method that leads to several improvements in deterministic gossiping. It combines communication techniques designed for both known as well as unknown ad hoc radio networks. First we show how to subsume the O(Dn)-time bound yield by the round-robin procedure proposing a new Õ(√Dn)-time gossiping algorithm.¹ Our algorithm is more efficient than the known Õ(n^3/2)-time gossiping algorithms [Proc. 41st IEEE Symp. on Found. of Computer Science, 2000, pp. 575-581; Proc. 13th ACM-SIAM Symp. on Discrete Algorithms, 2002], whenever D=O(n^α) and α < 1. For large values of maximum eccentricity D, we give another gossiping algorithm that works in time O(DΔ^3/2log³n) which subsumes the O(DΔ²log³n) upper bound presented in [Proc. 20th ACM Symp. on Principles of Distributed Computing, 2001, pp. 255-263]. Finally, we observe that for any so-called oblivious (i.e., non-adaptive) deterministic gossiping algorithm, any natural n and 1 ≤ D ≤ n-1, there is an unknown ad hoc radio network of size n and maximum eccentricity D which requires Ω(Dn) time-steps to complete gossiping.