Fast Structuring of Radio Networks Large for Multi-message Communications

Abstract

We introduce collision free layerings as a powerful way to structure radio networks. These layerings can replace hard-to-compute BFS-trees in many contexts while having an efficient randomized distributed construction. We demonstrate their versatility by using them to provide near optimal distributed algorithms for several multi-message communication primitives.

Designing efficient communication primitives for radio networks has a rich history that began 25 years ago when Bar-Yehuda et al. introduced fast randomized algorithms for broadcasting and for constructing BFS-trees. Their BFS-tree construction time was O(D log² n) rounds, where D is the network diameter and n is the number of nodes. Since then, the complexity of a broadcast has been resolved to be $T_{BC} = \Theta(D \log \frac{n}{D} + \log^2 n)$ rounds. On the other hand, BFS-trees have been used as a crucial building block for many communication primitives and their construction time remained a bottleneck for these primitives.

We introduce collision free layerings that can be used in place of BFS-trees and we give a randomized construction of these layerings that runs in nearly broadcast time, that is, w.h.p. in $T_{Lay} = O(D \log \frac{n}{D} + \log^{2+\epsilon} n)$ rounds for any constant ε > 0. We then use these layerings to obtain: (1) A randomized algorithm for gathering k messages running w.h.p. in O(T _Lay  + k) rounds. (2) A randomized k-message broadcast algorithm running w.h.p. in O(T _Lay  + k logn) rounds. These algorithms are optimal up to the small difference in the additive poly-logarithmic term between T _BC and T _Lay . Moreover, they imply the first optimal O(n logn) round randomized gossip algorithm.