Fast Structuring of Radio Networks for Multi-Message Communications

We introduce collision free layerings as a powerful way to structure distributed radio networks. These layerings can replace hard-to-compute BFS-trees in many contexts while having an efficient randomized 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 a BFS-tree. Their BFS-tree construction time was O(D log 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 TBC = Θ(D log 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 the BFS-tree 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, whp in TLay = O(D log n D + log n) rounds for any constant > 0. We then use these layerings to obtain: (1) A randomized kmessage broadcast running whp in O(TLay + k log n) rounds. (2) A randomized algorithm for gathering k messages running whp in O(TLay + k) rounds. These algorithms are optimal up to the small difference in the additive poly-logarithmic term between TBC and TLay . Moreover, they imply the first optimal O(n log n) round randomized gossip algorithm.