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(Dlog2n) 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(DlogDn+log2n) 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 TLay=O(DlogDn+log2+ϵn) 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(TLay+k) rounds. (2) A randomized k-message
broadcast algorithm running w.h.p. in O(TLay+klogn) 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(nlogn) round randomized gossip algorithm