We study a generalization of the well-known model of broadcasting on trees.
Consider a directed acyclic graph (DAG) with a unique source vertex X, and
suppose all other vertices have indegree d≥2. Let the vertices at
distance k from X be called layer k. At layer 0, X is given a random
bit. At layer k≥1, each vertex receives d bits from its parents in
layer k−1, which are transmitted along independent binary symmetric channel
edges, and combines them using a d-ary Boolean processing function. The goal
is to reconstruct X with probability of error bounded away from 1/2 using
the values of all vertices at an arbitrarily deep layer. This question is
closely related to models of reliable computation and storage, and information
flow in biological networks.
In this paper, we analyze randomly constructed DAGs, for which we show that
broadcasting is only possible if the noise level is below a certain degree and
function dependent critical threshold. For d≥3, and random DAGs with
layer sizes Ω(logk) and majority processing functions, we identify the
critical threshold. For d=2, we establish a similar result for NAND
processing functions. We also prove a partial converse for odd d≥3
illustrating that the identified thresholds are impossible to improve by
selecting different processing functions if the decoder is restricted to using
a single vertex.
Finally, for any noise level, we construct explicit DAGs (using expander
graphs) with bounded degree and layer sizes Θ(logk) admitting
reconstruction. In particular, we show that such DAGs can be generated in
deterministic quasi-polynomial time or randomized polylogarithmic time in the
depth. These results portray a doubly-exponential advantage for storing a bit
in DAGs compared to trees, where d=1 but layer sizes must grow exponentially
with depth in order to enable broadcasting.Comment: 33 pages, double column format. arXiv admin note: text overlap with
arXiv:1803.0752