  In this paper, we develop fast algorithms for two stochastic submodular
maximization problems. We start with the well-studied adaptive submodular
maximization problem subject to a cardinality constraint. We develop the first
linear-time algorithm which achieves a $(1-1/e-\epsilon)$ approximation ratio.
Notably, the time complexity of our algorithm is $O(n\log\frac{1}{\epsilon})$
(number of function evaluations) which is independent of the cardinality
constraint, where $n$ is the size of the ground set. Then we introduce the
concept of fully adaptive submodularity, and develop a linear-time algorithm
for maximizing a fully adaptive submoudular function subject to a partition
matroid constraint. We show that our algorithm achieves a
$\frac{1-1/e-\epsilon}{4-2/e-2\epsilon}$ approximation ratio using only
$O(n\log\frac{1}{\epsilon})$ number of function evaluations.
