  In this work, we give a new parallel algorithm for the problem of maximizing
a non-monotone diminishing returns submodular function subject to a cardinality
constraint. For any desired accuracy $\epsilon$, our algorithm achieves a $1/e
- \epsilon$ approximation using $O(\log{n} \log(1/\epsilon) / \epsilon^3)$
parallel rounds of function evaluations. The approximation guarantee nearly
matches the best approximation guarantee known for the problem in the
sequential setting and the number of parallel rounds is nearly-optimal for any
constant $\epsilon$. Previous algorithms achieve worse approximation guarantees
using $\Omega(\log^2{n})$ parallel rounds. Our experimental evaluation suggests
that our algorithm obtains solutions whose objective value nearly matches the
value obtained by the state of the art sequential algorithms, and it
outperforms previous parallel algorithms in number of parallel rounds,
iterations, and solution quality.
