  In group testing, the goal is to identify a subset of defective items within
a larger set of items based on tests whose outcomes indicate whether at least
one defective item is present. This problem is relevant in areas such as
medical testing, DNA sequencing, communication protocols, and many more. In
this paper, we study (i) a sparsity-constrained version of the problem, in
which the testing procedure is subjected to one of the following two
constraints: items are finitely divisible and thus may participate in at most
$\gamma$ tests; or tests are size-constrained to pool no more than $\rho$ items
per test; and (ii) a noisy version of the problem, where each test outcome is
independently flipped with some constant probability. Under each of these
settings, considering the for-each recovery guarantee with asymptotically
vanishing error probability, we introduce a fast splitting algorithm and
establish its near-optimality not only in terms of the number of tests, but
also in terms of the decoding time. While the most basic formulations of our
algorithms require $\Omega(n)$ storage for each algorithm, we also provide
low-storage variants based on hashing, with similar recovery guarantees.
