2 research outputs found
Geometric group testing
Group testing is concerned with identifying defective items in a set of
items, where each test reports whether a specific subset of items contains
at least one defective. In non-adaptive group testing, the subsets to be tested
are fixed in advance. By testing multiple items at once, the required number of
tests can be made much smaller than . In fact, for ,
the optimal number of (non-adaptive) tests is known to be .
In this paper, we consider the problem of non-adaptive group testing in a
geometric setting, where the items are points in -dimensional Euclidean
space and the tests are axis-parallel boxes (hyperrectangles). We present upper
and lower bounds on the required number of tests under this geometric
constraint. In contrast to the general, combinatorial case, the bounds in our
geometric setting are polynomial in . For instance, our results imply that
identifying a defective pair in a set of points in the plane always
requires tests, and there exist configurations of points
for which tests are sufficient, whereas to identify a
single defective point in the plane, tests are always
necessary and sometimes sufficient
On Adaptive Distance Estimation
We provide a static data structure for distance estimation which supports
{\it adaptive} queries. Concretely, given a dataset of
points in and , we construct a randomized data
structure with low memory consumption and query time which, when later given
any query point , outputs a -approximation of
with high probability for all . The main
novelty is our data structure's correctness guarantee holds even when the
sequence of queries can be chosen adaptively: an adversary is allowed to choose
the th query point in a way that depends on the answers reported by
the data structure for . Previous randomized Monte Carlo
methods do not provide error guarantees in the setting of adaptively chosen
queries. Our memory consumption is , slightly more
than the required to store in memory explicitly, but with the
benefit that our time to answer queries is only , much faster than the naive time obtained from a linear scan
in the case of and very large. Here hides
factors. We discuss applications to nearest neighbor search
and nonparametric estimation.
Our method is simple and likely to be applicable to other domains: we
describe a generic approach for transforming randomized Monte Carlo data
structures which do not support adaptive queries to ones that do, and show that
for the problem at hand, it can be applied to standard nonadaptive solutions to
norm estimation with negligible overhead in query time and a factor
overhead in memory.Comment: Minor correction in proof of Lemma B.