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Improved Lower Bound for Estimating the Number of Defective Items
Authors
Nader H. Bshouty
Publication date
15 August 2023
Publisher
View
on
arXiv
Abstract
Let
X
X
X
be a set of items of size
n
n
n
that contains some defective items, denoted by
I
I
I
, where
I
β
X
I \subseteq X
I
β
X
. In group testing, a {\it test} refers to a subset of items
Q
β
X
Q \subset X
Q
β
X
. The outcome of a test is
1
1
1
if
Q
Q
Q
contains at least one defective item, i.e.,
Q
β©
I
β
β
Q\cap I \neq \emptyset
Q
β©
I
ξ
=
β
, and
0
0
0
otherwise. We give a novel approach to obtaining lower bounds in non-adaptive randomized group testing. The technique produced lower bounds that are within a factor of
1
/
log
β‘
log
β‘
β―
k
log
β‘
n
1/{\log\log\stackrel{k}{\cdots}\log n}
1/
lo
g
lo
g
β―
k
β
lo
g
n
of the existing upper bounds for any constant~
k
k
k
. Employing this new method, we can prove the following result. For any fixed constants
k
k
k
, any non-adaptive randomized algorithm that, for any set of defective items
I
I
I
, with probability at least
2
/
3
2/3
2/3
, returns an estimate of the number of defective items
β£
I
β£
|I|
β£
I
β£
to within a constant factor requires at least
Ξ©
(
log
β‘
n
log
β‘
log
β‘
β―
k
log
β‘
n
)
\Omega\left(\frac{\log n}{\log\log\stackrel{k}{\cdots}\log n}\right)
Ξ©
(
lo
g
lo
g
β―
k
β
lo
g
n
lo
g
n
β
)
tests. Our result almost matches the upper bound of
O
(
log
β‘
n
)
O(\log n)
O
(
lo
g
n
)
and solves the open problem posed by Damaschke and Sheikh Muhammad [COCOA 2010 and Discrete Math., Alg. and Appl., 2010]. Additionally, it improves upon the lower bound of
Ξ©
(
log
β‘
n
/
log
β‘
log
β‘
n
)
\Omega(\log n/\log\log n)
Ξ©
(
lo
g
n
/
lo
g
lo
g
n
)
previously established by Bshouty [ISAAC 2019]
Similar works
Full text
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oai:arXiv.org:2308.07721
Last time updated on 18/08/2023