3 research outputs found
Non-adaptive Quantitative Group Testing Using Irregular Sparse Graph Codes
This paper considers the problem of Quantitative Group Testing (QGT) where
there are some defective items among a large population of items. We
consider the scenario in which each item is defective with probability ,
independently from the other items. In the QGT problem, the goal is to identify
all or a sufficiently large fraction of the defective items by testing groups
of items, with the minimum possible number of tests. In particular, the outcome
of each test is a non-negative integer which indicates the number of defective
items in the tested group. In this work, we propose a non-adaptive QGT scheme
for the underlying randomized model for defective items, which utilizes sparse
graph codes over irregular bipartite graphs with optimized degree profiles on
the left nodes of the graph as well as binary -error-correcting BCH codes.
We show that in the sub-linear regime, i.e., when the ratio vanishes as
grows unbounded, the proposed scheme with tests can identify all the defective items with probability
approaching , where and are the maximum and average left degree,
respectively, and depends only on and (and does not depend on
and ). For any , the testing and recovery algorithms of the
proposed scheme have the computational complexity of and , respectively. The proposed
scheme outperforms two recently proposed non-adaptive QGT schemes for the
sub-linear regime, including our scheme based on regular bipartite graphs and
the scheme of Gebhard et al., in terms of the number of tests required to
identify all defective items with high probability.Comment: 7 pages; This work was presented at the 57th Annual Allerton
Conference on Communication, Control, and Computing (Allerton'19),
Monticello, Illinois, USA, Sept 201
Quantitative group testing in the sublinear regime
The quantitative group testing (QGT) problem deals with efficiently
identifying a small number of infected individuals among a large population. To
this end, we can test groups of individuals where each test returns the total
number of infected individuals in the tested pool. For the regime where the
number of infected individuals is sublinear in the total population we derive
the information-theoretic threshold for the minimum number of tests required to
identify the infected individuals with high probability. Such a threshold was
so far only known for the case where the infected individuals are a constant
fraction of the population (Alaoui et al. 2019, Scarlett & Cevher 2017).
Moreover, we propose and analyze an efficient greedy reconstruction algorithm
that outperforms the best known algorithm (Karimi et al. 2019) for certain
sparsity regimes
Quantitative Group Testing and the rank of random matrices
Given a random Bernoulli matrix , an integer and the vector , where is of Hamming
weight , the objective in the {\em Quantitative Group Testing} (QGT)
problem is to recover . This problem is more difficult the smaller is.
For parameter ranges of interest to us, known polynomial time algorithms
require values of that are much larger than .
In this work, we define a seemingly easier problem that we refer to as {\em
Subset Select}. Given the same input as in QGT, the objective in Subset Select
is to return a subset of cardinality , such that for
all , if then . We show that if the square
submatrix of defined by the columns indexed by has nearly full rank,
then from the solution of the Subset Select problem we can recover in
polynomial-time the solution to the QGT problem. We conjecture that for
every polynomial time Subset Select algorithm, the resulting output matrix will
satisfy the desired rank condition. We prove the conjecture for some classes of
algorithms. Using this reduction, we provide some examples of how to improve
known QGT algorithms. Using theoretical analysis and simulations, we
demonstrate that the modified algorithms solve the QGT problem for values of that are smaller than those required for the original algorithms