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Added by
Tao Guo
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper, we propose algorithms that leverage a known community structure to make group testing more efficient. We consider a population organized in disjoint communities: each individual participates in a community, and its infection probability depends on the community (s)he participates in. Use cases include families, students who participate in several classes, and workers who share common spaces. Group testing reduces the number of tests needed to identify the infected individuals by pooling diagnostic samples and testing them together. We show that if we design the testing strategy taking into account the community structure, we can significantly reduce the number of tests needed for adaptive and non-adaptive group testing, and can improve the reliability in cases where tests are noisy.
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In this paper, we propose algorithms that leverage a known community structure to make group testing more efficient. We consider a population organized in connected communities: each individual participates in one or more communities, and the infection probability of each individual depends on the communities (s)he participates in. Use cases include students who participate in several classes, and workers who share common spaces. Group testing reduces the number of tests needed to identify the infected individuals by pooling diagnostic samples and testing them together. We show that making testing algorithms aware of the community structure, can significantly reduce the number of tests needed both for adaptive and non-adaptive group testing.
We will discuss superimposed codes and non-adaptive group testing designs arising from the potentialities of compressed genotyping models in molecular biology. The given paper was motivated by the 30th anniversary of Dyachkov-Rykov recurrent upper bound on the rate of superimposed codes published in 1982. We were also inspired by recent results obtained for non-adaptive threshold group testing which develop the theory of superimposed codes
We consider non-adaptive threshold group testing for identification of up to $d$ defective items in a set of $n$ items, where a test is positive if it contains at least $2 leq u leq d$ defective items, and negative otherwise. The defective items can be identified using $t = O left( left( frac{d}{u} right)^u left( frac{d}{d - u} right)^{d-u} left(u log{frac{d}{u}} + log{frac{1}{epsilon}} right) cdot d^2 log{n} right)$ tests with probability at least $1 - epsilon$ for any $epsilon > 0$ or $t = O left( left( frac{d}{u} right)^u left( frac{d}{d -u} right)^{d - u} d^3 log{n} cdot log{frac{n}{d}} right)$ tests with probability 1. The decoding time is $t times mathrm{poly}(d^2 log{n})$. This result significantly improves the best known results for decoding non-adaptive threshold group testing: $O(nlog{n} + n log{frac{1}{epsilon}})$ for probabilistic decoding, where $epsilon > 0$, and $O(n^u log{n})$ for deterministic decoding.
We consider an efficiently decodable non-adaptive group testing (NAGT) problem that meets theoretical bounds. The problem is to find a few specific items (at most $d$) satisfying certain characteristics in a colossal number of $N$ items as quickly as possible. Those $d$ specific items are called textit{defective items}. The idea of NAGT is to pool a group of items, which is called textit{a test}, then run a test on them. If the test outcome is textit{positive}, there exists at least one defective item in the test, and if it is textit{negative}, there exists no defective items. Formally, a binary $t times N$ measurement matrix $mathcal{M} = (m_{ij})$ is the representation for $t$ tests where row $i$ stands for test $i$ and $m_{ij} = 1$ if and only if item $j$ belongs to test $i$. There are three main objectives in NAGT: minimize the number of tests $t$, construct matrix $mathcal{M}$, and identify defective items as quickly as possible. In this paper, we present a strongly explicit construction of $mathcal{M}$ for when the number of defective items is at most 2, with the number of tests $t simeq 16 log{N} = O(log{N})$. In particular, we need only $K simeq N times 16log{N} = O(Nlog{N})$ bits to construct such matrices, which is optimal. Furthermore, given these $K$ bits, any entry in the matrix can be constructed in time $O left(ln{N}/ ln{ln{N}} right)$. Moreover, $mathcal{M}$ can be decoded with high probability in time $Oleft( frac{ln^2{N}}{ln^2{ln{N}}} right)$. When the number of defective items is greater than 2, we present a scheme that can identify at least $(1-epsilon)d$ defective items with $t simeq 32 C(epsilon) d log{N} = O(d log{N})$ in time $O left( d frac{ln^2{N}}{ln^2{ln{N}}} right)$ for any close-to-zero $epsilon$, where $C(epsilon)$ is a constant that depends only on $epsilon$.
The goal of threshold group testing is to identify up to $d$ defective items among a population of $n$ items, where $d$ is usually much smaller than $n$. A test is positive if it has at least $u$ defective items and negative otherwise. Our objective is to identify defective items in sublinear time the number of items, e.g., $mathrm{poly}(d, ln{n}),$ by using the number of tests as low as possible. In this paper, we reduce the number of tests to $O left( h times frac{d^2 ln^2{n}}{mathsf{W}^2(d ln{n})} right)$ and the decoding time to $O left( mathrm{dec}_0 times h right),$ where $mathrm{dec}_0 = O left( frac{d^{3.57} ln^{6.26}{n}}{mathsf{W}^{6.26}(d ln{n})} right) + O left( frac{d^6 ln^4{n}}{mathsf{W}^4(d ln{n})} right)$, $h = Oleft( frac{d_0^2 ln{frac{n}{d_0}}}{(1-p)^2} right)$ , $d_0 = max{u, d - u }$, $p in [0, 1),$ and $mathsf{W}(x) = Theta left( ln{x} - ln{ln{x}} right).$ If the number of tests is increased to $Oleft( h times frac{d^2ln^3{n}}{mathsf{W}^2(d ln{n})} right),$ the decoding complexity is reduced to $O left(mathrm{dec}_1 times h right),$ where $mathrm{dec}_1 = max left{ frac{d^2 ln^3{n}}{mathsf{W}^2(d ln{n})}, frac{ud ln^4{n}}{mathsf{W}^3(d ln{n})} right}.$ Moreover, our proposed scheme is capable of handling errors in test outcomes.
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