Non-Adaptive Multi-Stage Algorithm for Group Testing with Prior Statistics

We understand selection by intersection as distinguishing a single element of a set by the uniqueness of its occurrence in some other set. More precisely, given two sets A and B, if A ∩ B = {z}, then element z∈ A is selected by set B. Selectors are such families \(\mathcal{S}\) of sets B of some domain that allow to select many elements from sufficiently small subsets A of the domain. Selectors are used in communication protocols for the multiple-access channel, in implementations of distributed-computing primitives in radio networks, and in algorithms for group testing. We give new explicit (n,k,r)-selectors of size \(\mathcal{O}(min [n, \frac{k^2}{k-r+1} polylog~n])\), for any parameters r ≤ k ≤ n. We establish a lower bound \(\Omega(min [n, \frac{k^2}{k-r+1} \cdot \frac{log(n/k)}{log(k/(k-r+1))}])\) on the length of (n,k,r)-selectors, which demonstrates that our construction is within a polylog n factor close to optimal. The new selectors are applied to develop explicit implementations of selection resolution on the multiple-access channel, gossiping in radio networks and an algorithm for group testing with inhibitors.KeywordsRadio NetworkCollision DetectionLocal ClockBinary ArrayCombinatorial Group TestThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the k-junta testing problem, a tester has to efficiently decide whether a given function f: {0, 1}n → {0, 1} is a k-junta (i.e., depends on at most k of its input bits) or is ∊-far from any k-junta. Our main result is a quantum algorithm for this problem with query complexity and time complexity . This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of Ω(k1/3) queries for junta-testing (for constant ∊).

The goal of group testing is to efficiently identify a few specific items, called positives, in a large population of items via tests. A test is an action on a subset of items that returns positive if the subset contains at least one positive and negative otherwise. In non-adaptive group testing, all tests are independent, can be performed in parallel, and represented as a measurement matrix. In this work, we consider non-adaptive group testing with consecutive positives in which the items are linearly ordered and the positives are consecutive in that order. We present two algorithms for efficiently identifying consecutive positives. In particular, without storing measurement matrices, we can identify up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d$</tex> consecutive positives with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2 \log_{2}\frac{\mathrm{n}}{d}+2d (4\log_{2}\frac{n}{d}+2d,\ resp.)$</tex> tests in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log_{2}^{2}\frac{n}{d}+d)\ (O(\log_{2}\frac{n}{d}+d)$</tex> , resp.) time. These results significantly improve the state-of-the-art scheme in which it takes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$5 \log_{2}\frac{n}{d} +2d+21$</tex> tests to identify the positives in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\frac{n}{d}\log_{2}\frac{n}{d}+d^{2})$</tex> time with the measurement matrices associated with the scheme stored somewhere.

A non-adaptive group testing (NGT) algorithm is useful in many areas. One of the examples is the problem of DNA library screening. While a mathematical model of NGT algorithm is a d-disjunct matrix. In this paper, we construct a d-disjunct matrix in a simple way. By discussing the Hamming distance of B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> (delta <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">**</sup> (n, d, k)), where delta**(n, d, k) is the matrix by augmenting the matrix delta**(n, d, k) with delta <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sup> (n, alpha, k), 1 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">les</sub> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alpha</sub> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">les</sub> m + 1 and alpha isin Z, we imply that our construction gives one error-correcting and two error-detecting NGT algorithm.

Group testing is an efficient algorithmic approach to the infection identification problem, based on mixing the test samples and testing the mixed samples instead of individually testing each sample. In this paper, we consider the dynamic infection spread model that is based on the discrete SIR model, which assumes the disease to be spread over time via infected and non-isolated individuals. In our system, the main objective is not to minimize the number of required tests to identify every infection, but instead, to utilize the available, given testing capacity T at each time instance to efficiently control the infection spread. We introduce and study a novel performance metric, which we coin as ϵ-disease control time. This metric can be used to measure how fast a given algorithm can control the spread of a disease. We characterize the performance of the dynamic individual testing algorithm and introduce a novel dynamic SAFFRON-based group testing algorithm. We present theoretical results and implement the proposed algorithms to compare their performances.

Consider a finite population of N items, where item i has a probability pi to be defective. The goal is to identify all items by means of group testing. This is the generalized group testing problem (hereafter GGTP). In the case of p1 = … = pN = p, Yao and Hwang (1990) proved that the pairwise testing algorithm is the optimal nested algorithm, with respect to the expected number of tests, for all N if and only if (R‐range hereafter) (an optimal at the boundary values). In this note, we present a result that helps to define the generalized pairwise testing algorithm (hereafter GPTA) for the GGTP. We present two conjectures: (a) when all pi,i = 1,…,N belong to the R‐range, GPTA is the optimal procedure among nested procedures applied to pi of nondecreasing order; and (b) if all pi,i = 1,…,N belong to the R‐range, GPTA is the optimal nested procedure, that is, minimizes the expected total number of tests with respect to all possible testing orders in the class of nested procedures. Although these conjectures are logically reasonable, we were only able to empirically verify the first one up to a particular level of N. We also provide a short survey of GGTP.

Let MA(n,d) denote the maximum number of group tests for a group testing algorithm A to identify d defectives from a set of n items when d is known, and let MA(n∣d) denote the number when d is unknown. Define M(n,d) = minA MA(n,d). An algorithm A is called a competitive algorithm if there exist constants c and a such that for all n > d ≥ 0, MA(n∣d) ≤ cM(n,d) + a. In this paper, we present some competitive group testing algorithms.

In traditional Combinatorial Group Testing the problem is to identify up to d defective items from a set of n items on the basis of group tests. In this paper we describe a variant of the group testing problem above, which we call parity group testing. The problem is to identify up to d defective items from a set of n items as in the classical group test problem. The main difference is that we check the parity of the defective items in a subset. The test can be applied to an arbitrary subset of the n items with two possible outcomes. The test is positive if the number of defective items in the subset is odd, otherwise it is negative. In this paper we extend Hirschberg et al.’s method to the parity group testing scenario.

This article is the rejoinder to:

The dangers of COVID-19 remain ever-present worldwide. The asymptomatic nature of COVID-19 obfuscates the signs policy makers look for when deciding to reopen public areas or further quarantine. In much of the world, testing resources are often scarce, creating a need for testing potentially infected individuals that prioritizes efficiency. This report presents an advancement to Beigel and Kasif’s Approximate Counting Algorithm (ACA). ACA estimates the infection rate with a number of tests that is logarithmic in the population size. Our newer version of the algorithm provides an extra level of efficiency: each subject is tested exactly once. A simulation of the algorithm, created for and presented as part of this paper, can be used to find a linear regression of the results with R^2 > 0.999. This allows stakeholders and members of the biomedical community to estimate infection rates for varying population sizes and ranges of infection rates.

We consider Bernoulli nonadaptive group testing with $k = \\Theta(n^\\theta)$\ndefectives, for $\\theta \\in (0,1)$. The practical definite defectives (DD)\ndetection algorithm is known to be optimal for $\\theta \\geq 1/2$. We give a new\nupper bound on the rate of DD, showing that DD is strictly suboptimal for\n$\\theta &lt; 0.41$. We also show that the SCOMP algorithm and algorithms based on\nlinear programming achieve a rate at least as high as DD, so in particular are\nalso optimal for $\\theta \\geq 1/2$.\n

Given a finite population P consisting of g good and d defective objects, the problems of hypergeometric group testing (HGT) is concerned with the identification of the d defectives by means of the following test procedure. Any subset X⊂P can be tested with two possible results: (1) either all elements of X are good, or (2) at least one element of X is defective. In the latter case, we have no knowledge as to which ones or how many are defective. In recent years, various algorithms have been proposed to solve this problem with the aim of minimizing the maximum number of tests required. However, no optimal algorithm is known at present for general g and d> 1.We define an algorithm s to solve the HGT problem to be an l-set algorithm if throughout the entire process of implementing s, we only need to partition the still unclassified elements of P into at most l sets (P1. ... Pl) where objects in each Pi need not be differentiated from one another. Restricting s to be an l-set algorithm limits the range of useful tests we can make as the amount of information we can keep after each successive tests depends on l. We show that all previously known HGT algorithms are either 1 or 2-set algorithms. By increasing l, we are able to construct some new classes of HGT algorithms which are more efficient than all previously known algorithms. Finally, we are able to construct optimal HGT algorithms for l≤3.