A Construction of an Error-Correcting Pooling Design

Disjunct Matrices (DM) are a particular kind of binary matrices which have been especially applied to solve the Non-Adaptive Group Testing (NAGT) problem, where the task is to detect any configuration of t defectives out of a population of N items. Traditionally, the methods used to construct DM leverage on error-correcting codes and other related algebraic techniques. Here, we investigate the use of Evolutionary Algorithms to design DM and two of their generalizations, namely Resolvable Matrices (RM) and Almost Disjunct Matrices (ADM). After discussing the basic encoding used to represent the candidate solutions of our optimization problems, we define three fitness functions, each measuring the deviation of a generic binary matrix from being respectively a DM, an RM or an ADM. Next, we employ Estimation of Distribution Algorithms (EDA), Genetic Algorithms (GA), and Genetic Programming (GP) to optimize these fitness functions. The results show that GP achieves the best performances among the three heuristics, converging to an optimal solution on a wider range of problem instances. Although these results do not match those obtained by other state-of-the-art methods in the literature, we argue that our heuristic approach can generate solutions that are not expressible by currently known algebraic techniques, and sketch some possible ideas to further improve its performance.

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.

In group testing, simple binary-output tests are designed to identify a small number $t$ of defective items that are present in a large population of $N$ items. Each test takes as input a group of items and produces a binary output indicating whether the group is free of the defective items or contains one or more of them. In this paper, we study a relaxation of the combinatorial group testing problem. A matrix is called $(t,\epsilon )$ -disjunct if it gives rise to a nonadaptive group testing scheme with the property of identifying a uniformly random $t$ -set of defective subjects out of a population of size $N$ with false positive probability of an item at most $\epsilon $ . We establish a new connection between $(t,\epsilon )$ -disjunct matrices and error correcting codes based on the dual distance of the codes and derive estimates of the parameters of codes that give rise to such schemes. Our methods rely on the moments of the distance distribution of codes and inequalities for moments of sums of independent random variables. We also provide a new connection between group testing schemes and combinatorial designs.

Currently, coronavirus disease 2019 (COVID-19) has become the most severe infectious disease affecting the world, which has spread around the world to more than 200 countries in 2020. Until the number of COVID-19 vaccines is insufficient, nucleic acid testing is considered as an effective way to screen virus carriers and control the spread of the virus. Considering that the medical resources and infection rates are different across various countries and regions, if all infected areas adopt the traditional individual nucleic acid testing method, the workload will be heavy and time-consuming. Therefore, this will not lead to the control of the pandemic. After Wuhan completed a citywide nucleic acid testing in May 2020, China basically controlled the spread of COVID-19 and entered the post-epidemic period. Since then, although some cities in China, such as Qingdao, Xinjiang, Beijing, and Dalian, have experienced a local epidemic resurgence, the pandemic was quickly suppressed through wide-scale pooled nucleic acid testing methods. Combined with the successful experience of mass nucleic acid testing in China, this study introduces two main pooled testing methods used in two cities with a population of more than ten million people, Wuhan’s “five-in-one” and Qingdao’s “ten-in-one” rapid pooled testing methods. This study proposes an improved method for optimising the second round of “ten-in-one” pooled testing, known as “the pentagram mini-pooled testing method”, which speeds up the testing process (as a result of reducing the numbers of testing by 40%) and significantly reduces the cost. Qingdao’s optimised “ten-in-one” pooled testing method quickly screens out the infections by running fewer testing samples. This study also mathematically examines the probabilistic principles and applicability conditions for pooled testing of COVID-19. Herein, the study theoretically determines the optimal number of samples that could successfully be combined into a pool under different infection rates. Then, it quantitatively discusses the applicability and principles for choosing the pooled testing instead of individual testing. Overall, this research offers a reference for other countries with different infection rates to help them in implementing the mass testing for COVID-19 to reduce the spread of coronavirus.

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.

Coronavirus disease (COVID-19) has emerged as a global pandemic for public health due to the large scale outbreak, therefore there is an urgent need to detect the infected cases quickly and isolate them in order to suppress the further spread of the disease. This study tries to identify a suitable pool testing method and algorithm for COVID-19. This study tries to derive a general equation for the number of tests required for a pooled sample to detect every infected individual in the specific pool. The gain in pool testing over the normal procedure is quantified by the percentage of tests required compared to individual testing. The percentage of tests required by the pool testing strategy varies according to the different splitting procedures, the size of the pooled sample, and the probability of an individual being infected in the population. If the probability of infection is 0.05, then for a pool size of 32, only 14 tests are sufficient to detect every infected individual. The number of tests required to detect infected individuals by using the pooling method is much lower than individual testing. This may help us with increasing our testing capacity for COVID-19 by testing a large number of individuals in less time with limited resources.

Despite a large volume of research in group testing, explicit small-size group testing schemes are still difficult to construct, and the parameters of known combinatorial schemes are limited by the constraints of the problem. Relaxing the worst-case identification requirements to probabilistic localization of defectives enables one to expand the range of parameters, and yet the small-size practical constructions are sparse. Motivated by this question, we perform an experimental study of almost disjunct matrices constructed from low-weight codewords of binary BCH codes, and evaluate their performance in nonadaptive group testing. We observe that identification of defectives is much more stable in these schemes compared to the schemes constructed from random binary matrices. We derive an estimate of the error probability of identification in the constructed schemes which provides a partial explanation of their performance.

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 task of non-adaptive group testing is to identify up to $d$ defective items from $N$ items, where a test is positive if it contains at least one defective item, and negative otherwise. If there are $t$ tests, they can be represented as a $t \times N$ measurement matrix. We have answered the question of whether there exists a scheme such that a larger measurement matrix, built from a given $t\times N$ measurement matrix, can be used to identify up to $d$ defective items in time $O(t \log_2{N})$. In the meantime, a $t \times N$ nonrandom measurement matrix with $t = O \left(\frac{d^2 \log_2^2{N}}{(\log_2(d\log_2{N}) - \log_2{\log_2(d\log_2{N})})^2} \right)$ can be obtained to identify up to $d$ defective items in time $\mathrm{poly}(t)$. This is much better than the best well-known bound, $t = O \left( d^2 \log_2^2{N} \right)$. For the special case $d = 2$, there exists an efficient nonrandom construction in which at most two defective items can be identified in time $4\log_2^2{N}$ using $t = 4\log_2^2{N}$ tests. Numerical results show that our proposed scheme is more practical than existing ones, and experimental results confirm our theoretical analysis. In particular, up to $2^{7} = 128$ defective items can be identified in less than $16$s even for $N = 2^{100}$.

The main contribution of this paper is a non-trivial expression, that is called dual expression, of the posterior values for a non-adaptive group testing problem. The dual expression is useful for exact bitwise MAP estimation. We assume a simplest non-adaptive group testing scenario including N-objects with binary status and M-disjunctive tests. If a group contains a positive object, the test result for the group is assumed to be one; otherwise, the test result becomes zero. Our inference problem is to evaluate the posterior probabilities of the objects from the observation of M-test results and from our knowledge on the prior probabilities for objects. The derivation of the dual expression of posterior values can be naturally described based on a holographic transformation to the normal factor graph (NFG) representing the inference problem. In order to handle OR constraints in the NFG, we introduce a novel holographic transformation that converts an OR function to a function similar to an EQUAL function.

The decentralized web is currently a new trend in web technology, which operates based on the peer-to-peer network in which nodes communicate directly with each other without any central systems. Some nodes act as gateways of the network, some other nodes are used for data synchronization. However, this network architecture may exist malicious nodes that can perform security attacks to other nodes on the network. Hence, guaranteeing the availability of the main nodes plays an important role in the system. In this paper, we use the InterPlanetary File System (IPFS) platform to build the decentralized web system, and propose a novel security solution to detect Hot-IPs based on non-adaptive group testing at gateways and storage miner nodes of the decentralized web system. Hot-IPs are hosts appearing with high frequency in the network within a short period of time, caused by some threats such as (distributed) denial-of-service attacks or scanning Internet worm attacks. Detecting Hot-IPs in real time is the first important step that assists administrators in selecting security policies for the system.

The purchasing of customer databases, which is becoming more and more common, has led to a big problem: illegal distribution of purchased databases. An essential tool for identifying distributors is database fingerprinting. There are two basic problem in fingerprinting database: designing the fingerprint and embedding it. For the first problem, we have proven that Non-Adaptive Group Testing, which is used to identify specific items in a large population, can be used for fingerprinting and that it is secure against collusion attack efficiently. For the second problem, we have developed a solution that supports up to 262,144 fingerprints for 4,032 attributes, and that is secure against three types of attacks: attribute, collusion and complimentary. Moreover, illegal distributor can be identified within 0.15 seconds.

Publication in the conference proceedings of EUSIPCO, Lisbon, Portugal, 2014

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.

We revisit the well-known problem of boolean group testing which attempts to discover a sparse subset of faulty items in a large set of mostly good items using a small number of pooled (or grouped) tests. This problem originated during the second WorldWar, and has been the subject of active research during the 70's, and 80's. Recently, there has been a resurgence of interest due to the striking parallels between group testing and the now highly popular field of compressed sensing. In fact, boolean group testing is nothing but compressed sensing in a different algebra - with boolean `AND' and `OR' operations replacing vector space multiplication and addition. In this paper we review existing solutions for non-adaptive (batch) group testing and propose a linear programming relaxation solution, which has a resemblance to the basis pursuit algorithm for sparse recovery in linear models. We compare its performance to alternative methods for group testing.