Sterrett Procedure for the Generalized Group Testing Problem

Abstract

Group testing is a useful method that has broad applications in medicine, engineering, and even in airport security control. Consider a finite population of N items, where item i has a probability p i to be defective. The goal is to identify all items by means of group testing. This is the generalized group testing problem. The optimum procedure, with respect to the expected total number of tests, is unknown even in case when all p i are equal. Hwang (1975) proved that an ordered partition (with respect to p i ) is the optimal for the Dorfman procedure (procedure D), and obtained an optimum solution (i.e., found an optimal partition) by dynamic programming. In this paper, we investigate the Sterrett procedure (procedure S). We provide close form expression for the expected total number of tests, which allows us to find the optimum arrangement of the items in the particular group. We also show that an ordered partition is not optimal for the procedure S or even for a slightly modified Dorfman procedure (procedure D ′ ). This discovery implies that finding an optimal procedure S appears to be a hard computational problem. However, by using an optimal ordered partition for all procedures, we show that procedure D ′ is uniformly better than procedure D, and based on numerical comparisons, procedure S is uniformly and significantly better than procedures D and D ′ .

An Optimal Design for Hierarchical Generalized Group Testing

“…The ordering assumption is necessary to make the problem tractable since optimizing with respect to all possible permutations of p 1 ,…, p N is impossible even for small N . Additional discussion regarding the issue of ordering can be found in Malinovsky (2019a) and references therein.…”

“…We denote these algorithms as procedures D ′ and S respectively. In the GGTP setting, the optimum ordered configuration for both algorithms has been characterized by Malinovsky (2019a). For the CRC procedure, we obtained the expected total number of tests by using R functions provided by Black et al .…”

“…In Table , we present the mean and standard deviation of the mean (1000 simulated realizations of p 1 ,…, p N ) of the expected total number of tests for D ′ , S, HDP and the CRC procedure for population sizes of 20 and 100. In addition, we present the lower bound (Shannon entropy H ( p )) of the optimal group testing algorithm among the general class (see the discussion and references in Malinovsky (2019a)). For N =20, we performed the CRC procedure with four stages since the performance of the algorithm was better than that with three stages in this case.…”

“…if each previous group test contains either both of them or none of them (Sobel and Groll, ; Hwang et al ., ). Additional non‐optimal hierarchical algorithms for the GGTP have been developed (Litvak et al ., ; Bilder et al ., ; Malinovsky, 2019a).…”

Conjectures on optimal nested generalized group testing algorithm

“…12 Since its introduction, GGTP has been investigated. 10,11,[17][18][19][20][21][22][23][24][25][26]29 Even for a particular nested group testing algorithm, the optimal regime (or, order in which groups/units will be tested) is known only for for the Dorfman procedure 2 because of Hwang. 26,29 For the GGTP, Hwang 26,29 proved that under Dorfman's procedure an optimal partition is an ordered partition (ie, each pair of subsets has the property such that the numbers in one subset are all greater or equal to every number in the other subset).…”