A Simple and Efficient Strategy for the Coin Weighing Problem with a Spring Scale

Abstract

This paper considers a generalized version of the coin weighing problem with a spring scale that lies at the intersection of group testing and compressed sensing problems. Given a collection of n ≥ 2 coins of total weight d (for a known integer d), where the weight of each coin is an unknown integer in the range of {0, 1, . . . , k} (for a known integer k ≥ 1), the problem is to determine the weight of each coin by weighing subsets of coins in a spring scale. The goal is to minimize the average number of weighings over all possible weight configurations. For d = k = 1, an adaptive bisecting weighing strategy is known to be optimal. However, even the case of d = k = 2, which is the simplest non-trivial case of the problem, is still open. For this case, we propose and analyze a simple and effective adaptive weighing strategy. A numerical evaluation of the exact recursive formulas, derived for the analysis of the proposed strategy, shows that this strategy requires about 1.365 log 2 n − 0.5 weighings on average. To the best of our knowledge, this is the first non-trivial achievable upper bound on the minimum expected required number of weighings for the case of d = k = 2. As n grows unbounded, the proposed strategy, when compared to an optimal strategy within the commonly-used class of nested strategies, requires about 31.75% less number of weighings on average; and in comparison with the information-theoretic lower bound, it requires at most about 8.16% extra number of weighings on average.

Sparse Graph Codes for Non-adaptive Quantitative Group Testing

“…This problem was first introduced by Shapiro in [6]. Several non-adaptive and adaptive QGT strategies have been previously proposed, see, e.g., [3], [7], [8]. It was shown in [9] that any non-adaptive algorithm must perform at least (2K log 2 (N/K))/log 2 K tests.…”

Non-adaptive Quantitative Group Testing Using Irregular Sparse Graph Codes

“…The QGT problem can be traced back to the seminal work by Shapiro in [6]. To date, several adaptive and nonadaptive QGT strategies have been proposed, see, e.g., [3]- [5], [7]- [9] and references therein. Using a simple information theoretic argument, one can easily show the informationtheoretic lower bound log K N K ≈ (K log(N/K))/log K on the minimum number of tests for any adaptive QGT scheme.…”

“…the combinatorial model), the exact number of defective items is known, whereas in the randomized model (a.k.a. the probabilistic model), each item is defective with some probability, independent of the other items [1]- [5]. In this work, we consider the randomized model in which each item is defective with probability K N , independently from the other items, where N is the The authors are with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843 USA (E-mail: {esmaeil.karimi, fatemeh.kazemi, anoosheh, krn, spalex}@tamu.edu).…”

“…the combinatorial model), the exact number of defective items is known, whereas in the randomized model (a.k.a. the probabilistic model), each item is defective with some probability, independent of the other items [1]- [5]. In this work, we consider the randomized model in which each item is defective with probability This material is based upon work supported by the National Science Foundation under Grants No.…”

Scheduling Improves the Performance of Belief Propagation for Noisy Group Testing

“…in contrast to adaptive schemes, in which the design of each test depends on the results of the previous tests [12]- [14]. In most practical applications, when compared to adaptive group testing schemes, non-adaptive schemes are preferred because all tests can be executed at once in parallel.…”