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