A Simple and Efficient Strategy for the Coin Weighing Problem with a Spring Scale
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 goal is to determine the weight of each coin by weighing subsets of coins in a spring scale. The problem is to devise a weighing strategy that minimizes 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 simplest non-trivial case of the problem, i.e., d = k = 2, is still open. For this case, we propose and analyze a simple and effective adaptive weighing strategy. Our analysis shows that the proposed strategy requires about 1.365log2n-0.5 weighings on average. 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 more »
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NSF-PAR ID:
10076574
Journal Name:
IEEE International Symposium on Information Theory (ISIT)
Page Range or eLocation-ID:
1730 to 1734
3. Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set N , a weight function $$w: N \rightarrow \mathbb {R}_+$$ w : N → R + , r monotone submodular functions $$f_1,f_2,\ldots ,f_r$$ f 1 , f 2 , … , f r over N and requirements $$k_1,k_2,\ldots ,k_r$$ k 1 , k 2 , … , k r the goal is to find a minimum weight subset $$S \subseteq N$$ S ⊆ N such that $$f_i(S) \ge k_i$$ f i ( S ) ≥ k i for $$1 \le i \le r$$ 1 ≤more »