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

TL;DR

This paper introduces a simple adaptive weighing strategy for the coin weighing problem with a spring scale, specifically addressing the challenging case where total weight and individual coin weights are both two, achieving near-optimal efficiency.

Contribution

It proposes and analyzes the first effective strategy for the $d=k=2$ case, providing a new upper bound on the expected number of weighings needed.

Findings

Requires about 1.365 log2 n weighings on average for large n

Achieves approximately 31.75% fewer weighings than nested strategies

Within 8.16% of the information-theoretic lower bound

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 $n\geq 2$ coins of total weight $d$ (for a known integer $d$), where the weight of each coin is an unknown integer in the range of $\{0,1,\dots,k\}$ (for a known integer $k\geq 1$), 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.