Jeong-Han Kim (김정한), How to find counterfeit coins? An algorithmic version.

How to find counterfeit coins? An algorithmic version.
Jeong-Han Kim (김정한)
Dept. of Mathematics, Yonsei University, Seoul, Korea.
2012/05/03 Thu 4PM-5PM
We consider a well-known combinatorial search problem. Suppose that there are n identical looking coins and some of them are counterfeit. The weights of all authentic coins are the same and known a priori. The weights of counterfeit coins vary but different from the weight of an authentic coin. Without loss of generality, we may assume the weight of authentic coins is 0. The problem is to find all counterfeit coins by weighing sets of coins (queries) on a spring scale. Finding the optimal number of queries is difficult even when there are only 2 counterfeit coins.
We introduce a polynomial time randomized algorithm to find all counterfeit coins when the number of them is known to be at most m≥2 and the weight w(c) of each counterfeit coin c satisfies α≤|w(c)|≤β for fixed constants α, β>0. The query complexity of the algorithm is O((m log n)/log m), which is optimal up to a constant factor. The algorithm uses, in part, random walks.
We will also discuss the problem of finding edges of a hidden weighted graph using a certain type of queries.

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