On the Coin Weighing Problem with the Presence of Noise

Abstract

Abstract. In this paper we consider the following coin weighing problem: Given n coins for which some of them are counterfeit with the same weight. The problem is: given the weights of the counterfeit coin and the authentic coin, detect the counterfeit coins a with minimal number of weighings. This problem has many applications in computational learning theory, compressed sensing and multiple access adder channels.An old optimal non-adaptive polynomial time algorithm of Lindstrom can detect the counterfeit coins with O(n/ log n) weighings. An information theoretic proof shows that Lindstrom's algorithm is optimal. In this paper we study non-adaptive algorithms for this problem when some of the answers of the weighings received are incorrect or unknown.We show that no coin weighing algorithm exists that can detect the counterfeit coins when the number of incorrect weighings is more than 1/4 fraction of the number of weighings. We also give the tight bound Θ(n/ log n) for the number of weighings when the number of incorrect answers is less than 1/4 fraction of the number of weighings.We then give a non-adaptive polynomial time algorithm that detects the counterfeit coins with k = O(n/ log log n) weighings even if some constant fraction of the answers of the weighings received are incorrect. This improves Bshouty and Mazzawi's algorithm [7] that uses O(n) weighings. This is the first sublinear algorithm for this problem.

Exact learning from an honest teacher that answers membership queries

“…The class r-LF is studied in [37,78,79,113,145,200,204,264,266]. It is shown that OPT AD (r-LF) = OPT NAD (r-LF) = Θ r log(n/r) log r .…”

“…Note here that in the literature they use log(n/r) to mean log(2n/r). In [37], Bshouty shows that it is optimally adaptively learnable. The problem is still open for the non-adaptive learning.…”

“…All the learning algorithms for this class are either for restricted subclasses or randomized algorithms with success probability that depend on r or non-optimal. See other subclasses in [37,145,225,236]. Similar problems are studied in other areas such as coding theory [229] compressed sensing [183] Multiple Access Channels [50] (e.g., adder channels [98]) and combinatorial group testing [113,114] (e.g., coin weighing problem [37]).…”

“…The class r-LF is the class (r, {0, 1})-LF and LF is the class n-LF. Learning (r, V )-LF is equivalent to coin weighing problem [37] and signature coding problem [50].…”

“…answers to the first t queries in this round. In the non-adaptive model, the learner knows p or some upper bound for p. [37]: In the adaptive model, if the algorithm asks T queries then the teacher can give at most pT "?" answers.…”