Minimum average-case queries of q + 1 -ary search game with small sets

Given a search space S={1,2,...,n}, an unknown element x*∈S and fixed integers ℓ≥1 and q≥1, a q+1-ary ℓ-restricted query is of the following form: which one of the set {A 0,A 1,...,A q} is the x* in?, where (A 0,A 1,...,A q) is a partition of S and | Ai|≤ℓ for i=1,2,...,q. The problem of finding x* from S with q+1-ary size-restricted queries is called as a q+1-ary search game with small sets. In this paper, we consider sequential algorithms for the above problem, and establish the minimum number of average-case sequential queries when x* satisfies the uniform distribution on S. © 2011 Elsevier B.V. All rights reserved

Optimal detection of two counterfeit coins with two-arms balance

AbstractWe consider the following coin-weighing problem: suppose among the given n coins there are two counterfeit coins, which are either heavier or lighter than other n−2 good coins, this is not known beforehand. The weighing device is a two-arms balance. Let NA(k) be the number of coins from which k weighings suffice to identify the two counterfeit coins by algorithm A and U(k)=max{n|n(n−1)⩽3k} be the information-theoretic upper bound of the number of coins then NA(k)⩽U(k). We establish a new method of reducing the above original problem to another identity problem of more simple configurations. It is proved that the information-theoretic upper bound U(k) are always achievable for all even integer k⩾1. For odd integer k⩾1, our general results can be used to approximate arbitrarily the information-theoretic upper bound. The ideas and techniques of this paper can be easily employed to settle other models of two counterfeit coins