Finding Weighted Graphs by Combinatorial Search

We consider the problem of finding edges of a hidden weighted graph using a certain type of queries. Let G be a weighted graph with n vertices. In the most general setting, the n vertices are known and no other information about G is given. The problem is finding all edges of G and their weights using additive queries, where, for an additive query, one chooses a set of vertices and asks the sum of the weights of edges with both ends in the set. This model has been extensively used in bioinformatics including genom sequencing. Extending recent results of Bshouty and Mazzawi [11], and Choi and Kim [17], we present a polynomial time randomized algorithm to find the hidden weighted graph G when the number of edges in G is known to be at most m ≥ 2 and the weight w(e) of each edge e satisfies α ≤ |w(e)| ≤ β for fixed constants α, β > 0. The query complexity of the algorithm is O(m logn logm ), which is optimal up to a constant factor. The algorithm heavily relies on a well-known combinatorial search problem, which may be of independent interest. 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 on a spring scale. We introduce a polynomial time randomized algorithm to find all counterfeit coins when the number of them is know 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. The query complexity of the algorithm is O(m logn logm ), which is optimal up to a constant factor. The algorithm uses, in part, random walks.