Extensions of Generalized Binary Search to Group Identification and Exponential Costs

Generalized Binary Search (GBS) is a well known greedy algorithm for identifying an unknown object while minimizing the number of "yes" or "no" questions posed about that object, and arises in problems such as active learning and active diagnosis. Here, we provide a coding-theoretic interpretation for GBS and show that GBS can be viewed as a top-down algorithm that greedily minimizes the expected number of queries required to identify an object. This interpretation is then used to extend GBS in two ways. First, we consider the case where the objects are partitioned into groups, and the objective is to identify only the group to which the object belongs. Then, we consider the case where the cost of identifying an object grows exponentially in the number of queries. In each case, we present an exact formula for the objective function involving Shannon or Renyi entropy, and develop a greedy algorithm for minimizing it.