Searching for majority with k-tuple queries

Diagnosing the quality of components in fault-tolerant computer systems often requires numerous tests with limited resources. It is usually the case that repeated tests on a selected, limited number of components are performed and the results are taken into account so as to infer a diagnostic property of the computer system as a whole. In this paper we abstract fault-tolerant testing as the following problem concerning the color of the majority in a set of colored balls. Given a set of balls each colored with one of two colors, the majority problem is to determine whether or not there is a majority in one of the two colors. In case there is such a majority, the aim is to output a ball of the majority color, otherwise to declare that there is no majority. We propose algorithms for solving the majority problem by repeatedly testing only k-tuple queries. Namely, successive answers of an oracle (which accepts as input only k-tuples) to a sequence of k-tuple queries are assembled so as to determine whether or not the majority problem has a solution. An issue is to design an algorithm which minimizes the number of k-tuple queries needed in order to solve the majority problem on any possible input of n balls. In this paper we consider three querying models: Output, Counting, and General, reflecting the amount and type of information provided by the oracle on each test for a k-tuple.