Searching for majority with k-tuple queries

Abstract

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.

Majority problems of large query size

“…Theorem 3. (Borzyszkowski [7], Theorem 1) For all 2 ≤ k, n with 2k − 1 ≤ n we have n/2 + k − 2 ≤ A(GM, k, n).…”

“…• Answer : is a number i ≤ k/2 such that the query has exactly i balls of one color General (or Yes-No) Model = GM, [7]:…”

“…In the so-called Counting Model the answer to a query gives the difference between the cardinalities of the color classes in the query.Our goal is to show a ball of the larger color class, or prove that the color classes are of the same size, using as few queries as possible. In this paper we improve the bounds given by De Marco and Kranakis [7] for the number of queries needed. arXiv:1610.09114v2 [math.CO] 30 Aug 2018 (or showing that there is no such ball) the Majority Problem.…”

“…We call the problem of finding a majority ball * Research supported by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences. De Marco and Kranakis [7] introduced three different models, and gave upper and lower bounds for all of them. Eppstein and Hirschberg [11] improved three of those six bounds, here we improve the other three.…”

“…In this paper we deal with two possible generalizations of the pairing model, the Counting and the General model. Both of them deals with queries of size greater than two.The first model that generalized the pairing model to larger queries was introduced and investigated by De Marco, Kranakis and Wiener [8], then many results appeared in the literature [4,7,11,15]. We note that for k ≥ 3 it is possible that in some model one can not solve the Majority Problem for small n or even for any n (see such a model in [15]).…”