9 research outputs found
On non-adaptive majority problems of large query size
We are given balls and an unknown coloring of them with two colors. Our
goal is to find a ball that belongs to the larger color class, or show that the
color classes have the same size. We can ask sets of balls as queries, and
the problem has different variants, according to what the answers to the
queries can be. These questions has attracted several researchers, but the
focus of most research was the adaptive version, where queries are decided
sequentially, after learning the answer to the previous query. Here we study
the non-adaptive version, where all the queries have to be asked at the same
time.Comment: 12 page
Finding a non-minority ball with majority answers
Suppose we are given a set of balls each colored
either red or blue in some way unknown to us. To find out some information
about the colors, we can query any triple of balls
. As an answer to such a query we obtain (the
index of) a {\em majority ball}, that is, a ball whose color is the same as the
color of another ball from the triple. Our goal is to find a {\em non-minority
ball}, that is, a ball whose color occurs at least times among the
balls. We show that the minimum number of queries needed to solve this
problem is in the adaptive case and in the
non-adaptive case. We also consider some related problems
Majority problems of large query size
We study two models of the Majority problem. We are given n balls and an
unknown coloring of them with two colors. We can ask sets of balls of size k as
queries, and in the so-called General Model the answer to a query shows if all
the balls in the set are of the same color or not. 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 for the number of queries needed.Comment: We cut the non-adaptive results from the first version to publish
separatel
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. </jats:p