Finding non-minority balls with majority and plurality queries

Chang, Huilan; Gerbner, Dániel; Patkós, Balázs

doi:10.1016/j.dam.2020.03.023

Cited by 4 publications

(2 citation statements)

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“…In this paper we deal with generalizations of the pairing model, when we ask queries of larger size. The first model of this kind was introduced and investigated by De Marco, Kranakis and Wiener [6], then many related results appeared in the literature [2,3,5,7,10,11]. However, most of them studied only the adaptive case.…”

Section: Modelsmentioning

confidence: 99%

On non-adaptive majority problems of large query size

Gerbner¹,

Vizer²

2021

Discrete Mathematics &Amp; Theoretical Computer Science

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We are given $n$ 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 $k$ 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.

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Section: Modelsmentioning

confidence: 99%

On non-adaptive majority problems of large query size

Gerbner¹,

Vizer²

2021

Discrete Mathematics &Amp; Theoretical Computer Science

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Section: Modelsmentioning

confidence: 99%

Majority problems of large query size

Gerbner

Vizer

2019

Discrete Applied Mathematics

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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 [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 would like to determine the minimum number of queries needed in the worst case, when our adversary -we will call him Adversary in the following -, who tells us the answers for the queries wants to postpone the solution of the Majority Problem.A model of the Majority Problem is given by the number of balls [n], the number of colors, the size of the queries (that we will denote by k), and the possible answers of Adversary. Sometimes we will use the hypergraph language, so we will speak about the hypergraph of the queries (i.e. the hypergraph, where the vertices are the balls and the edges are the asked queries up to a certain round.).The first Majority Problem model, the so-called pairing model -when the query size is two, and the answer of Adversary is yes if the two balls have the same color and no otherwise -was investigated by Fisher and Salzberg [12], who proved that if we do not have any restriction on the number of colors, 3n/2 − 2 queries are necessary and sufficient to solve the Majority Problem. If the number of colors is two, then Saks and Werman [19] proved that the minimum number of queries needed is n−b(n), where b(n) is the number of 1's in the diadic form of n (we note that there are simpler proofs of this result, see [1,20]).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]). We also note that there are other possibilities to generalize the pairing model of the Majority Problem, e.g. the Plurality Problem [1,2,6,14,16], some random scenarios [9, 10, 13], or investigate the case when Adversary can lie [1,5], etc.Structure of the paper The rest of the paper is organized as follows: in the next section first we define the models and state the known results, then we state our new results. In Section 3 and Section 4 we prove our theorems, and in Section 5 we finish the article with some...

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