Finding a non-minority ball with majority answers

Gerbner, Dániel; Keszegh, Balázs; Pálvölgyi, Dömötör; Patkós, Balázs; Vizer, Máté; Wiener, Gábor

doi:10.1016/j.dam.2016.11.012

Cited by 8 publications

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“…In the third phase, we denote the colors by 0 and 1, following [8]. We are already given k balls of color 0 and k balls of color 1, from S. Let us consider two balls x, y ∈ X \ S. We ask two queries: x and y together with k balls of color 0 and k − 1 balls of color 1, and after that x and y together with k − 1 balls of color 0 and k balls of color 1.…”

Section: Large Odd Queries and Two Colorsmentioning

confidence: 99%

“…In this paper we are going to study and generalize a variant introduced by Gerbner, Keszegh, Pálvölgyi, Patkós, Vizer and Wiener [8]. Before describing it in details, we introduce the following basic notions.…”

Section: Introductionmentioning

confidence: 99%

“…Gerbner, Keszegh, Pálvölgyi, Patkós, Vizer and Wiener [8] considered the following variant. The answer to a query Q is YES and a majority ball inside Q if such a ball exists, and NO if there is no majority color inside Q.…”

Section: Introductionmentioning

confidence: 99%

“…Using the above notation the results of [8] can be summarized as MB(n, 3, 2) = O(n), MB(n, 2k, 2) = O(n) and MB(n, 2k + 1, 2) = O(n 2 ). Note that in case of two colors, M and P are the same, while B and C are the same as well.…”

Section: Introductionmentioning

confidence: 99%

“…Note that in case of two colors, M and P are the same, while B and C are the same as well. It was conjectured in [8] that MB(n, 2k + 1, 2) = O(n). Here we will prove this.…”

Section: Introductionmentioning

confidence: 99%

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Finding non-minority balls with majority and plurality queries

Chang

Gerbner

Patkós

2020

Discrete Applied Mathematics

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Given a set of n colored balls, a majority, non-minority or plurality ball is one whose color class has size more than n/2, at least n/2 or larger than any other color class, respectively. We describe linear time algorithms for finding non-minority balls using query sets of size q of the following form: the answer to a majority/plurality query Q is a majority/plurality ball in Q or the statement that there is no such ball in Q.

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Section: Large Odd Queries and Two Colorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Note that in case of two colors, M and P are the same, while B and C are the same as well. It was conjectured in [8] that MB(n, 2k + 1, 2) = O(n). Here we will prove this.…”

Section: Introductionmentioning

confidence: 99%

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Finding non-minority balls with majority and plurality queries

Chang

Gerbner

Patkós

2020

Discrete Applied Mathematics

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From Discrepancy to Majority

Eppstein¹,

Hirschberg²

2017

Algorithmica

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We show how to select an item with the majority color from n two-colored items, given access to the items only through an oracle that returns the discrepancy of subsets of k items. We use n/ k 2 + O(k) queries, improving a previous method by De Marco and Kranakis that used n − k + k 2 /2 queries. We also prove a lower bound of n/(k − 1) − O(n 1/3 ) on the number of queries needed, improving a lower bound of n/k by De Marco and Kranakis.David Eppstein was supported in part by NSF grant CCF-1228639.

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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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Finding a non-minority ball with majority answers

Cited by 8 publications

References 16 publications

Finding non-minority balls with majority and plurality queries

Finding non-minority balls with majority and plurality queries

From Discrepancy to Majority

Majority problems of large query size

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