On a Combinatorial Problem in Number Theory

Lindström, Bernt

doi:10.4153/cmb-1965-034-2

Cited by 86 publications

(60 citation statements)

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“…and Lindström [10] gave a more precise estimate (stated and proved below) which has not been improved in 37 years. Ruzsa [12] gave a new proof of it, using an easy but interesting lemma, which we prove via the overlapping theorem.…”

Section: Sidon Sets a Set Of Integers A Is Called A Sidon Set If Allmentioning

confidence: 96%

An overlapping theorem with applications

Cilleruelo¹,

Tenenbaum²

2007

Publicacions Matemàtiques

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Section: Sidon Sets a Set Of Integers A Is Called A Sidon Set If Allmentioning

confidence: 96%

An overlapping theorem with applications

Cilleruelo¹,

Tenenbaum²

2007

Publicacions Matemàtiques

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“…Lindström [26] and independently Cantor and Mills [10] gave a non-adaptive polynomial time algorithm for the problem with query complexity that matches the lower bound. Simplifications appear in [29,1,5].…”

Section: Introductionmentioning

confidence: 86%

“…We introduce the signature coding problem for the multiple access adder channels [26,10,11,23,17,15,25,3].…”

Section: Applicationsmentioning

confidence: 99%

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On the Coin Weighing Problem with the Presence of Noise

Bshouty

2012

Lecture Notes in Computer Science

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Abstract. In this paper we consider the following coin weighing problem: Given n coins for which some of them are counterfeit with the same weight. The problem is: given the weights of the counterfeit coin and the authentic coin, detect the counterfeit coins a with minimal number of weighings. This problem has many applications in computational learning theory, compressed sensing and multiple access adder channels.An old optimal non-adaptive polynomial time algorithm of Lindstrom can detect the counterfeit coins with O(n/ log n) weighings. An information theoretic proof shows that Lindstrom's algorithm is optimal. In this paper we study non-adaptive algorithms for this problem when some of the answers of the weighings received are incorrect or unknown.We show that no coin weighing algorithm exists that can detect the counterfeit coins when the number of incorrect weighings is more than 1/4 fraction of the number of weighings. We also give the tight bound Θ(n/ log n) for the number of weighings when the number of incorrect answers is less than 1/4 fraction of the number of weighings.We then give a non-adaptive polynomial time algorithm that detects the counterfeit coins with k = O(n/ log log n) weighings even if some constant fraction of the answers of the weighings received are incorrect. This improves Bshouty and Mazzawi's algorithm [7] that uses O(n) weighings. This is the first sublinear algorithm for this problem.

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“…The upper bound was derandomized by Lindström [Lin64,Lin65] and, independently, by Cantor and Mills [CM66]. That is, for 2-color black-pegs Mastermind a deterministic non-adaptive winning strategy using (2 + o(1))n/ log n guesses exists, and no non-adaptive strategy can do better.…”

Section: Non-adaptive Strategiesmentioning

confidence: 99%

Playing Mastermind With Many Colors

Doerr

Spöhel

et al. 2016

J. ACM

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We analyze the general version of the classic guessing game Mastermind with n positions and k colors. Since the case k ≤ n 1−ε , ε > 0 a constant, is well understood, we concentrate on larger numbers of colors. For the most prominent case k = n, our results imply that Codebreaker can find the secret code with O(n log log n) guesses. This bound is valid also when only black answer-pegs are used. It improves the O(n log n) bound first proven by Chvátal (Combinatorica 3 (1983), 325-329). We also show that if both black and white answer-pegs are used, then the O(n log log n) bound holds for up to n 2 log log n colors. These bounds are almost tight as the known lower bound of Ω(n) shows. Unlike for k ≤ n 1−ε , simply guessing at random until the secret code is determined is not sufficient. In fact, we show that an optimal non-adaptive strategy (deterministic or randomized) needs Θ(n log n) guesses.

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On a Combinatorial Problem in Number Theory

Abstract: Given an integer k ≤ 2 and a finite set M of rational integers. Let vi (i = 1, 2, …, n) be m-dimensional (column-)vectors with all components from M and such that the kn sums1.1are all different. Then we shall say that {v1, v2, …, vn} is a detecting set of vectors.

Cited by 86 publications

References 4 publications

An overlapping theorem with applications

An overlapping theorem with applications

On the Coin Weighing Problem with the Presence of Noise

Playing Mastermind With Many Colors

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