Query complexity of mastermind variants

Berger, Aaron J.; Chute, Christopher G.; Stone, Matthew

doi:10.1016/j.disc.2017.11.004

Cited by 4 publications

(3 citation statements)

References 11 publications

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“…El Ouali et al [5] combined the upper bound with a lower bound of n queries, which is implied by a codemaker (cheating) strategy. It improved the lower bound of n − log log n by Berger et al [2]. However, a gap between Ω(n) and O(n log 2 n) remains for this Mastermind variant.…”

Section: Introductionmentioning

confidence: 84%

The Exact Query Complexity of Yes-No Permutation Mastermind

Ouali

Sauerland

2020

Games

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Mastermind is famous two-players game. The first player (codemaker ) chooses a secret code which the second player (codebreaker ) is supposed to crack within a minimum number of code guesses (queries). Therefore, codemaker's duty is to help codebreaker by providing a well-defined error measure between the secret code and the guessed code after each query. We consider a variant, called Yes-No AB-Mastermind, where both secret code and queries must be repetition-free and the provided information by codemaker only indicates if a query contains any correct position at all. For this Mastermind version with n positions and k ≥ n colors and ℓ := k + 1 − n we prove a lower bound of k j=ℓ log 2 j and an upper bound of n log 2 n + k on the number of queries necessary to break the secret code. For the important case k = n, where both secret code and queries represent permutations, our results imply an exact asymptotic complexity of Θ(n log n) queries.

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Section: Introductionmentioning

confidence: 84%

The Exact Query Complexity of Yes-No Permutation Mastermind

Ouali

Sauerland

2020

Games

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“…From this perspective, Jiang and Polyanskii [JP19] recently showed that the minimum number of queries is (2 + o(1))n log k log n ) for any constant k. For k ≤ n, the non-adaptive complexity of black-white-peg Mastermind is the same as black-peg Mastermind, since the non-adaptive strategy using only black-peg queries has reached the entropy lower bound. Contrarily, for k > n it is still not clear whether the non-adaptive strategy with black-white-peg queries can reduce currently the best bound O(k log k) achieved by the one with only black-peg queries [DDST16,BCS18].…”

Section: Related Workmentioning

confidence: 99%

Playing Mastermind on quantum computers

Li¹,

Luo²,

Xu³

2022

Preprint

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From the 1970s up to now, Mastermind, a classic two-player game, has attracted plenty of attention, not only from the public as a popular game, but also from the academic community as a scientific issue. Mastermind with n positions and k colors is formally described as: the codemaker privately chooses a secret s ∈ [k] n , and the coderbreaker want to determine s in as few queries like f s (x) as possible to the codemaker, where f s (x) is called a black-peg query if f s (x) = B s (x) = |{i : s i = x i }|, and a black-white-peg query if f s (x) = {B s (x), W s (x)} with W s (x) = max σ∈Pn |{i : s i = x σ(i) }| − B s (x) and P n denoting the set of all permutations of {1, 2, . . . , n}. The complexity of a strategy is measured by the number of queries used. In this work we have a systematic study on quantum strategies for playing Mastermind, obtaining a full characterization of the quantum complexity and optimal quantum algorithms in both non-adaptive and adaptive settings.(i) The quantum complexity is proved to be Θ(k) in the non-adaptive setting. Two non-adaptive quantum algorithms are constructed for Mastermind with n positions and k colors that returns the secret with certainty: one with O(k log k)complexity and the other with O(k)-complexity. In addition, an algorithm with O(1) queries is constructed for the case k = 2. The algorithm-design skills in the three algorithms have substantial differences, and may be helpful for solving other problems. (ii) The quantum complexity is proved to be Θ( √ k) in the adaptive setting. An optimal adaptive quantum algorithm is constructed for Mastermind with n positions and k colors that uses O( √ k) queries and returns the secret with certainty.Our results show that the codebreaker wins greatly more on quantum computers than on classical computers. In the non-adaptive setting, when k ≤ n the classical complexity is Θ( n log k max{log(n/k),1} ) monotonically increasing with respect to n, and when k > n the current best classical algorithm has complexity O(k log k) which is * The authors are ordered alphabetically.

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“…The original version was completely characterized by [Knu77], who showed upper and lower bounds of 5 queries for deterministic strategies. The n-position k-color generalization of the game was studied in [Chv83], which sparked a line of research that lead to progressive improvement in upper and lower bounds for this problem, both in the original version of the game as well as in related variants of the game [BCS18]. As these variants are not the focus of this work, we refer the reader to the expositions of [DDST16,BCS18] for more details on this literature.…”

Section: Introductionmentioning

confidence: 99%

The Query Complexity of Mastermind with $\ell_p$ Distances

Fernandez,

Woodruff,

Yasuda

2019

Preprint

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Consider a variant of the Mastermind game in which queries are ℓp distances, rather than the usual Hamming distance. That is, a codemaker chooses a hidden vector y ∈ {−k, −k + 1, . . . , k − 1, k} n and answers to queries of the form y − x p where x ∈ {−k, −k + 1, . . . , k − 1, k} n . The goal is to minimize the number of queries made in order to correctly guess y.Motivated by this question, in this work, we develop a nonadaptive polynomial time algorithm that works for a natural class of separable distance measures, i.e. coordinate-wise sums of functions of the absolute value. This in particular includes distances such as the smooth max (LogSumExp) as well as many widely-studied M -estimator losses, such as ℓp norms, the ℓ1-ℓ2 loss, the Huber loss, and the Fair estimator loss. When we apply this result to ℓp queries, we obtain an upper bound of O min n, n log k log n queries for any real 1 ≤ p < ∞. We also show matching lower bounds up to constant factors for the ℓp problem, even for adaptive algorithms for the approximation version of the problem, in which the problem is to output y ′ such that y ′ − y p ≤ R for any R ≤ k 1−ε n 1/p for constant ε > 0. Thus, essentially any approximation of this problem is as hard as finding the hidden vector exactly, up to constant factors. Finally, we show that for the noisy version of the problem, i.e. the setting when the codemaker answers queries with any q = (1 ± ε) y − x p , there is no query efficient algorithm.

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Query complexity of mastermind variants

Cited by 4 publications

References 11 publications

The Exact Query Complexity of Yes-No Permutation Mastermind

The Exact Query Complexity of Yes-No Permutation Mastermind

Playing Mastermind on quantum computers

The Query Complexity of Mastermind with $\ell_p$ Distances

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