scholarly journals The Exact Query Complexity of Yes-No Permutation Mastermind

Games ◽  
2020 ◽  
Vol 11 (2) ◽  
pp. 19
Author(s):  
Mourad El Ouali ◽  
Volkmar Sauerland
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Important Case ◽  
Query Complexity ◽  
Correct Position ◽  
Error Measure ◽  
Minimum Number ◽  
Player Game ◽  
Secret Code ◽  
Asymptotic Complexity

Mastermind is famous two-player 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, the codemaker’s duty is to help the 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 the 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 ∑ j = ℓ k 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.

BALANCED SETS AND THE VECTOR GAME

2008 ◽  
Vol 04 (03) ◽  
pp. 339-347 ◽  
Author(s):  
ZHIVKO NEDEV ◽  
ANTHONY QUAS
Keyword(s):  
Field Theory ◽  
Lower Bound ◽  
Finite Field ◽  
Upper Bound ◽  
Log P ◽  
Prime Factorization ◽  
Minimum Cardinality ◽  
Player Game

We consider the notion of a balanced set modulo N. A nonempty set S of residues modulo N is balanced if for each x ∈ S, there is a d with 0 < d ≤ N/2 such that x ± d mod N both lie in S. We define α(N) to be the minimum cardinality of a balanced set modulo N. This notion arises in the context of a two-player game that we introduce and has interesting connections to the prime factorization of N. We demonstrate that for p prime, α(p) = Θ( log p), giving an explicit algorithmic upper bound and a lower bound using finite field theory and show that for N composite, α(N) = min p|Nα(p).

scholarly journals On Compact Encoding of Pagenumber $k$ Graphs

2008 ◽  
Vol Vol. 10 no. 3 ◽  
Author(s):  
Cyril Gavoille ◽  
Nicolas Hanusse
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Constant Time ◽  
Worst Case ◽  
Encoding Scheme ◽  
Information Theoretic ◽  
Auxiliary Table ◽  
Minimum Number ◽  
International Audience

International audience In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.

scholarly journals On Bipartite Distinct Distances in the Plane

10.37236/9687 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Surya Mathialagan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Crossing Number ◽  
Minimum Number

Given sets $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^2$ of sizes $m$ and $n$ respectively, we are interested in the number of distinct distances spanned by $\mathcal{P} \times \mathcal{Q}$. Let $D(m, n)$ denote the minimum number of distances determined by sets in $\mathbb{R}^2$ of sizes $m$ and $n$ respectively, where $m \leq n$. Elekes showed that $D(m, n) = O(\sqrt{mn})$ when $m \leqslant n^{1/3}$. For $m \geqslant n^{1/3}$, we have the upper bound $D(m, n) = O(n/\sqrt{\log n})$ as in the classical distinct distances problem.In this work, we show that Elekes' construction is tight by deriving the lower bound of $D(m, n) = \Omega(\sqrt{mn})$ when $m \leqslant n^{1/3}$. This is done by adapting Székely's crossing number argument. We also extend the Guth and Katz analysis for the classical distinct distances problem to show a lower bound of $D(m, n) = \Omega(\sqrt{mn}/\log n)$ when $m \geqslant n^{1/3}$.

scholarly journals Recasting Mermin's multi-player game into the framework of pseudo-telepathy

2005 ◽  
Vol 5 (7) ◽  
pp. 538-550
Author(s):  
G. Brassard ◽  
A. Broadbent ◽  
A. Tapp
Keyword(s):  
Information Processing ◽  
Lower Bound ◽  
Upper Bound ◽  
Quantum Mechanical ◽  
Computer Scientist ◽  
Player Game ◽  
Classical Strategy ◽  
Distributed Tasks

Entanglement is perhaps the most non-classical manifestation of quantum \mbox{mechanics}. Among its many interesting applications to information processing, it can be harnessed to \emph{reduce} the amount of communication required to process a variety of distributed computational tasks. Can it be used to eliminate communication altogether? Even though it cannot serve to signal information between remote parties, there are distributed tasks that can be performed without any need for communication, provided the parties share prior entanglement: this is the realm pseudo-telepathy. One of the earliest uses of multi-party entanglement was presented by Mermin in 1990. Here we recast his idea in terms of pseudo-telepathy: we provide a new computer-scientist-friendly analysis of this game. We prove an upper bound on the best possible classical strategy for attempting to play this game, as well as a novel, matching lower bound. This leads us to considerations on how well imperfect quantum-mechanical apparatus must perform in order to exhibit a behaviour that would be classically impossible to explain. Our results include improved bounds that could help vanquish the infamous detection loophole.

A (5,5)-Colouring of Kn with Few Colours

2018 ◽  
Vol 27 (6) ◽  
pp. 892-912
Author(s):  
ALEX CAMERON ◽  
EMILY HEATH
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Minimum Number ◽  
Edge Colouring

For fixed integers p and q, let f(n,p,q) denote the minimum number of colours needed to colour all of the edges of the complete graph Kn such that no clique of p vertices spans fewer than q distinct colours. Any edge-colouring with this property is known as a (p,q)-colouring. We construct an explicit (5,5)-colouring that shows that f(n,5,5) ≤ n1/3 + o(1) as n → ∞. This improves upon the best known probabilistic upper bound of O(n1/2) given by Erdős and Gyárfás, and comes close to matching the best known lower bound Ω(n1/3).

scholarly journals An Explicit Edge-Coloring of $K_n$ with Six Colors on Every $K_5$

10.37236/7852 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Alex Cameron
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Edge Coloring ◽  
Minimum Number ◽  
Positive Integers

Let $p$ and $q$ be positive integers such that $1 \leq q \leq {p \choose 2}$. A $(p,q)$-coloring of the complete graph on $n$ vertices $K_n$ is an edge coloring for which every $p$-clique contains edges of at least $q$ distinct colors. We denote the minimum number of colors needed for such a $(p,q)$-coloring of $K_n$ by $f(n,p,q)$. This is known as the Erdös-Gyárfás function. In this paper we give an explicit $(5,6)$-coloring with $n^{1/2+o(1)}$ colors. This improves the best known upper bound of $f(n,5,6)=O\left(n^{3/5}\right)$ given by Erdös and Gyárfás, and comes close to matching the order of the best known lower bound, $f(n,5,6) = \Omega\left(n^{1/2}\right)$.

Some bounds on the minimum number of queries required for quantum channel perfect discrimination

2012 ◽  
Vol 12 (1&2) ◽  
pp. 138-148
Author(s):  
Cheng Lu ◽  
Jianxin Chen ◽  
Runyao Duan
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Quantum Channel ◽  
Necessary Condition ◽  
Quantum Channels ◽  
Sequential Scheme ◽  
Minimum Number ◽  
Perfect Discrimination ◽  
Sufficient And Necessary Condition

We prove a lower bound on the $q$-maximal fidelities between two quantum channels $\E_0$ and $\E_1$ and an upper bound on the $q$-maximal fidelities between a quantum channel $\E$ and an identity $\I$. Then we apply these two bounds to provide a simple sufficient and necessary condition for sequential perfect distinguishability between $\E$ and $\I$ and provide both a lower bound and an upper bound on the minimum number of queries required to sequentially perfectly discriminating $\E$ and $\I$. Interestingly, in the $2$-dimensional case, both bounds coincide. Based on the optimal perfect discrimination protocol presented in \cite{DFY09}, we can further generalize the lower bound and upper bound to the minimum number of queries to perfectly discriminating $\E$ and $I$ over all possible discrimination schemes. Finally the two lower bounds are shown remain working for perfectly discriminating general two quantum channels $\E_0$ and $\E_1$ in sequential scheme and over all possible discrimination schemes respectively.

scholarly journals The number of images of a point source in an N-point gravitational lens

2019 ◽  
Keyword(s):  
Algebraic Geometry ◽  
Lower Bound ◽  
Point Source ◽  
Upper Bound ◽  
Gravitational Lensing ◽  
Gravitational Lens ◽  
Abscissa Axis ◽  
Analysis Theory ◽  
Minimum Number ◽  
Special Case

An important problem of the theory of gravitational lensing is the problem of studying images of a given source in a given lens. A special case of this problem is the problem of the number of images of a point source in a planar N-point gravitational lens. On this issue, several papers have been published. Most of the works are devoted to the upper bound on the number of images. However, there is no work on the lower bound on the number of images. The present work is devoted to this question. The article calculates what the minimum number of images of a point source in an N-point gravitational lens is equal to. Proven the theorem about infimum of a number of point source images in the N-point gravitational lens. Is proved that this limit is being reached. In particular, it is established that a point source has a minimum number of images in the lens if all point masses are equal and located on the abscissa axis. Besides, the source is also on the abscissa axis. Regular and non-regular cases are considered. Using the theorem that was proved in the paper and the previously known results, a classification theorem about the number of images of a point source in an N-point gravitational lens is formulated. The theorem proved in this paper is illustrated by an example of point source images in a binary lens. The point masses in this lens are the same and are located on the abscissa axis symmetrically with regard to the origin of the coordinates. The minimum number of point source images, in this case, is three, and the maximum is five. A point source has a minimum number of images if it is located on the abscissa axis. The paper used methods of mathematical analysis, theory of functions of real variables and algebraic geometry.

scholarly journals On the Minimum Number of Completely 3-Scrambling Permutations

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Jun Tarui
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Explicit Construction ◽  
Minimum Size ◽  
Minimum Number ◽  
International Audience

International audience A family $\mathcal{P} = \{\pi_1, \ldots , \pi_q\}$ of permutations of $[n]=\{1,\ldots,n\}$ is $\textit{completely}$ $k$-$\textit{scrambling}$ [Spencer, 1972; Füredi, 1996] if for any distinct $k$ points $x_1,\ldots,x_k \in [n]$, permutations $\pi_i$'s in $\mathcal{P}$ produce all $k!$ possible orders on $\pi_i (x_1),\ldots, \pi_i(x_k)$. Let $N^{\ast}(n,k)$ be the minimum size of such a family. This paper focuses on the case $k=3$. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison. $\frac{2}{ \log _2e} \log_2 n \leq N^{\ast}(n,3) \leq 2\log_2n + (1+o(1)) \log_2 \log _2n$. We also prove the existence of $\lim_{n \to \infty} N^{\ast}(n,3) / \log_2 n = c_3$. Determining the value $c_3$ and proving the existence of $\lim_{n \to \infty} N^{\ast}(n,k) / \log_2 n = c_k$ for $k \geq 4$ remain open.

scholarly journals NOTE ON THE NUMBER OF OBTUSE ANGLES IN POINT SETS

2014 ◽  
Vol 24 (03) ◽  
pp. 177-181 ◽  
Author(s):  
RUY FABILA-MONROY ◽  
CLEMENS HUEMER ◽  
EULÀLIA TRAMUNS
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
General Position ◽  
Crossing Number ◽  
The Other ◽  
Point Sets ◽  
Convex Position ◽  
Minimum Number

In 1979 Conway, Croft, Erdős and Guy proved that every set S of n points in general position in the plane determines at least [Formula: see text] obtuse angles and also presented a special set of n points to show the upper bound [Formula: see text] on the minimum number of obtuse angles among all sets S. We prove that every set S of n points in convex position determines at least [Formula: see text] obtuse angles, hence matching the upper bound (up to sub-cubic terms) in this case. Also on the other side, for point sets with low rectilinear crossing number, the lower bound on the minimum number of obtuse angles is improved.

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