Abstract
We propose a continuous version of the classical Gale–Berlekamp switching game. The main results of this paper concern growth estimates for the corresponding optimization problems.
Key words
Gale–Berlekamp switching game; unbalancing lights problem; game theory; Khinchin inequality
Introduction
Designed1 2010 Mathematics Subject Classification. Primary 91A, 26D15. independently by Elwyn Berlekamp and David Gale in the 1960’s, the Gale–Berlekamp switching game – also known as the unbalancing lights problem – represents a classic in the field of combinatorics and its applications, with deep connections to theoretical Computer Science. This single-player game consists of an square matrix of light bulbs set-up at an initial light configuration. The goal is to turn off as many lights as possible using row and column switches, which invert the state of each bulb in the corresponding row or column.
For an initial pattern of lights , let denote the smallest final number of on-lights achievable by row and column switches starting from The smallest possible number of remaining on-lights , starting from the worst initial pattern, is then
Sometimes this optimization problem is posed as finding the maximum of the difference between the number of lights that are on and the number that are off, often denoted by . Obviously both problems are equivalent as .The original problem introduced by Berlekamp asks for the exact value of and it was proved in Carlson & Stolarski that (and thus ) Several related questions pertaining to the original problem have been investigated in depth, see e.g. Brualdi & Meyer 2015BRUALDI RA & MEYER SA. 2015. Gale-Berlekamp permutation-switching problem. European J Combin 44: pat A, 43-56., Carlson & Stolarski 2004CARLSON J & STOLARSKI D. 2004. The correct solution to Berlekamp’s switching game. Discrete Math 287: 145-150., Fishburn & Sloane 1989FISHBURN PC & SLOANE NJA. 1989. The solution to Berlekamp’s switching game. Discrete Math 74: 262-290. and Schauz 2011; in particular the hardness of solving the Gale-Berlekamp switching game was studied in Roth & Viswanathan 2008ROTH RM & VISWANATHAN K. 2008. On the hardness of decoding the Gale-Berlekamp code. IEEE Trans. Inform Theory 54(3): 1050-1060..
In this paper we propose a continuous version of the Gale–Berlekamp switching game. We are interested in a continuous version of the game for which vectors replace light bulbs and knobs substitute the discrete switches used to invert the state of the bulbs in the original problem. In our approach, we also allow the game–board not to be square.
To explain the new proposed game, we initially notice that by associating to the on–lights and to the off–lights from the array of lights the goal of the original game can be understood mathematically as to determine
where and denote the switches of the row and of the column , respectively.
The new optimization problem herein proposed involves a matrix with rows and columns whose elements are unit vectors of the plane, The initial direction pattern of each vectors is set up at the beginning of the game. In each row and each column there are knobs and , respectively. Rotating the knob by an angle it rotates all vectors of the row by the same angle . Analogously, when the knob is rotated by an angle the same happens with all the vectors of the column (see Figure 1).
The game consists of maximizing the Euclidean norm of the sum of all vectors in the final stage. More precisely, for an initial pattern of unit vectors, let be the supremum of the (Euclidean) norms of the sums of all vectors achievable by row and column adjusts. The extremal problem is to determine
Our main result estimates the asymptotic growth of :
Theorem 1. For all positive integers , we have
We conclude this introduction by commenting on the ideas and techniques used to prove Theorem 1, which are of particular interest. We observe that due to the combinatorial complexity of this kind of problems, growth estimates as in Theorem 1 are often obtained by non-deterministic techniques, see for instance Alon & Spencer 1992ALON N & SPENCER J. 1992. The Probabilistic Method, Wiley. (Second Edition, 2000, Third Edition 2008)., Araújo & Pellegrino 2019ARAÚJO G & PELLEGRINO D. 2019. A Gale-Berlekamp permutation-switching problem in higher dimensions. European J Combin 77: 17-30. and Bennett et al. 1975BENNETT G, GOODMAN V & NEWMAN CM. 1975. Norms of random matrices. Pacific J Math 59(2): 359-365.. A main novelty proposed in this article regards a deterministic approach to estimating , which yields improved, more precise estimates than those obtained by non-deterministic methods. We believe that the methods herein developed are likely to be applicable in an array of other problems and to exemplify the depth of these new ideas, we also prove analogues of (1) in higher-dimensional configurations.
Proof of Theorem 1
Initially, it is more convenient to conceive the vectors in the game as complex numbers with modulus , which represent the elements of the array . In this case, when the player rotates a knob, the action is modeled by the multiplication by unimodular complex numbers.
There is no loss of generality in supposing that . We start off the proof of Theorem 1 by reminding that a consequence of the Krein–Milman Theorem assures that for all , defined by
where , with exactly at the -th position, there holds i.e., the supremum norm of is attained at the extreme points of the closed unit balls of and . Thus we can easily observe that and our task is then to estimate the infimum of over all bilinear forms with unimodular coefficients.Once the problem has been described as above, the upper bound in Theorem 1 can be obtained by means of an argument from the seminal paper of Bohnenblust & Hille 1931BOHNENBLUST HF & HILLE E. 1931. On absolute convergence of Dirichlet series. Ann of Math 32: 600-622., Theorem II, page 608. We shall explain the necessary adaptations when we deliver the proof of Theorem 2.
As for the lower estimate, we shall make use of Khinchin inequality, which we revise for the sake of completeness.
Khinchin inequality
To motivate, let’s state the following question: suppose that we have real numbers and a fair coin. When we flip the coin, if it comes up heads, you choose , and if it comes up tails, you choose When we play for the second time, if it comes up heads, you choose and, if it comes up tails, you choose . Repeating the process, after having flipped the coin times we have
if it comes up heads and if it comes up tails. After steps, what should be the expected value of Khinchin’s inequality, see for instance Diestel et al. 1995DIESTEL J, JARCHOW H & TONGE A. 1995. Absolutely summing operators, Cambridge Stud. Adv Math 43., page 10, shows that the “ average” where and , behaves as the -norm of More precisely, it asserts that for any there are constants such that for all sequences of scalars and all positive integers The natural counterpart for the average in the complex framework is It is well known that in this new context we also have a Khinchin-type inequality, called Khinchin inequality for Steinhaus variables, which asserts that there exist constants and such that for every positive integer and all scalars .Back to the proof of Theorem 1, for the purpose of establishing a lower estimate for the growth of , we are interested in the case and only in the left hand side of (2). In König 2014KÖNIG H. 2014. On the best constants in the Khintchine inequality for Steinhaus variables. Israel J Math 203: 23-57. it is proven that . For a bilinear form given by
with we have Since denoting the topological dual of by and its closed unit ball by we have where in the last equality we have used the isometric isomorphism with defined by Finally, since , we conclude that Hence, as , we haveThe game in higher dimensions
The Gale–Berlekamp switching game has a natural extension to higher dimensions. Let be an integer and let an array of lights be given, each either on () or off (). Let us also suppose that for each and each there is a switch so that if the switch is pulled () all of the corresponding lights (with fixed) are switched: on to off or off to on. The goal is to maximize the difference between the number of lights that are on and the number of lights that are off. As in the two-dimensional case, maximizing the difference between the number of on-lights and off-lights is equivalent to estimating
and the extremal problem consists of estimating As in the bilinear case, with The anisotropic case allows to consider arrays, not necessarily square arrays and, in this case, we write From a recent result of Albuquerque & Rezende 2019ALBUQUERQUE N & REZENDE L. 2019. Asymptotic estimates for unimodular multilinear forms with small norms on sequence spaces. Bull Braz Math Soc New Series 52(2021): 23-39., we can easily obtainFollowing the notation of Araújo and Pellegrino 2019, let be an integer and be an array of complex scalars such that . For , let
where the maximum is evaluated over all such that for all It is not difficult to prove that with Denoting where minimum is evaluated over all unimodular -linear forms , the best information we can collect (combining results from Araújo & Pellegrino 2019ARAÚJO G & PELLEGRINO D. 2019. A Gale-Berlekamp permutation-switching problem in higher dimensions. European J Combin 77: 17-30. and Pellegrino et al. 2020PELLEGRINO D, SERRANO-RODRÍGUEZ D & SILVA J. 2020. On unimodular multilinear forms with small norms on sequence spaces. Lin Algebra Appl 595: 24-32.) is the following:where is obtained by interpolation (via the Riesz–Thorin Theorem) of the constant (the constant when ) and (the constant when ).
The above solution rests in a non-deterministic tool. We shall show in what follows that for we can find deterministic solutions with better constants.
We begin with a matrix whose elements are unit vectors in the Euclidean space The initial direction pattern of each vectors is set up at the beginning of the game. For each , we have control knobs . When the knob is rotated by an angle the same happens with all the vectors with fixed. Defining and as in the two-dimensional case, the extremal problem is to determine
It is worth mentioning that, as a consequence of the Krein-Milman Theorem, we know that coincides with , as defined in (3). We prove the following:Theorem 2. For all positive integers , we have
Moreover, the universal upper bound cannot be improved.
The proof that, in general, the upper bound cannot be improved is trivial — just consider and note that in this case
The case was investigated in Araújo & Pellegrino , but the techniques used by the authors do not provide good estimates for the upper constants: for instance, if we follow the arguments from Araújo & Pellegrino we just obtain , due to Kahane–Salem–Zygmund inequality, instead of the universal sharp constant .Proof of Theorem 2
Our task is to estimate , where the infimum runs over all -linear forms with unimodular coefficients.
With no loss of generality, we suppose . For the upper bound, consider, for all , a matrix with
A simple computation shows that All the matrices are completed with zeros (if necessary) in order to get a square matrix Define by and note that, since , the coefficients of all monomials with are unimodular. For consider defined by and so on. We have Thus Since we have Thus Since we conclude that and repeating this procedure we finally obtain ThusThe lower estimate is an adaptation of the bilinear case, using this well-know extension of inequality (2), in the case , to multiple sums as follows:
where .ACKNOWLEDGMENTS
The authors thank Fernando Costa J\’unior for kindly designing and providing Figure 1. The authors also thank the referee for important comments that helped to improve the final version of this paper. D. Pellegrino is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Grant 307327/2017-5 and Grant 2019/0014 Fundação de Apoio à Pesquisa do Estado da Paraíba (FAPESQ). J. Silva is supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).
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2010 Mathematics Subject Classification. Primary 91A, 26D15.
- ALBUQUERQUE N & REZENDE L. 2019. Asymptotic estimates for unimodular multilinear forms with small norms on sequence spaces. Bull Braz Math Soc New Series 52(2021): 23-39.
- ALON N & SPENCER J. 1992. The Probabilistic Method, Wiley. (Second Edition, 2000, Third Edition 2008).
- ARAÚJO G & PELLEGRINO D. 2019. A Gale-Berlekamp permutation-switching problem in higher dimensions. European J Combin 77: 17-30.
- BENNETT G, GOODMAN V & NEWMAN CM. 1975. Norms of random matrices. Pacific J Math 59(2): 359-365.
- BOHNENBLUST HF & HILLE E. 1931. On absolute convergence of Dirichlet series. Ann of Math 32: 600-622.
- BRUALDI RA & MEYER SA. 2015. Gale-Berlekamp permutation-switching problem. European J Combin 44: pat A, 43-56.
- CARLSON J & STOLARSKI D. 2004. The correct solution to Berlekamp’s switching game. Discrete Math 287: 145-150.
- DIESTEL J, JARCHOW H & TONGE A. 1995. Absolutely summing operators, Cambridge Stud. Adv Math 43.
- FISHBURN PC & SLOANE NJA. 1989. The solution to Berlekamp’s switching game. Discrete Math 74: 262-290.
- KÖNIG H. 2014. On the best constants in the Khintchine inequality for Steinhaus variables. Israel J Math 203: 23-57.
- PELLEGRINO D, SERRANO-RODRÍGUEZ D & SILVA J. 2020. On unimodular multilinear forms with small norms on sequence spaces. Lin Algebra Appl 595: 24-32.
- ROTH RM & VISWANATHAN K. 2008. On the hardness of decoding the Gale-Berlekamp code. IEEE Trans. Inform Theory 54(3): 1050-1060.
- SHAUZ U. 2011. Colorings and nowhere-zero flows of graphs in terms of Berlekamp’s switching game. Electron J Combin 18: no. 1, Paper 65, 33 pp.
Publication Dates
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Publication in this collection
08 May 2023 -
Date of issue
2023
History
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Received
30 Apr 2020 -
Accepted
26 Oct 2020