On a continuous Gale-Berlekamp switching game

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.


Introduction
Designed 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 trademark in the field of combinatorics and its applications, with special interests in the theory of Computer Sciences.This single-player game consists of an n × n square matrix of light bulbs set-up at an initial light configuration.The goal is to turn off as many lights as possible using n row and n column switches, which invert the state of each bulb in the corresponding row or column.
For an initial pattern of lights Θ, let i(Θ) denote the smallest final number of on-lights achievable by row and column switches starting from Θ.The smallest possible number of remaining on-lights R n , starting from a worse initial pattern, is then Sometimes this optimization problem is posed as to find the maximum of the difference between lights on and off, often denoted by G n .Obviously both problems are equivalent as R n = 1  2 n 2 − G n .The original problem introduced by Berlekamp asks for the exact value of R 10 and it was proved in [10] that R 10 = 35 (and thus G 10 = 30).Several related questions pertaining to the original problem have been investigated in depth, see e.g.[8,10,11,16], in particular the hardness of solving the Gale-Berlekamp game was studied in [15].
In this paper we propose a continuous version of the Elwyn Berlekamp switching game.We are interested in a continuous version of the game for which vectors replace light bulbs and direction knobs substitute the discrete switches, used to invert the state of the bulbs in the original problem.We also allow for non-square game-boards.
To explain the new proposed game, we initially notice that by associating +1 to the on-lights and −1 to the off-lights from the array of lights (a ij ) n i,j=1 the original game can be treated mathematically as where x i and y j denote the switches of row i and of column j, respectively.
The new optimization problem herein proposed involves a matrix (a ij ) with n 1 rows and n 2 columns whose elements are unit vectors of the plane, R 2 .The initial direction pattern of each n 1 n 2 vectors is set up at the beginning of the game.For each row i and each column j there are knobs x i and y j , respectively.Rotating knob x i by an angle θ i , it affects all vectors a ij of the row i. Analogously, when knob y j is rotated by an angle θ j , the same happens with all the vectors a ij of the column j (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 Θ of unit vectors, let s(Θ) be the supremum of the (Euclidean) norms of the sums of all n 1 n 2 vectors achievable by row and column adjusts.The extremal problem, that is if one starts with the worse possible initial pattern, is to determine n 1 n 2 := min{s(Θ) : Θ an n 1 × n 2 pattern}.Regarding this problem, our main result provides an asymptotic growth for G (1) n 1 n 2 , as indicated in the following theorem: Theorem 1.1.For all integers n 1 , n 2 , we have In this paper we are also interested in a weighted version of the above problem.More precisely, we want to understand growth estimates when the knobs in the n 2 columns may be used not only to rotate column j , but also multiply the vectors a ij by a choice of real numbers b j verifying Again the game consists of maximizing the Euclidean norm of the sum of all vectors.In other words, for an initial pattern Θ of unit vectors, let s(Θ) be the supremum of the Euclidean norms of the sums of all n 1 n 2 vectors achievable by row and column adjusts.The extremal problem, starting at the worse possible initial pattern, is to determine n 1 n 2 := min{s(Θ) : Θ an n 1 × n 2 pattern}.The weighted feature of the second game yields a different freedom to the player.Surprisingly, in this case, we shall obtain a definitive, constructive solution to the problem.Indeed, the second main result we prove in this article is: We conclude this introduction by commenting on the ideas and techniques used to prove both Theorems 1.1 and 1.2, which are of particular interest.We observe due to the combinatorial complexity of this kind of problems, growth estimates as in Theorem 1.1 and Theorem 1.2 are often obtained by non-deterministic techniques, see for instance [2,3,5].A main novelty proposed in this article regards a deterministic approach to estimate both G (1) n 1 n 2 , which yields improved, more precise estimates than non-deterministic methods.We believe 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) and (2) in higher-dimensional spaces.

Proof of Theorem 1.1
Initially, it is more convenient to conceive the vectors in the game as complex numbers a ij with norm 1, which represent the elements of the array (a ij ) n 1 ,n 2 i,j=1 .In this case, when the player rotates a knob, the action is modeled by multiplication by unimodular complex numbers.In the weighted game, to be treated in the next section, the action of rotating and multiplying by real numbers b j , with n 2 j=1 b 2 j = 1, will then be modeled as multiplication by certain complex numbers with norm less than or equal to 1.
We start off the proof of Theorem 1.1 by reminding that a consequence of the Krein-Milman Theorem assures that for all A : a j 1 j 2 x (1) and thus, G , where the infimum runs over all bilinear forms A : Once the problem has been described as above, the upper bound in Theorem 1.1 can be obtained by means of an argument from the seminal paper of Bohnenblust and Hille [7, Theorem II, page 608].We shall explain the necessary adaptations when we deliver the proof of Theorem 4.1, on Section 4.
As for the lower estimate, we shall make use of Khinchin inequality, which we revise for the sake of completeness.
2.1.Khinchin inequality.To motivate, let's state the following question: suppose that we have n real numbers a 1 , . . ., a n and a fair coin.When we flip the coin, if it comes up heads, you choose β 1 = a 1 , and if it comes up tails, you choose β 1 = −a 1 .When we play for the second time, if it comes up heads, you choose β 2 = β 1 + a 2 and, if it comes up tails, you choose Repeating the process, after having flipped the coin k times we have if it comes up heads and if it comes up tails.After n steps, what should be the expected value of Khinchin's inequality, see for instance [9, page 10], shows that the "average" behaves as the 2 -norm of (a n ) .More precisely, it asserts that for any p > 0 there are constants for all sequence of scalars (a i ) n i=1 and all positive integers n.
Back to the proof of Theorem 1.1, for the purpose of establishing a lower estimate for the growth of G (1) n 1 n 2 , we are interested in the case p = 1.The natural counterpart for the average 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 A 1 and B 1 such that (5) for every positive integer n and scalars a 1 , . . ., a n .In [13] it is proven that A 1 = √ π 2 and, for our purposes, from now on we shall only be interested in the left hand side of (5).For a bilinear form j 1 e j 1 , e j 2

  dt
(1) denoting the topological dual of n ∞ by ( n ∞ ) * and its closed unit ball by B ( n ∞ ) * , we have where in the last equality we have used the isometric isometry Finally, since |A (e j 1 , e j 2 )| = 1, we conclude that 1 .

In fact, by the orthogonality of Rademacher functions r
When n 1 > n 2 we were not able to design a constructive, definitive answer to the problem.Even the correct asymptotic growth behavior of G (2) n 1 n 2 in this case seems an interesting theoretical question.

The game in higher dimensions
The Gale-Berlekamp switching game has a natural extension to higher dimensions.Let m ≥ 2 be an integer and let an n × • • • × n array (a j 1 •••jm ) of lights be given each either on (a j 1 •••jm = 1) or off (a j 1 •••jm = −1).Let us also suppose that for each j k there is a switch x (k) j k so that if the switch is pulled (x (k) j k = −1) all of the corresponding lights are switched: on to off or off to on.The goal is to maximize the difference between the lights on and off.As in the two-dimensional case, maximize the difference between on-lights on and off-lights is equivalent to estimate max and the extremal problem consists of estimating As in the bilinear case, The anisotropic case allows for n 1 × • • • × n m arrays not necessarily square arrays and, in this case, we write From a recent result of [1], we can easily obtain (7) 1 Following the notation of [3], let m ≥ 2 be an integer and (a i 1 •••im ) and n × • • • × n be an array of complex scalars such that where the maximum is evaluated over all x (j)

Denoting
G m,n (p) = min g C m,n (p), where minimum is evaluated over all unimodular m-linear forms A : n p × • • • × n p → C, the best information we can collect (combining results from [3,14]) is the following: where C m is obtained by interpolation (via the Riesz-Thorin Theorem) of the constant 1 (the constant when p = 1) and 8 √ m! log(1 + 4m) (the constant when p = 2).Moreover, defining G m,n (p 1 , . . ., p m ) = min g C m,n (p 1 , . . ., p m ), from [14], it is possible to show that where D m , K m behave essentially as the constants from (8).
The above solution rests in a non-deterministic tool.We shall show in what follows that for the case of complex scalars we can find deterministic solutions with better constants, which happen to be optimal in same meaningful cases.
4.1.The anisotropic, continuous game.We begin with a matrix (a j 1 ...jm ) n 1 ,...,nm j 1 ,...,jm=1 whose elements are unit vectors in the Euclidean space R 2 .The initial direction pattern of each n 1 • • • n m vectors is set up at the beginning of the game.For each k ∈ {1, . . ., m}, we have n k knobs x j k , the same happens with all the vectors a j 1 ...jm with j k fixed.Defining Θ and s(Θ) as in the two-dimensional case, the extremal problem is to determine G (1) n 1 ...nm := min{s(Θ) : Θ an n 1 × • • • × n m pattern}.We prove the following: Theorem 4.1.For all positive integers m ≥ 2, n 1 , . . ., n m ≥ 1 we have Moreover, the universal upper bound 1 cannot be improved.