Berlekamp switching game

History

Berlekamp worked at Bell Labs in Murray Hill, New Jersey from 1966 to 1971. While there, he constructed a physical instance of this game for the case ${\displaystyle 10\times 10}$ in the Mathematics Department commons room. David Gale also invented the game independently, some time prior to 1971.

Early research on related problems included publications by Andrew M. Gleason (1960), whose computer experiments can be interpreted as asking, for the ${\displaystyle 15\times 15}$ game, how well the second player can do against a first player who plays randomly, and by J. W. Moon and Leo Moser (1966), who address Gleason's question theoretically, showing that for almost all choices of the first player, in the limit of large game board sizes, the optimal game value is close to ${\displaystyle \frac{1}{2}{n}^2}$.

Analysis

Mathematically, one can describe the lights turned on by the first player's move as a set ${\displaystyle S}$, and the smallest number of lights that can be achieved by the best play for the second player as a number ${\displaystyle f(S)}$. The best play for the first player is to choose a set ${\displaystyle S}$ that maximizes ${\displaystyle f(S)}$. Therefore, one can describe the largest number of lights that can be achieved by the best play for the first player as a number $R_{a,b}=\underset{S}{max}f(S)$. Beyond the question of how to play well in an individual game, a broader question that has been the object of mathematical research is to characterize the value of ${\displaystyle R_{a,b}}$ in general, as a function of ${\displaystyle a}$ and ${\displaystyle b}$, to determine its behavior as a function, or to calculate its value for as many combinations of ${\displaystyle a}$ and ${\displaystyle b}$ as possible.

The case of a square ${\displaystyle n\times n}$ array has been solved for ${\displaystyle n\le 12}$. Additionally, lower bounds for ${\displaystyle R_{n,n}}$ have been found for ${\displaystyle n\le 20}$. These numbers are:

Asymptotically, these numbers grow as ${\displaystyle \frac{n^2}{2}-\mathrm{\Theta }(n^{3/2})}$.

Computational complexity

Because there are exponentially many choices for which switches to flip, an exhaustive search for the optimal choice is not possible for large ${\displaystyle n}$, setting up the question of how well computationally-limited players can play this game.

The first player can cause the expected game value to be ${\displaystyle \frac{n^2}{2}-O(n^{3/2})}$ by playing randomly. Similarly, the second player can obtain a value whose expected distance from ${\displaystyle \frac{n^2}{2}}$ is ${\displaystyle \mathrm{\Omega }(n^{3/2})}$ by playing randomly; this value might either be larger or smaller than ${\displaystyle \frac{n^2}{2}}$, but if it is larger the second player can flip all row switches to get a value that is smaller by the same amount. This random strategy for the second player can be made non-random using the method of conditional probabilities, giving a polynomial time algorithm that obtains the same solution value guarantees. A different derandomization gives a parallel algorithm in the complexity class NC.

Finding the optimal choice for the second player in the game, once the first player has chosen which bulbs to light, is an NP-hard problem. However, there is a polynomial-time approximation scheme for the ${\displaystyle n\times n}$ game that can find a choice for the second player that leaves only ${\displaystyle (1+\epsilon )}$ times the minimum possible number of lit bulbs, for any ${\displaystyle \epsilon >0}$, in time ${\displaystyle O(n^2)+2^{O(1/{\epsilon }^2)}}$.

Connection to coding theory

The Berlekamp switching game can be used in coding theory as a demonstration of the covering radius of a certain binary linear code. A binary linear code of length ${\displaystyle n}$ and dimension ${\displaystyle d}$ is defined as a ${\displaystyle d}$-dimensional linear subspace of the ${\displaystyle n}$-dimensional vector space ${\displaystyle {\mathbb{F}}_2^n}$ over the finite field with two elements, ${\displaystyle {\mathbb{F}}_2}$. The elements of the subspace are called codewords, and the covering radius is the smallest number ${\displaystyle r}$ such that every point of ${\displaystyle {\mathbb{F}}_2^n}$ is within Hamming distance ${\displaystyle r}$ of a codeword.

Let ${\displaystyle n=ab}$ and ${\displaystyle d=a+b-1}$. For these parameter values, the vector space ${\displaystyle {\mathbb{F}}_2^n}$ describes all possible patterns of lit bulbs on the ${\displaystyle a\times b}$ array of lightbulbs, with a vector addition operation that combines two patterns by lighting the bulbs that appear in exactly one of the two patterns (the symmetric difference operation on the sets of lit bulbs). One can define a linear subspace consisting of all patterns that the second player can turn completely off, or equivalently of all patterns that the second player could create starting with a board that is completely off. Although the second player has ${\displaystyle 2^{a+b}}$ choices for how to set the second bank of switches, this subspace has ${\displaystyle 2^{a+b-1}}$ elements, giving it dimension ${\displaystyle a+b-1}$, because flipping all of the second player's switches has no effect on the pattern of lit bulbs.

Then ${\displaystyle R_{a,b}}$ is the covering radius of this code. The set of lit bulbs chosen by the first player, with best play, gives a point of ${\displaystyle {\mathbb{F}}_2^n}$ that is as far as possible from the linear subspace. The set of bulbs whose state is changed by the second player, with best play, gives the closest point in the linear subspace. The set of bulbs that remain lit after these choices are the ones whose number defines the Hamming distance between these two points.