Berlekamp switching game

The Berlekamp switching game is a mathematical game proposed by American mathematician Elwyn Berlekamp.[1] It has also been called the Gale–Berlekamp switching game, after David Gale, who discovered the same game independently,[2] or the unbalancing lights game.[3] It involves a system of lightbulbs controlled by two banks of switches, with one game player trying to turn many lightbulbs on and the other trying to keep as many as possible off. It can be used to demonstrate the concept of covering radius in coding theory.

Rules

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The equipment for playing the game consists of a room containing rectangular array of lightbulbs, of dimensions   for some numbers   and  . A bank of   switches on one side of the room controls each lightbulb individually. Flipping one of these switches changes its lightbulb from off to on or from on to off, depending on its previous state. On the other side of the room is another bank of   switches, one for each row or column of lightbulbs. Whenever any of these switches is flipped, every lightbulb in the row or column that it controls changes from off to on or from on to off, depending on its previous state. When flipping more than one switch, the order in which the switches are flipped does not make a difference to the outcome: the same lightbulbs will be lit at the end of the sequence of flips no matter what order they are flipped.

The game is played in two rounds. In the first round, the first player uses the switches that control individual lights, to set the lights on or off arbitrarily. In the second round, the second player uses the switches that control rows or columns of lights, changing the pattern of lights set by the first player into another pattern (or, possibly, leaving it unchanged). The goal of the first player is to have as many lights remaining lit at the end of the game as possible, and the goal of the second player is to have as few lights remaining lit as possible. Therefore, the first player should choose a pattern of lights for which the second player cannot turn off many lights.

History

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Berlekamp worked at Bell Labs in Murray Hill, New Jersey from 1966 to 1971.[4] While there, he constructed a physical instance of this game for the case   in the Mathematics Department commons room.[1][2] David Gale also invented the game independently, some time prior to 1971.[5]

Early research on related problems included publications by Andrew M. Gleason (1960), whose computer experiments can be interpreted as asking, for the   game, how well the second player can do against a first player who plays randomly,[6] 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  .[7]

Analysis

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Mathematically, one can describe the lights turned on by the first player's move as a set  , and the smallest number of lights that can be achieved by the best play for the second player as a number  . The best play for the first player is to choose a set   that maximizes  . Therefore, one can describe the largest number of lights that can be achieved by the best play for the first player as a number  . 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   in general, as a function of   and  , to determine its behavior as a function, or to calculate its value for as many combinations of   and   as possible.

The case of a square   array has been solved for  . Additionally, lower bounds for   have been found for  .[8][9][10][11] These numbers are:

Solutions for  
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
  0 1 2 4 7 11 16 22 27 35 43 54 ≥ 60 ≥ 71 ≥ 83 ≥ 96 ≥ 107 ≥ 122 ≥ 139 ≥ 148

Asymptotically, these numbers grow as  .[2][5][12]

Computational complexity

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Because there are exponentially many choices for which switches to flip, an exhaustive search for the optimal choice is not possible for large  , 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   by playing randomly. Similarly, the second player can obtain a value whose expected distance from   is   by playing randomly; this value might either be larger or smaller than  , but if it is larger the second player can flip all row switches to get a value that is smaller by the same amount.[2][5][12] 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.[13]

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.[14] However, there is a polynomial-time approximation scheme for the   game that can find a choice for the second player that leaves only   times the minimum possible number of lit bulbs, for any  , in time  .[15]

Connection to coding theory

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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   and dimension   is defined as a  -dimensional linear subspace of the  -dimensional vector space   over the finite field with two elements,  . The elements of the subspace are called codewords, and the covering radius is the smallest number   such that every point of   is within Hamming distance   of a codeword.

Let   and  . For these parameter values, the vector space   describes all possible patterns of lit bulbs on the   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   choices for how to set the second bank of switches, this subspace has   elements, giving it dimension  , because flipping all of the second player's switches has no effect on the pattern of lit bulbs.

Then   is the covering radius of this code. The set of lit bulbs chosen by the first player, with best play, gives a point of   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.[1]

See also

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  • Lights Out (game), a different puzzle involving turning off lightbulbs using switches that control multiple bulbs

References

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  1. ^ a b c Sloane, N. J. A. (1987). "Unsolved problems related to the covering radius of codes". In Cover, Thomas M.; Gopinath, B. (eds.). Open Problems in Communication and Computation. New York: Springer. pp. 51–56. doi:10.1007/978-1-4612-4808-8_11.
  2. ^ a b c d Spencer, Joel (1994). "Lecture 6: Chaos from order". Ten Lectures on the Probabilistic Method. CBMS-NSF Regional Conference Series in Applied Mathematics. Vol. 64 (Second ed.). Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics. pp. 45–50. doi:10.1137/1.9781611970074. ISBN 0-89871-325-0. MR 1249485.
  3. ^ Araújo, Gustavo; Pellegrino, Daniel (2019). "A Gale–Berlekamp permutation-switching problem in higher dimensions". European Journal of Combinatorics. 77: 17–30. arXiv:1801.09194. doi:10.1016/j.ejc.2018.10.007. MR 3872901. S2CID 57760841.
  4. ^ Sanders, Robert (April 18, 2019). "Elwyn Berlekamp, game theorist and coding pioneer, dies at 78". Berkeley News. University of California, Berkeley.
  5. ^ a b c Brown, Thomas A.; Spencer, Joel H. (1971). "Minimization of   matrices under line shifts". Colloquium Mathematicum. 23: 165–171, 177. doi:10.4064/cm-23-1-165-171. MR 0307944.
  6. ^ Gleason, Andrew M. (1960). "A search problem in the  -cube". In Bellman, Richard; Hall, Marshall Jr. (eds.). Combinatorial Analysis. Proceedings of Symposia in Applied Mathematics. Vol. 10. Providence, Rhode Island: American Mathematical Society. pp. 175–178. MR 0114323.
  7. ^ Moon, J. W.; Moser, L. (1966). "An extremal problem in matrix theory". Matematički Vesnik. 3(18) (37): 209–211. MR 0207570.
  8. ^ Carlson, Jordan; Stolarski, Daniel (October 2004). "The correct solution to Berlekamp's switching game". Discrete Mathematics. 287 (1–3): 145–150. doi:10.1016/j.disc.2004.06.015. MR 2094708.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A005311 (Solution to Berlekamp's switching game (or lightbulb game) on an n X n board)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  10. ^ Pellegrino, D.; Raposo Jr, A. (2022). "Constants of the Kahane–Salem–Zygmund inequality asymptotically bounded by 1". Journal of Functional Analysis. 282 (2): 109293. arXiv:2006.12892. doi:10.1016/j.jfa.2021.109293. S2CID 231895733.
  11. ^ Pellegrino, D.; Raposo Jr, A. (2021). "Upper bounds for the constants of Bennett's inequality and the Gale-Berlekamp switching game". arXiv:2111.00445v3 [math.CO].
  12. ^ a b Komlós, J.; Sulyok, M. (1970). "On the sum of elements of   matrices". Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969). pp. 721–728. MR 0299500.
  13. ^ Berger, Bonnie (1997). "The fourth moment method". SIAM Journal on Computing. 26 (4): 1188–1207. doi:10.1137/S0097539792240005. MR 1460721. S2CID 14313557.
  14. ^ Roth, Ron M.; Viswanathan, Krishnamurthy (2008). "On the hardness of decoding the Gale–Berlekamp code". IEEE Transactions on Information Theory. 54 (3): 1050–1060. doi:10.1109/TIT.2007.915716. MR 2445050.
  15. ^ Karpinski, Marek; Schudy, Warren (2009). "Linear time approximation schemes for the Gale–Berlekamp game and related minimization problems". In Mitzenmacher, Michael (ed.). Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009. ACM. pp. 313–322. arXiv:0811.3244. doi:10.1145/1536414.1536458.