The space complexity of mirror games
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We consider a simple streaming game between two players Alice and Bob, which we call the mirror game. In this game, Alice and Bob take turns saying numbers belonging to the set {1, 2, ...,2N}. A player loses if they repeat a number that has already been said. Bob, who goes second, has a very simple (and memoryless) strategy to avoid losing: whenever Alice says x, respond with 2N+1-x. The question is: does Alice have a similarly simple strategy to win that avoids remembering all the numbers said by Bob? The answer is no. We prove a linear lower bound on the space complexity of any deterministic winning strategy of Alice. Interestingly, this follows as a consequence of the Eventown-Oddtown theorem from extremal combinatorics. We additionally demonstrate a randomized strategy for Alice that wins with high probability that requires only Õ(√(N)) space (provided that Alice has access to a random matching on K_2N). We also investigate lower bounds for a generalized mirror game where Alice and Bob alternate saying 1 number and b numbers each turn (respectively). When 1+b is a prime, our linear lower bounds continue to hold, but when 1+b is composite, we show that the existence of a o(N) space strategy for Bob implies the existence of exponential-sized matching vector families over Z^N_1+b.
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