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The Space Complexity of Mirror Games

Authors Sumegha Garg, Jon Schneider



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LIPIcs.ITCS.2019.36.pdf
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Author Details

Sumegha Garg
  • Princeton University, Princeton, USA
Jon Schneider
  • Google Research, New York, USA

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Sumegha Garg and Jon Schneider. The Space Complexity of Mirror Games. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 36:1-36:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ITCS.2019.36

Abstract

We consider the following game between two players Alice and Bob, which we call the mirror game. Alice and Bob take turns saying numbers belonging to the set {1, 2, ...,N}. A player loses if they repeat a number that has already been said. Otherwise, after N turns, when all the numbers have been spoken, both players win. When N is even, Bob, who goes second, has a very simple (and memoryless) strategy to avoid losing: whenever Alice says x, respond with N+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 O~(sqrt N) space (provided that Alice has access to a random matching on K_N). 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 (when N != 0 mod (1+b)) implies the existence of exponential-sized matching vector families over Z^N_{1+b}.

Subject Classification

ACM Subject Classification
  • Theory of computation → Interactive computation
Keywords
  • Mirror Games
  • Space Complexity
  • Eventown-Oddtown

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References

  1. Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. Google Scholar
  2. Elwyn R Berlekamp, John Horton Conway, and Richard K Guy. Winning ways for your mathematical plays, volume 3. AK Peters Natick, 2003. Google Scholar
  3. ER Berlekamp. On subsets with intersections of even cardinality. Canad. Math. Bull, 12(4):471-477, 1969. Google Scholar
  4. Abhishek Bhowmick, Zeev Dvir, and Shachar Lovett. New bounds for matching vector families. SIAM Journal on Computing, 43(5):1654-1683, 2014. Google Scholar
  5. Zeev Dvir, Parikshit Gopalan, and Sergey Yekhanin. Matching vector codes. SIAM Journal on Computing, 40(4):1154-1178, 2011. Google Scholar
  6. Zeev Dvir and Sivakanth Gopi. 2-Server PIR with Subpolynomial Communication. Journal of the ACM (JACM), 63(4):39, 2016. Google Scholar
  7. Klim Efremenko. 3-query locally decodable codes of subexponential length. SIAM Journal on Computing, 41(6):1694-1703, 2012. Google Scholar
  8. Peter Frankl and Richard M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357-368, 1981. Google Scholar
  9. Vince Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20(1):71-86, 2000. Google Scholar
  10. S Muthukrishnan. Data Streams: Algorithms and Applications (Foundations and Trends in Theoretical Computer Science). Foundations and Trends in Theoretical Computer Science, 2005. Google Scholar
  11. Noam Nisan. On read-once vs. multiple access to randomness in logspace. In Structure in Complexity Theory Conference, 1990, Proceedings., Fifth Annual, pages 179-184. IEEE, 1990. Google Scholar
  12. Tibor Szabó. http://discretemath.imp.fu-berlin.de/DMII-2011-12/linalgmethod.pdf, 2011. Google Scholar
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