{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11963","name":"The Space Complexity of Mirror Games","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?\nThe 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).\nWe 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}.","keywords":["Mirror Games","Space Complexity","Eventown-Oddtown"],"author":[{"@type":"Person","name":"Garg, Sumegha","givenName":"Sumegha","familyName":"Garg","affiliation":"Princeton University, Princeton, USA"},{"@type":"Person","name":"Schneider, Jon","givenName":"Jon","familyName":"Schneider","affiliation":"Google Research, New York, USA"}],"position":36,"pageStart":"36:1","pageEnd":"36:14","dateCreated":"2019-01-08","datePublished":"2019-01-08","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Garg, Sumegha","givenName":"Sumegha","familyName":"Garg","affiliation":"Princeton University, Princeton, USA"},{"@type":"Person","name":"Schneider, Jon","givenName":"Jon","familyName":"Schneider","affiliation":"Google Research, New York, USA"}],"copyrightYear":"2019","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ITCS.2019.36","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6327","volumeNumber":124,"name":"10th Innovations in Theoretical Computer Science Conference (ITCS 2019)","dateCreated":"2019-01-08","datePublished":"2019-01-08","editor":{"@type":"Person","name":"Blum, Avrim","givenName":"Avrim","familyName":"Blum"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11963","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6327"}}}