{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11983","name":"Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle","abstract":"We define a model of size-S R-way branching programs with oracles that can make up to S distinct oracle queries over all of their possible inputs, and generalize a lower bound proof strategy of Beame [SICOMP 1991] to apply in the case of random oracles. Through a series of succinct reductions, we prove that the following problems require randomized algorithms where the product of running time and space usage must be Omega(n^2\/poly(log n)) to obtain correct answers with constant nonzero probability, even for algorithms with constant-time access to a uniform random oracle (i.e., a uniform random hash function): \n- Given an unordered list L of n elements from [n] (possibly with repeated elements), output [n]-L. \n- Counting satisfying assignments to a given 2CNF, and printing any satisfying assignment to a given 3CNF. Note it is a major open problem to prove a time-space product lower bound of n^{2-o(1)} for the decision version of SAT, or even for the decision problem Majority-SAT. \n- Printing the truth table of a given CNF formula F with k inputs and n=O(2^k) clauses, with values printed in lexicographical order (i.e., F(0^k), F(0^{k-1}1), ..., F(1^k)). Thus we have a 4^k\/poly(k) lower bound in this case. \n- Evaluating a circuit with n inputs and O(n) outputs. \n As our lower bounds are based on R-way branching programs, they hold for any reasonable model of computation (e.g. log-word RAMs and multitape Turing machines).","keywords":["branching programs","random oracles","time-space tradeoffs","lower bounds","SAT","counting complexity"],"author":[{"@type":"Person","name":"McKay, Dylan M.","givenName":"Dylan M.","familyName":"McKay","affiliation":"EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA"},{"@type":"Person","name":"Williams, Richard Ryan","givenName":"Richard Ryan","familyName":"Williams","sameAs":"https:\/\/orcid.org\/0000-0003-2326-2233","affiliation":"EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA","funding":"Parts of this work were performed while visiting the Simons Institute for the Theory of Computing and the EECS department at UC Berkeley."}],"position":56,"pageStart":"56:1","pageEnd":"56:20","dateCreated":"2019-01-08","datePublished":"2019-01-08","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"McKay, Dylan M.","givenName":"Dylan M.","familyName":"McKay","affiliation":"EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA"},{"@type":"Person","name":"Williams, Richard Ryan","givenName":"Richard Ryan","familyName":"Williams","sameAs":"https:\/\/orcid.org\/0000-0003-2326-2233","affiliation":"EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA","funding":"Parts of this work were performed while visiting the Simons Institute for the Theory of Computing and the EECS department at UC Berkeley."}],"copyrightYear":"2019","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ITCS.2019.56","funding":"Supported by NSF CCF-1741615 (CAREER: Common Links in Algorithms and Complexity).","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.4086\/toc.2005.v001a008","http:\/\/arxiv.org\/abs\/1007.1146"],"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":"#article11983","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"}}}