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Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle McKay, Dylan M. Williams, Richard Ryan branching programs random oracles time-space tradeoffs lower bounds SAT counting complexity 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): - Given an unordered list L of n elements from [n] (possibly with repeated elements), output [n]-L. - 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. - 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. - Evaluating a circuit with n inputs and O(n) outputs. 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). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Dylan M. McKay and Richard Ryan Williams 2019 Is Part Of LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2019.56 urn:nbn:de:0030-drops-101493 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.56 eng https://creativecommons.org/licenses/by/3.0/legalcode