Quadratic time-space lower bounds for computing natural functions with a random oracle
Name
LIPIcs-ITCS-2019-56.pdf
Description
Published version
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556.28 KB
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Unknown
Checksum (MD5)
6de0debc74eee10972565f4de5556ff3
Author(s)
McKay, Dylan M.
Williams, Richard Ryan
Date Issued
January 2019
Journal
Leibniz International Proceedings in Informatics, LIPIcs
Citation
2019. "Quadratic time-space lower bounds for computing natural functions with a random oracle." Leibniz International Proceedings in Informatics, LIPIcs, 124.
Version
Final published version
Abstract
© Dylan M. McKay and Richard Ryan Williams. 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 Ω(n2/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 n2−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(2k) clauses, with values printed in lexicographical order (i.e., F(0k), F(0k−11), . . ., F(1k)). Thus we have a 4k/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).
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.4230/LIPIcs.ITCS.2019.56
