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DOI: 10.4230/LIPIcs.ITCS.2019.56
URN: urn:nbn:de:0030-drops-101493
URL: http://drops.dagstuhl.de/opus/volltexte/2018/10149/
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McKay, Dylan M. ; Williams, Richard Ryan

Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle

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LIPIcs-ITCS-2019-56.pdf (0.5 MB)


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): - 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).

BibTeX - Entry

@InProceedings{mckay_et_al:LIPIcs:2018:10149,
  author =	{Dylan M. McKay and Richard Ryan Williams},
  title =	{{Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle}},
  booktitle =	{10th Innovations in Theoretical Computer Science  Conference (ITCS 2019)},
  pages =	{56:1--56:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{124},
  editor =	{Avrim Blum},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/10149},
  URN =		{urn:nbn:de:0030-drops-101493},
  doi =		{10.4230/LIPIcs.ITCS.2019.56},
  annote =	{Keywords: branching programs, random oracles, time-space tradeoffs, lower bounds, SAT, counting complexity}
}

Keywords: branching programs, random oracles, time-space tradeoffs, lower bounds, SAT, counting complexity
Seminar: 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)
Issue Date: 2018
Date of publication: 21.12.2018


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