License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2019.56
URN: urn:nbn:de:0030-drops-101493
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/10149/
McKay, Dylan M. ;
Williams, Richard Ryan
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):
- 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}
}
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Keywords: |
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branching programs, random oracles, time-space tradeoffs, lower bounds, SAT, counting complexity |
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Collection: |
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10th Innovations in Theoretical Computer Science Conference (ITCS 2019) |
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Issue Date: |
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2018 |
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Date of publication: |
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08.01.2019 |
