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

Authors Dylan M. McKay, Richard Ryan Williams

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LIPIcs.ITCS.2019.56.pdf
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Dylan M. McKay
  • EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA
Richard Ryan Williams
  • EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA

Funding

Supported by NSF CCF-1741615 (CAREER: Common Links in Algorithms and Complexity).
  • Williams, Richard Ryan: Parts of this work were performed while visiting the Simons Institute for the Theory of Computing and the EECS department at UC Berkeley.

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Dylan M. McKay and Richard Ryan Williams. Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ITCS.2019.56

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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Oracles and decision trees
Keywords
  • branching programs
  • random oracles
  • time-space tradeoffs
  • lower bounds
  • SAT
  • counting complexity

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