Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle
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).
branching programs
random oracles
time-space tradeoffs
lower bounds
SAT
counting complexity
Theory of computation~Circuit complexity
Theory of computation~Oracles and decision trees
56:1-56:20
Regular Paper
Supported by NSF CCF-1741615 (CAREER: Common Links in Algorithms and Complexity).
Dylan M.
McKay
Dylan M. McKay
EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA
Richard Ryan
Williams
Richard Ryan Williams
EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA
https://orcid.org/0000-0003-2326-2233
Parts of this work were performed while visiting the Simons Institute for the Theory of Computing and the EECS department at UC Berkeley.
10.4230/LIPIcs.ITCS.2019.56
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Dylan M. McKay and Richard Ryan Williams
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