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|a © 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).
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