Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle
Innovations in Theoretical Computer Science
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(n 2 /poly(log n)) to obtain correct answers with constant nonzero
... 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).