Constructive derandomization of query algorithms
share
We give efficient deterministic algorithms for converting randomized query algorithms into deterministic ones. We first give an algorithm that takes as input a randomized q-query algorithm R with description length N and a parameter ε, runs in time poly(N) · 2^O(q/ε), and returns a deterministic O(q/ε)-query algorithm D that ε-approximates the acceptance probabilities of R. These parameters are near-optimal: runtime N + 2^Ω(q/ε) and query complexity Ω(q/ε) are necessary. Next, we give algorithms for instance-optimal and online versions of the problem: ∘ Instance optimal: Construct a deterministic q^_R-query algorithm D, where q^_R is minimum query complexity of any deterministic algorithm that ε-approximates R. ∘ Online: Deterministically approximate the acceptance probability of R for a specific input x in time poly(N,q,1/ε), without constructing D in its entirety. Applying the techniques we develop for these extensions, we constructivize classic results that relate the deterministic, randomized, and quantum query complexities of boolean functions (Nisan, STOC 1989; Beals et al., FOCS 1998). This has direct implications for the Turing machine model of computation: sublinear-time algorithms for total decision problems can be efficiently derandomized and dequantized with a subexponential-time preprocessing step.
READ FULL TEXT