A Quadratic Time-Space Tradeoff for Unrestricted Deterministic Decision Branching Programs
<|reference_start|>A Quadratic Time-Space Tradeoff for Unrestricted Deterministic Decision Branching Programs: For a decision problem from coding theory, we prove a quadratic expected time-space tradeoff of the form $\eT\eS=\Omega(\tfrac{n^2}{q})$ for $q$-way deterministic decision branching programs, where $q\geq 2$. Here $\eT$ is the expected computation time and $\eS$ is the expected space, when all inputs are equally likely. This bound is to our knowledge, the first such to show an exponential size requirement whenever $\eT = O(n^2)$. Previous exponential size tradeoffs for Boolean decision branching programs were valid for time-restricted models with $T=o(n\log_2{n})$. Proving quadratic time-space tradeoffs for unrestricted time decision branching programs has been a major goal of recent research -- this goal has already been achieved for multiple-output branching programs two decades ago. We also show the first quadratic time-space tradeoffs for Boolean decision branching programs verifying circular convolution, matrix-vector multiplication and discrete Fourier transform. Furthermore, we demonstrate a constructive Boolean decision function which has a quadratic expected time-space tradeoff in the Boolean deterministic decision branching program model. When $q$ is a constant the tradeoff results derived here for decision functions verifying various functions are order-comparable to previously known tradeoff bounds for calculating the corresponding multiple-output functions.<|reference_end|>
@article{santhi2006a,
    title={A Quadratic Time-Space Tradeoff for Unrestricted Deterministic Decision
  Branching Programs},
    author={Nandakishore Santhi, Alexander Vardy},
    journal={arXiv preprint arXiv:cs/0608085},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0608085},
    primaryClass={cs.CC cs.DM cs.IT math.IT}
}