A Quadratic Time-Space Tradeoff for Unrestricted Deterministic Decision Branching Programs

TL;DR

This paper establishes a fundamental quadratic time-space tradeoff for unrestricted deterministic decision branching programs, demonstrating exponential size requirements at quadratic time and extending results to various computational problems.

Contribution

It proves the first quadratic time-space tradeoff for unrestricted deterministic decision branching programs and applies this to several important computational functions.

Findings

Quadratic expected time-space tradeoff $ ext{T} ext{S} = ext{Omega}(n^2/q)$ for $q$-way deterministic decision branching programs.

Exponential size requirement when expected time $ ext{T} = O(n^2)$ for these programs.

First such tradeoffs shown for functions verifying circular convolution, matrix-vector multiplication, and discrete Fourier transform.

Abstract

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.