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

The branching program is a fundamental model of nonuniform computation, which conveniently captures both time and space restrictions. In this paper we prove a quadratic expected time-space tradeoff of the form TS = Ω ( n2 q ) for q-way deterministic decision branching programs, where q > 2. Here T is the expected computation time and S 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 T = O(n2). Previous exponential size tradeoffs for Boolean decision branching programs were valid for time-restricted models with T = o(n log2 n). Proving quadratic timespace 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 a few decades ago. The decision branching programs we consider are related to families of good linear codes. Our results also imply the first quadratic time-space tradeoffs for Boolean decision branching programs verifying circular convolution, matrix-vector multiplication and discrete Fourier transform. A quadratic tradeoff is the largest possible for all these problems. Using the constructive family of Justesen codes which are asymptotically good, we also demonstrate a constructive Boolean decision function which has a quadratic expected time-space tradeoff in the Boolean deterministic decision branching program model. For q-way programs where 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. In deriving these bounds we use several bounding techniques and introduce a few new ideas. These include a particular measure of progress which is specific to the decision function considered, partitioning the computational paths into disjoint sets and obtaining tradeoffs for each class separately and extensive use of linear constraints to obtain probability bounds.