Sublinear-time Reductions for Big Data Computing

Abstract

With the rapid popularization of big data, the dichotomy between tractable and intractable problems in big data computing has been shifted. Sublinear time, rather than polynomial time, has recently been regarded as the new standard of tractability in big data computing. This change brings the demand for new methodologies in computational complexity theory in the context of big data. Based on the prior work for sublinear-time complexity classes [9], this paper focuses on sublineartime reductions specialized for problems in big data computing. First, the pseudo-sublinear-time reduction is proposed and the complexity classes P and PsT are proved to be closed under it. To establish PsT-intractability for certain problems in P, we find the first problem in P \ PsT. Using the pseudo-sublinear-time reduction, we prove that the nearest edge query is in PsT but the algebraic equation root problem is not. Then, the pseudopolylog-time reduction is introduced and the complexity class PsPL is proved to be closed under it. The PsT-completeness under it is regarded as an evidence that some problems can not be solved in polylogarithmic time after a polynomial-time preprocessing, unless PsT = PsPL. We prove that all PsT-complete problems are also P-complete, which gives a further direction for identifying PsT-complete problems.