Sublinear-time Reductions for Big Data Computing

TL;DR

This paper introduces new sublinear-time reductions for big data problems, establishing complexity classes and demonstrating the limits of fast algorithms in big data computing.

Contribution

It proposes pseudo-sublinear-time and pseudo-polylog-time reductions, defining new complexity classes and analyzing problem tractability in big data contexts.

Findings

Pseudo-sublinear-time reduction preserves certain complexity classes.

Identified problems that are in extit{PsT} but not in extit{PsT} for intractability.

Proved extit{PsT}-completeness implies extit{P}-completeness under these reductions.

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 \cite{DBLP:journals/tcs/GaoLML20}, this paper focuses on sublinear-time reductions specialized for problems in big data computing. First, the pseudo-sublinear-time reduction is proposed and the complexity classes \Pproblem and \PsT are proved to be closed under it. To establish \PsT-intractability for certain problems in \Pproblem, we find the first problem in $\Pproblem \setminus \PsT$. 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 pseudo-polylog-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 \Pproblem-complete, which gives a further direction for identifying \PsT-complete problems.