Sublinear-time reductions for big data computing

Abstract

The inefficiency of polynomial-time (PTIME) algorithms in the context of big data indicates a departure from the traditional view on tractability. In recent years, sublinear-time algorithms have been regarded as the new standard for tractability of big data computing problems. Based on the prior work on sublinear-time complexity classes [1], this paper focuses on designing appropriate reductions specialized for big data computing problems. In particular, a series of pseudo-sublinear-time reductions are proposed, and their properties are systematically investigated. It is proved that PsT is properly contained in P, and any PsT-complete problem and PsPLi-complete problem that is not P-complete must be a witness in P∖NC. Several examples are also given to illustrate the usefulness of the proposed reductions. Then, to cope with problems that are intractable in sublinear time even after a PTIME preprocessing, the L-reduction is extended to pseudo-sublinear time. Finally, it is proved that there is no PPL-complete problem under DLOGTIME reduction.