Recognizing the Tractability in Big Data Computing

TL;DR

This paper introduces new complexity classes based on sublinear time computation using random-access Turing machines to better understand tractability in big data computing, challenging traditional polynomial time standards.

Contribution

It proposes pure-tractable and pseudo-tractable classes for big data, analyzing their structure and relationships, and extends complexity theory to sublinear time models.

Findings

Pure-tractable classes $ ext{PL}$ and $ ext{ST}$ are structurally distinct.

Pseudo-tractable classes $ ext{PTR}$ and $ ext{PTE}$ are defined with preprocessing.

Relationships among classes are formally established, linking to classical complexity classes.

Abstract

Due to the limitation on computational power of existing computers, the polynomial time does not works for identifying the tractable problems in big data computing. This paper adopts the sublinear time as the new tractable standard to recognize the tractability in big data computing, and the random-access Turing machine is used as the computational model to characterize the problems that are tractable on big data. First, two pure-tractable classes are first proposed. One is the class $\mathrm{PL}$ consisting of the problems that can be solved in polylogarithmic time by a RATM. The another one is the class $\mathrm{ST}$ including all the problems that can be solved in sublinear time by a RATM. The structure of the two pure-tractable classes is deeply investigated and they are proved $\mathrm{PL^i} \subsetneq \mathrm{PL^{i+1}}$ and $\mathrm{PL} \subsetneq \mathrm{ST}$. Then, two pseudo-tractable classes, $\mathrm{PTR}$ and $\mathrm{PTE}$, are proposed. $\mathrm{PTR}$ consists of all the problems that can solved by a RATM in sublinear time after a PTIME preprocessing by reducing the size of input dataset. $\mathrm{PTE}$ includes all the problems that can solved by a RATM in sublinear time after a PTIME preprocessing by extending the size of input dataset. The relations among the two pseudo-tractable classes and other complexity classes are investigated and they are proved that $\mathrm{PT} \subseteq \mathrm{P}$, $\sqcap'\mathrm{T^0_Q} \subsetneq \mathrm{PTR^0_Q}$ and $\mathrm{PT_P} = \mathrm{P}$.