Making queries tractable on big data with preprocessing: through the eyes of complexity theory

Abstract

A query class is traditionally considered tractable if there exists a polynomial-time (PTIME) algorithm to answer its queries. When it comes to big data, however, PTIME al- gorithms often become infeasible in practice. A traditional and e ective approach to coping with this is to preprocess data o -line, so that queries in the class can be subsequently evaluated on the data e ciently. This paper aims to pro- vide a formal foundation for this approach in terms of com- putational complexity. (1) We propose a set of -tractable queries, denoted by T0 Q, to characterize classes of queries that can be answered in parallel poly-logarithmic time (NC) after PTIME preprocessing. (2) We show that several natu- ral query classes are -tractable and are feasible on big data. (3) We also study a set TQ of query classes that can be ef- fectively converted to -tractable queries by re-factorizing its data and queries for preprocessing. We introduce a form of NC reductions to characterize such conversions. (4) We show that a natural query class is complete for TQ. (5) We also show that T0 Q P unless P = NC, i.e., the set T0 Q of all -tractable queries is properly contained in the set P of all PTIME queries. Nonetheless, TQ = P, i.e., all PTIME query classes can be made -tractable via proper re- factorizations. This work is a step towards understanding the tractability of queries in the context of big data.