Tractable queries on big data via preprocessing with logarithmic-size output

To provide a dichotomy between those queries that are feasible on big data after appropriate preprocessing and those for which preprocessing does not help, Fan et al. developed the \(\sqcap \)-tractability theory, which provides a formal foundation on the tractability of query classes in the context of big data. Inspired by some technologies used to deal with big data, we introduce a novel notion of \(\sqcap '\)-tractability in this paper. We place a restriction on preprocessing functions, which limits the functions to produce relatively short outputs, at most logarithmic-size of the inputs. We set a complexity class to denote the classes of Boolean queries that are \(\sqcap '\)-tractable and conclude that it is properly contained in that of \(\sqcap \)-tractable query classes, after discovering that a \(\sqcap \)-tractable query class is not \(\sqcap '\)-tractable. With an existing reduction, which does not allow re-factorizing data and query parts, we define complete query classes for the complexity class and give an efficient way to detect such query classes. We also investigate the query classes that can be made \(\sqcap '\)-tractable and prove that all PTIME classes of Boolean queries can be made \(\sqcap '\)-tractable.