QuickLexSort: An efficient algorithm for lexicographically sorting nested restrictions of a database

Abstract

Lexicographical sorting is a fundamental problem with applications to contingency tables, databases, Bayesian networks, and more. A standard method to lexicographically sort general data is to iteratively use a stable sort -- a sort which preserves existing orders. Here we present a new method of lexicographical sorting called QuickLexSort. Whereas a stable sort based lexicographical sorting algorithm operates from the least important to most important features, in contrast, QuickLexSort sorts from the most important to least important features, refining the sort as it goes. QuickLexSort first requires a one-time modest pre-processing step where each feature of the data set is sorted independently. When lexicographically sorting a database, QuickLexSort (including pre-processing) has comparable running time to using a stable sort based approach. For a data base with $m$ rows and $n$ columns, and a sorting algorithm running in time $O(mlog(m))$, a stable sort based lexicographical sort and QuickLexSort will both take time $O(nmlog(m))$. However in many applications one has the need to lexicographically sort nested data, e.g.\ all possible sub-matrices up to a certain cardinality of columns. In such cases we show QuickLexSort gives a performance improvement of a log factor of the database length (rows in matrix) over using a standard stable sort based approach. E.g.\ to sort all sub-matrices up to cardinality $k$, QuickLexSort has running time $O(mn^k)$ whereas a stable sort based lexicographical sort will take time $O(mlog(m)n^k)$. After the pre-processing step that is run only once for the entire matrix, QuickLexSort has a running time linear in the number of nested sub-matrices to sort. We conclude with an application to Bayesian network scoring to detect epistasis using SNP marker data.