Practical Algorithms for Finding Extremal Sets

The minimal sets within a collection of sets are defined as the ones that do not have a proper subset within the collection, and the maximal sets are the ones that do not have a proper superset within the collection. Identifying extremal sets is a fundamental problem with a wide range of applications in SAT solvers, data mining, and social network analysis. In this article, we present two novel improvements of the high-quality extremal set identification algorithm, AMS-Lex, described by Bayardo and Panda. The first technique uses memoization to improve the execution time of the single-threaded variant of the AMS-Lex, while our second improvement uses parallel programming methods. In a subset of the presented experiments, our memoized algorithm executes more than 400 times faster than the highly efficient publicly available implementation of AMS-Lex. Moreover, we show that our modified algorithm's speedup is not bounded above by a constant and that it increases as the length of the common prefixes in successive input itemsets increases. We provide experimental results using both real-world and synthetic datasets, and show our multithreaded variant algorithm outperforming AMS-Lex by 3 to 6 times. We find that on synthetic input datasets, when executed using 16 CPU cores of a 32-core machine, our multithreaded program executes about as fast as the state-of-the-art parallel GPU-based program using an NVIDIA GTX 580 graphics processing unit.