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The minimal sets within a collection of sets are defined as the ones which do not have a proper subset within the collection, and the maximal sets are the ones which 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 paper, we present two novel improvements of the high-quality extremal set identification algorithm, \textit{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, whilst 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 \textit{itemsets} increases. We provide experimental results using both real-world and synthetic data sets, and show our multi-threaded variant algorithm out-performing 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 multi-threaded program executes about as fast as the state of the art parallel GPU-based program using an NVIDIA GTX 580 graphics processing unit.

The subitemset isomorphism problem is really important and there are excellent practical solutions described in the literature. However, the computational complexity analysis and classification of the BZ (Bundala and Zavodny) subitemset isomorphism problem is currently an open problem. In this paper we prove that checking whether two sorting networks are BZ isomorphic to each other is GI-Complete; the general GI (Graph Isomorphism) problem is known to be in NP and LWPP, but widely believed to be neither P nor NP-Complete; recent research suggests that the problem is in QP. Moreover, we state the BZ sorting network isomorphism problem as a general isomorphism problem on itemsets --- because every sorting network is represented by Bundala and Zavodny as an itemset. The complexity classification presented in this paper applies sorting networks, as well as the general itemset isomorphism problem. The main consequence of our work is that currently no polynomial-time algorithm exists for solving the BZ sorting network subitemset isomorphism problem; however the CM (Choi and Moon) sorting network isomorphism problem can be efficiently solved in polynomial time.

In this paper we extend the knowledge on the problem of empirically searching for sorting networks of minimal depth. We present new search space pruning techniques for the last four levels of a candidate sorting network by considering only the output set representation of a network. We present an algorithm for checking whether an $n$-input sorting network of depth $d$ exists by considering the minimal up to permutation and reflection itemsets at each level and using the pruning at the last four levels. We experimentally evaluated this algorithm to find the optimal depth sorting networks for all $n \leq 12$., Comment: arXiv admin note: substantial text overlap with arXiv:1502.04748

A complete set of filters $F_n$ for the optimal-depth $n$-input sorting network problem is such that if there exists an $n$-input sorting network of depth $d$ then there exists one of the form $C \oplus C'$ for some $C \in F_n$. Previous work on the topic presents a method for finding complete set of filters $R_{n, 1}$ and $R_{n, 2}$ that consists only of networks of depths one and two respectively, whose outputs are minimal and representative up to permutation and reflection. Our main contribution is a practical approach for finding a complete set of filters $R_{n, 3}$ containing only networks of depth three whose outputs are minimal and representative up to permutation and reflection. In previous work, we have developed a highly efficient algorithm for finding extremal sets ( i.e. outputs of comparator networks; itemsets; ) up to permutation. In this paper we present a modification to this algorithm that identifies the representative itemsets up to permutation and reflection. Hence, the presented practical approach is the successful combination of known theory and practice that we apply to the domain of sorting networks. For $n < 17$, we empirically compute the complete set of filters $R_{n, 2}$, $R_{n, 3}$, $R_{n, 2} \upharpoonright w $ and $R_{n, 3}^w$ of the representative minimal up to permutation and reflection $n$-input networks, where all but $R_{n, 2}$ are novel to this work.