Parallel Computation of the Minimal Elements of a Poset

Computing the minimal elements of a partially ordered finite set (poset) is a fundamental problem in combinatorics with numerous applications such as polynomial expression optimization, transversal hypergraph generation and redundant component removal, to name a few. We propose a divide-and-conquer algorithm which is not only cache-oblivious but also can be parallelized free of determinacy races. We have implemented it in Cilk++ targeting multicores. For our test problems of sufficiently large input size our code demonstrates a linear speedup on 32 cores.National Science Foundation (U.S.). (Grant number CNS-0615215)National Science Foundation (U.S.). (Grant number CCF- 0621511

E-CLoG: Counting edge-centric local graphlets

In recent years, graphlet counting has emerged as an important task in topological graph analysis. However, the existing works on graphlet counting obtain the graphlet counts for the entire network as a whole. These works capture the key graphical patterns that prevail in a given network but they fail to meet the demand of the majority of real-life graph related prediction tasks such as link prediction, edge/node classification, etc., which require to build features for an edge (or a vertex) of a network. To meet the demand for such applications, efficient algorithms are needed for counting local graphlets within the context of an edge (or a vertex). In this work, we propose an efficient method, titled E-CLOG, for counting all 3,4 and 5 size local graphlets with the context of a given edge for its all different edge orbits. We also provide a shared-memory, multi-core implementation of E-CLOG, which makes it even more scalable for very large real-world networks. In particular, We obtain strong scaling on a variety of graphs (14x-20x on 36 cores). We provide extensive experimental results to demonstrate the efficiency and effectiveness of the proposed method. For instance, we show that E-CLOG is faster than existing work by multiple order of magnitudes; for the Wordnet graph E-CLOG counts all 3,4 and 5-size local graphlets in 1.5 hours using a single thread and in only a few minutes using the parallel implementation, whereas the baseline method does not finish in more than 4 days. We also show that local graphlet counts around an edge are much better features for link prediction than well-known topological features; our experiments show that the former enjoys between 10% to 45% of improvement in the AUC value for predicting future links in three real-life social and collaboration networks

Invariants of Triangular Lie Algebras

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.Comment: LaTeX2e, 16 pages; misprints are corrected, some proofs are extende