Parallel Peeling Algorithms

The analysis of several algorithms and data structures can be framed as a peeling process on a random hypergraph: vertices with degree less than k are removed until there are no vertices of degree less than k left. The remaining hypergraph is known as the k-core. In this paper, we analyze parallel peeling processes, where in each round, all vertices of degree less than k are removed. It is known that, below a specific edge density threshold, the k-core is empty with high probability. We show that, with high probability, below this threshold, only (log log n)/log(k-1)(r-1) + O(1) rounds of peeling are needed to obtain the empty k-core for r-uniform hypergraphs. Interestingly, we show that above this threshold, Omega(log n) rounds of peeling are required to find the non-empty k-core. Since most algorithms and data structures aim to peel to an empty k-core, this asymmetry appears fortunate. We verify the theoretical results both with simulation and with a parallel implementation using graphics processing units (GPUs). Our implementation provides insights into how to structure parallel peeling algorithms for efficiency in practice.Comment: Appears in SPAA 2014. Minor typo corrections relative to previous versio

Unsupervised Bump Hunting Using Principal Components

Principal Components Analysis is a widely used technique for dimension reduction and characterization of variability in multivariate populations. Our interest lies in studying when and why the rotation to principal components can be used effectively within a response-predictor set relationship in the context of mode hunting. Specifically focusing on the Patient Rule Induction Method (PRIM), we first develop a fast version of this algorithm (fastPRIM) under normality which facilitates the theoretical studies to follow. Using basic geometrical arguments, we then demonstrate how the PC rotation of the predictor space alone can in fact generate improved mode estimators. Simulation results are used to illustrate our findings.Comment: 24 pages, 9 figure

Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable

There has been significant recent interest in parallel graph processing due to the need to quickly analyze the large graphs available today. Many graph codes have been designed for distributed memory or external memory. However, today even the largest publicly-available real-world graph (the Hyperlink Web graph with over 3.5 billion vertices and 128 billion edges) can fit in the memory of a single commodity multicore server. Nevertheless, most experimental work in the literature report results on much smaller graphs, and the ones for the Hyperlink graph use distributed or external memory. Therefore, it is natural to ask whether we can efficiently solve a broad class of graph problems on this graph in memory. This paper shows that theoretically-efficient parallel graph algorithms can scale to the largest publicly-available graphs using a single machine with a terabyte of RAM, processing them in minutes. We give implementations of theoretically-efficient parallel algorithms for 20 important graph problems. We also present the optimizations and techniques that we used in our implementations, which were crucial in enabling us to process these large graphs quickly. We show that the running times of our implementations outperform existing state-of-the-art implementations on the largest real-world graphs. For many of the problems that we consider, this is the first time they have been solved on graphs at this scale. We have made the implementations developed in this work publicly-available as the Graph-Based Benchmark Suite (GBBS).Comment: This is the full version of the paper appearing in the ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 201

Sympiler: Transforming Sparse Matrix Codes by Decoupling Symbolic Analysis

Sympiler is a domain-specific code generator that optimizes sparse matrix computations by decoupling the symbolic analysis phase from the numerical manipulation stage in sparse codes. The computation patterns in sparse numerical methods are guided by the input sparsity structure and the sparse algorithm itself. In many real-world simulations, the sparsity pattern changes little or not at all. Sympiler takes advantage of these properties to symbolically analyze sparse codes at compile-time and to apply inspector-guided transformations that enable applying low-level transformations to sparse codes. As a result, the Sympiler-generated code outperforms highly-optimized matrix factorization codes from commonly-used specialized libraries, obtaining average speedups over Eigen and CHOLMOD of 3.8X and 1.5X respectively.Comment: 12 page