Round Compression for Parallel Matching Algorithms

For over a decade now we have been witnessing the success of {\em massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately capture the nature of large-scale computation. In particular, compared to the classic distributed algorithms or PRAM models, these frameworks allow for much more local computation. The fundamental question that arises in this context is though: can we leverage this additional power to obtain even faster parallel algorithms? A prominent example here is the {\em maximum matching} problem---one of the most classic graph problems. It is well known that in the PRAM model one can compute a 2-approximate maximum matching in $O(\log{n})$ rounds. However, the exact complexity of this problem in the MPC framework is still far from understood. Lattanzi et al. showed that if each machine has $n^{1+\Omega(1)}$ memory, this problem can also be solved $2$-approximately in a constant number of rounds. These techniques, as well as the approaches developed in the follow up work, seem though to get stuck in a fundamental way at roughly $O(\log{n})$ rounds once we enter the near-linear memory regime. It is thus entirely possible that in this regime, which captures in particular the case of sparse graph computations, the best MPC round complexity matches what one can already get in the PRAM model, without the need to take advantage of the extra local computation power. In this paper, we finally refute that perplexing possibility. That is, we break the above $O(\log n)$ round complexity bound even in the case of {\em slightly sublinear} memory per machine. In fact, our improvement here is {\em almost exponential}: we are able to deliver a $(2+\epsilon)$-approximation to maximum matching, for any fixed constant $\epsilon>0$, in $O((\log \log n)^2)$ rounds

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

Parallel algorithms for normalization

Given a reduced affine algebra A over a perfect field K, we present parallel algorithms to compute the normalization \bar{A} of A. Our starting point is the algorithm of Greuel, Laplagne, and Seelisch, which is an improvement of de Jong's algorithm. First, we propose to stratify the singular locus Sing(A) in a way which is compatible with normalization, apply a local version of the normalization algorithm at each stratum, and find \bar{A} by putting the local results together. Second, in the case where K = Q is the field of rationals, we propose modular versions of the global and local-to-global algorithms. We have implemented our algorithms in the computer algebra system SINGULAR and compare their performance with that of the algorithm of Greuel, Laplagne, and Seelisch. In the case where K = Q, we also discuss the use of modular computations of Groebner bases, radicals, and primary decompositions. We point out that in most examples, the new algorithms outperform the algorithm of Greuel, Laplagne, and Seelisch by far, even if we do not run them in parallel.Comment: 19 page

Main Memory Adaptive Indexing for Multi-core Systems

Adaptive indexing is a concept that considers index creation in databases as a by-product of query processing; as opposed to traditional full index creation where the indexing effort is performed up front before answering any queries. Adaptive indexing has received a considerable amount of attention, and several algorithms have been proposed over the past few years; including a recent experimental study comparing a large number of existing methods. Until now, however, most adaptive indexing algorithms have been designed single-threaded, yet with multi-core systems already well established, the idea of designing parallel algorithms for adaptive indexing is very natural. In this regard only one parallel algorithm for adaptive indexing has recently appeared in the literature: The parallel version of standard cracking. In this paper we describe three alternative parallel algorithms for adaptive indexing, including a second variant of a parallel standard cracking algorithm. Additionally, we describe a hybrid parallel sorting algorithm, and a NUMA-aware method based on sorting. We then thoroughly compare all these algorithms experimentally; along a variant of a recently published parallel version of radix sort. Parallel sorting algorithms serve as a realistic baseline for multi-threaded adaptive indexing techniques. In total we experimentally compare seven parallel algorithms. Additionally, we extensively profile all considered algorithms. The initial set of experiments considered in this paper indicates that our parallel algorithms significantly improve over previously known ones. Our results suggest that, although adaptive indexing algorithms are a good design choice in single-threaded environments, the rules change considerably in the parallel case. That is, in future highly-parallel environments, sorting algorithms could be serious alternatives to adaptive indexing.Comment: 26 pages, 7 figure

Totally parallel multilevel algorithms

Four totally parallel algorithms for the solution of a sparse linear system have common characteristics which become quite apparent when they are implemented on a highly parallel hypercube such as the CM2. These four algorithms are Parallel Superconvergent Multigrid (PSMG) of Frederickson and McBryan, Robust Multigrid (RMG) of Hackbusch, the FFT based Spectral Algorithm, and Parallel Cyclic Reduction. In fact, all four can be formulated as particular cases of the same totally parallel multilevel algorithm, which are referred to as TPMA. In certain cases the spectral radius of TPMA is zero, and it is recognized to be a direct algorithm. In many other cases the spectral radius, although not zero, is small enough that a single iteration per timestep keeps the local error within the required tolerance

Shared-Memory Parallel Maximal Clique Enumeration

We present shared-memory parallel methods for Maximal Clique Enumeration (MCE) from a graph. MCE is a fundamental and well-studied graph analytics task, and is a widely used primitive for identifying dense structures in a graph. Due to its computationally intensive nature, parallel methods are imperative for dealing with large graphs. However, surprisingly, there do not yet exist scalable and parallel methods for MCE on a shared-memory parallel machine. In this work, we present efficient shared-memory parallel algorithms for MCE, with the following properties: (1) the parallel algorithms are provably work-efficient relative to a state-of-the-art sequential algorithm (2) the algorithms have a provably small parallel depth, showing that they can scale to a large number of processors, and (3) our implementations on a multicore machine shows a good speedup and scaling behavior with increasing number of cores, and are substantially faster than prior shared-memory parallel algorithms for MCE.Comment: 10 pages, 3 figures, proceedings of the 25th IEEE International Conference on. High Performance Computing, Data, and Analytics (HiPC), 201