A medium-grain method for fast 2D bipartitioning of sparse matrices

We present a new hypergraph-based method, the medium-grain method, for solving the sparse matrix partitioning problem. This problem arises when distributing data for parallel sparse matrix-vector multiplication. In the medium-grain method, each matrix nonzero is assigned to either a row group or a column group, and these groups are represented by vertices of the hypergraph. For an m x n sparse matrix, the resulting hypergraph has m + n vertices and m + n hyperedges. Furthermore, we present an iterative refinement procedure for improvement of a given partitioning, based on the medium-grain method, which can be applied as a cheap but effective postprocessing step after any partitioning method. The medium-grain method is able to produce fully two-dimensional bipartitionings, but its computational complexity equals that of one-dimensional methods. Experimental results for a large set of sparse test matrices show that the medium-grain method with iterative refinement produces bipartitionings with lower communication volume compared to current state-of-the-art methods, and is faster at producing them

Reordering matrices for optimal sparse matrix bipartitioning

Sparse-matrix vector multiplication (SpMV) is one of the widely used and extensively studied kernels in today’s scientific computing and high-performance computing domains. The efficiency and scalability of this kernel is extensively investigated on single-core, multi-core, many-core processors and accelerators, and on distributed memory. In general, a good mapping of an application’s tasks to the processing units in a distributed environment is important since communication among these tasks is the main bottleneck on scalability. A fundamental approach to solve this problem is modeling the application via a graph/hypergraph and partitioning it. For SpMV, several graph/hypergraph models have been proposed. These approaches consider the problem as a balanced partitioning problem where the vertices (tasks) are partitioned (assigned) to the parts (processors) in a way that the total vertex weight (processor load) is balanced and the total communication incurred among the processors is minimized. The partitioning problem is NP-Hard and all the existing studies and tools use heuristics to solve the problem. For graphs, the literature on optimal partitioning contains a number of notable studies; however for hypergraphs, very little work has been done. Unfortunately, it has been shown that unlike graphs, hypergraphs can exactly model the total communication for SpMV. Recently, Pelt and Bisseling proposed a novel, purely combinatorial branch-and-bound-based approach for the sparse-matrix bipartitioning problem which can tackle relatively larger hypergraphs that were impossible to optimally partition into two by using previous methods. This work can be considered as an extension to their approach with two ideas. We propose to use; 1) matrix ordering techniques to use more information in the earlier branches of the tree, and 2) a machine learning approach to choose the best ordering based on matrix features. As our experiments on various matrices will show, these enhancements make the optimal bipartitioning process much faster

Open Problems in (Hyper)Graph Decomposition

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition

A Recursive Hypergraph Bipartitioning Framework for Reducing Bandwidth and Latency Costs Simultaneously

Intelligent partitioning models are commonly used for efficient parallelization of irregular applications on distributed systems. These models usually aim to minimize a single communication cost metric, which is either related to communication volume or message count. However, both volume- and message-related metrics should be taken into account during partitioning for a more efficient parallelization. There are only a few works that consider both of them and they usually address each in separate phases of a two-phase approach. In this work, we propose a recursive hypergraph bipartitioning framework that reduces the total volume and total message count in a single phase. In this framework, the standard hypergraph models, nets of which already capture the bandwidth cost, are augmented with message nets. The message nets encode the message count so that minimizing conventional cutsize captures the minimization of bandwidth and latency costs together. Our model provides a more accurate representation of the overall communication cost by incorporating both the bandwidth and the latency components into the partitioning objective. The use of the widely-adopted successful recursive bipartitioning framework provides the flexibility of using any existing hypergraph partitioner. The experiments on instances from different domains show that our model on the average achieves up to 52 percent reduction in total message count and hence results in 29 percent reduction in parallel running time compared to the model that considers only the total volume. © 2016 IEEE

Hypergraph Partitioning in the Cloud

The thesis investigates the partitioning and load balancing problem which has many applications in High Performance Computing (HPC). The application to be partitioned is described with a graph or hypergraph. The latter is of greater interest as hypergraphs, compared to graphs, have a more general structure and can be used to model more complex relationships between groups of objects such as non-symmetric dependencies. Optimal graph and hypergraph partitioning is known to be NP-Hard but good polynomial time heuristic algorithms have been proposed. In this thesis, we propose two multi-level hypergraph partitioning algorithms. The algorithms are based on rough set clustering techniques. The first algorithm, which is a serial algorithm, obtains high quality partitionings and improves the partitioning cut by up to 71\% compared to the state-of-the-art serial hypergraph partitioning algorithms. Furthermore, the capacity of serial algorithms is limited due to the rapid growth of problem sizes of distributed applications. Consequently, we also propose a parallel hypergraph partitioning algorithm. Considering the generality of the hypergraph model, designing a parallel algorithm is difficult and the available parallel hypergraph algorithms offer less scalability compared to their graph counterparts. The issue is twofold: the parallel algorithm and the complexity of the hypergraph structure. Our parallel algorithm provides a trade-off between global and local vertex clustering decisions. By employing novel techniques and approaches, our algorithm achieves better scalability than the state-of-the-art parallel hypergraph partitioner in the Zoltan tool on a set of benchmarks, especially ones with irregular structure. Furthermore, recent advances in cloud computing and the services they provide have led to a trend in moving HPC and large scale distributed applications into the cloud. Despite its advantages, some aspects of the cloud, such as limited network resources, present a challenge to running communication-intensive applications and make them non-scalable in the cloud. While hypergraph partitioning is proposed as a solution for decreasing the communication overhead within parallel distributed applications, it can also offer advantages for running these applications in the cloud. The partitioning is usually done as a pre-processing step before running the parallel application. As parallel hypergraph partitioning itself is a communication-intensive operation, running it in the cloud is hard and suffers from poor scalability. The thesis also investigates the scalability of parallel hypergraph partitioning algorithms in the cloud, the challenges they present, and proposes solutions to improve the cost/performance ratio for running the partitioning problem in the cloud. Our algorithms are implemented as a new hypergraph partitioning package within Zoltan. It is an open source Linux-based toolkit for parallel partitioning, load balancing and data-management designed at Sandia National Labs. The algorithms are known as FEHG and PFEHG algorithms

High-Quality Hypergraph Partitioning

This dissertation focuses on computing high-quality solutions for the NP-hard balanced hypergraph partitioning problem: Given a hypergraph and an integer $k$, partition its vertex set into $k$ disjoint blocks of bounded size, while minimizing an objective function over the hyperedges. Here, we consider the two most commonly used objectives: the cut-net metric and the connectivity metric. Since the problem is computationally intractable, heuristics are used in practice - the most prominent being the three-phase multi-level paradigm: During coarsening, the hypergraph is successively contracted to obtain a hierarchy of smaller instances. After applying an initial partitioning algorithm to the smallest hypergraph, contraction is undone and, at each level, refinement algorithms try to improve the current solution. With this work, we give a brief overview of the field and present several algorithmic improvements to the multi-level paradigm. Instead of using a logarithmic number of levels like traditional algorithms, we present two coarsening algorithms that create a hierarchy of (nearly) $n$ levels, where $n$ is the number of vertices. This makes consecutive levels as similar as possible and provides many opportunities for refinement algorithms to improve the partition. This approach is made feasible in practice by tailoring all algorithms and data structures to the $n$-level paradigm, and developing lazy-evaluation techniques, caching mechanisms and early stopping criteria to speed up the partitioning process. Furthermore, we propose a sparsification algorithm based on locality-sensitive hashing that improves the running time for hypergraphs with large hyperedges, and show that incorporating global information about the community structure into the coarsening process improves quality. Moreover, we present a portfolio-based initial partitioning approach, and propose three refinement algorithms. Two are based on the Fiduccia-Mattheyses (FM) heuristic, but perform a highly localized search at each level. While one is designed for two-way partitioning, the other is the first FM-style algorithm that can be efficiently employed in the multi-level setting to directly improve $k$-way partitions. The third algorithm uses max-flow computations on pairs of blocks to refine $k$-way partitions. Finally, we present the first memetic multi-level hypergraph partitioning algorithm for an extensive exploration of the global solution space. All contributions are made available through our open-source framework KaHyPar. In a comprehensive experimental study, we compare KaHyPar with hMETIS, PaToH, Mondriaan, Zoltan-AlgD, and HYPE on a wide range of hypergraphs from several application areas. Our results indicate that KaHyPar, already without the memetic component, computes better solutions than all competing algorithms for both the cut-net and the connectivity metric, while being faster than Zoltan-AlgD and equally fast as hMETIS. Moreover, KaHyPar compares favorably with the current best graph partitioning system KaFFPa - both in terms of solution quality and running time

A geometric partitioning method for distributed tomographic reconstruction

Tomography is a powerful technique for 3D imaging of the interior of an object. With the growing sizes of typical tomographic data sets, the computational requirements for algorithms in tomography are rapidly increasing. Parallel and distributed-memory methods for tomographic reconstruction are therefore becoming increasingly common. An underexposed aspect is the effect of the data distribution on the performance of distributed-memory reconstruction algorithms. In this work, we introduce a geometric partitioning method, which takes into account the acquisition geometry and aims to minimize the necessary communication between nodes for distributed-memory forward projection and back projection operations. These operations are crucial subroutines for an important class of reconstruction methods. We show that the choice of data distribution has a significant impact on the runtime of these methods. With our novel partitioning method we reduce the communication volume drastically compared to straightforward distributions, by up to 90% for a number of cases, and furthermore we guarantee a specified load balance

Scalable High-Quality Graph and Hypergraph Partitioning

The balanced hypergraph partitioning problem (HGP) asks for a partition of the node set of a hypergraph into $k$ blocks of roughly equal size, such that an objective function defined on the hyperedges is minimized. In this work, we optimize the connectivity metric, which is the most prominent objective function for HGP. The hypergraph partitioning problem is NP-hard and there exists no constant factor approximation. Thus, heuristic algorithms are used in practice with the multilevel scheme as the most successful approach to solve the problem: First, the input hypergraph is coarsened to obtain a hierarchy of successively smaller and structurally similar approximations. The smallest hypergraph is then initially partitioned into $k$ blocks, and subsequently, the contractions are reverted level-by-level, and, on each level, local search algorithms are used to improve the partition (refinement phase). In recent years, several new techniques were developed for sequential multilevel partitioning that substantially improved solution quality at the cost of an increased running time. These developments divide the landscape of existing partitioning algorithms into systems that either aim for speed or high solution quality with the former often being more than an order of magnitude faster than the latter. Due to the high running times of the best sequential algorithms, it is currently not feasible to partition the largest real-world hypergraphs with the highest possible quality. Thus, it becomes increasingly important to parallelize the techniques used in these algorithms. However, existing state-of-the-art parallel partitioners currently do not achieve the same solution quality as their sequential counterparts because they use comparatively weak components that are easier to parallelize. Moreover, there has been a recent trend toward simpler methods for partitioning large hypergraphs that even omit the multilevel scheme. In contrast to this development, we present two shared-memory multilevel hypergraph partitioners with parallel implementations of techniques used by the highest-quality sequential systems. Our first multilevel algorithm uses a parallel clustering-based coarsening scheme, which uses substantially fewer locking mechanisms than previous approaches. The contraction decisions are guided by the community structure of the input hypergraph obtained via a parallel community detection algorithm. For initial partitioning, we implement parallel multilevel recursive bipartitioning with a novel work-stealing approach and a portfolio of initial bipartitioning techniques to compute an initial solution. In the refinement phase, we use three different parallel improvement algorithms: label propagation refinement, a highly-localized direct $k$-way FM algorithm, and a novel parallelization of flow-based refinement. These algorithms build on our highly-engineered partition data structure, for which we propose several novel techniques to compute accurate gain values of node moves in the parallel setting. Our second multilevel algorithm parallelizes the $n$-level partitioning scheme used in the highest-quality sequential partitioner KaHyPar. Here, only a single node is contracted on each level, leading to a hierarchy with approximately $n$ levels where $n$ is the number of nodes. Correspondingly, in each refinement step, only a single node is uncontracted, allowing a highly-localized search for improvements. We show that this approach, which seems inherently sequential, can be parallelized efficiently without compromises in solution quality. To this end, we design a forest-based representation of contractions from which we derive a feasible parallel schedule of the contraction operations that we apply on a novel dynamic hypergraph data structure on-the-fly. In the uncoarsening phase, we decompose the contraction forest into batches, each containing a fixed number of nodes. We then uncontract each batch in parallel and use highly-localized versions of our refinement algorithms to improve the partition around the uncontracted nodes. We further show that existing sequential partitioning algorithms considerably struggle to find balanced partitions for weighted real-world hypergraphs. To this end, we present a technique that enables partitioners based on recursive bipartitioning to reliably compute balanced solutions. The idea is to preassign a small portion of the heaviest nodes to one of the two blocks of each bipartition and optimize the objective function on the remaining nodes. We integrated the approach into the sequential hypergraph partitioner KaHyPar and show that our new approach can compute balanced solutions for all tested instances without negatively affecting the solution quality and running time of KaHyPar. In our experimental evaluation, we compare our new shared-memory (hyper)graph partitioner Mt-KaHyPar to $25$ different graph and hypergraph partitioners on over $800$ (hyper)graphs with up to two billion edges/pins. The results indicate that already our fastest configuration outperforms almost all existing hypergraph partitioners with regards to both solution quality and running time. Our highest-quality configuration ($n$-level with flow-based refinement) achieves the same solution quality as the currently best sequential partitioner KaHyPar, while being almost an order of magnitude faster with ten threads. In addition, we optimize our data structures for graph partitioning, which improves the running times of both multilevel partitioners by almost a factor of two for graphs. As a result, Mt-KaHyPar also outperforms most of the existing graph partitioning algorithms. While the shared-memory graph partitioner KaMinPar is still faster than Mt-KaHyPar, its produced solutions are worse by $10\%$ in the median. The best sequential graph partitioner KaFFPa-StrongS computes slightly better partitions than Mt-KaHyPar (median improvement is $1\%$), but is more than an order of magnitude slower on average

Parallel sparse matrix-vector multiplies and iterative solvers

Cataloged from PDF version of article.Sparse matrix-vector multiply (SpMxV) operations are in the kernel of many scientific computing applications. Therefore, efficient parallelization of SpMxV operations is of prime importance to scientific computing community. Previous works on parallelizing SpMxV operations consider maintaining the load balance among processors and minimizing the total message volume. We show that the total message latency (start-up time) may be more important than the total message volume. We also stress that the maximum message volume and latency handled by a single processor are important communication cost metrics that should be minimized. We propose hypergraph models and hypergraph partitioning methods to minimize these four communication cost metrics in one dimensional and two dimensional partitioning of sparse matrices. Iterative methods used for solving linear systems appear to be the most common context in which SpMxV operations arise. Usually, these iterative methods apply a technique called preconditioning. Approximate inverse preconditioning—which can be applied to a large class of unsymmetric and symmetric matrices—replaces an SpMxV operation by a series of SpMxV operations. That is, a single SpMxV operation is only a piece of a larger computation in the iterative methods that use approximate inverse preconditioning. In these methods, there are interactions in the form of dependencies between the successive SpMxV operations. These interactions necessitate partitioning the matrices simultaneously in order to parallelize a full step of the subject class of iterative methods efficiently. We show that the simultaneous partitioning requirement gives rise to various matrix partitioning models depending on the iterative method used. We list the partitioning models for a number of widely used iterative methods. We propose operations to build a composite hypergraph by combining the previously proposed hypergraph models and show that partitioning the composite hypergraph models addresses the simultaneous matrix partitioning problem. We strove to demonstrate how the proposed partitioning methods—both the one that addresses multiple communication cost metrics and the other that addresses the simultaneous partitioning problem—help in practice. We implemented a library and investigated the performances of the partitioning methods. These practical investigations revealed a problem that we call message ordering problem. The problem asks how to organize the send operations to minimize the completion time of a certain class of parallel programs. We show how to solve the message ordering problem optimally under reasonable assumptions.Uçar, BoraPh.D