Abstract. The inverse of an irreducible sparse matrix is structurally full, so that it is impractical to think of computing or storing it. However, there are several applications where a subset of the entries of the inverse is required. Given a factorization of the sparse matrix held in out-of-core storage, we show how to compute such a subset e ciently, by accessing only parts of the factors. When there are many inverse entries to compute, we need to guarantee that the overall computation scheme has reasonable memory requirements, while minimizing the cost of loading the factors. This leads to a partitioning problem that we prove is NP-complete. We also show that we cannot get a close approximation to the optimal solution in polynomial time. We thus need to develop heuristic algorithms, and we propose: (i) a lower bound on the cost of an optimum solution; (ii) an exact algorithm for a particular case; (iii) two other heuristics for a more general case; and (iv) hypergraph partitioning models for the most general setting. We illustrate the performance of our algorithms in practice using the MUMPS software package on a set of real-life problems as well as some standard test matrices. We show that our techniques can improve the execution time by a factor of 50. Key words. Sparse matrices, direct methods for linear systems and matrix inversion, multifrontal method, graphs and hypergraphs. AMS subject classi cations. 05C50, 05C65, 65F05, 65F50 1. Introduction. We

In this paper, we present parallel multilevel algorithms for the hypergraph partitioning problem. In particular, we describe schemes for parallel coarsening, parallel greedy k-way refinement and parallel multi-phase refinement. Using an asymptotic theoretical performance model, we derive the isoefficiency function for our algorithms and hence show that they are technically scalable when the maximum vertex and hyperedge degrees are small. We conduct experiments on hypergraphs from six different application domains to investigate the empirical scalability of our algorithms both in terms of runtime and partition quality. Our findings confirm that the quality of partition produced by our algorithms is stable as the number of processors is increased while being competitive with those produced by a state-of-the-art serial multilevel partitioning tool. We also validate our theoretical performance model through an isoefficiency study. Finally, we evaluate the impact of introducing parallel multi-phase refinement into our parallel multilevel algorithm in terms of the trade off between improved partition quality and higher runtime cost.

Abstract. The efficiency of the PageRank computation is important since the constantly evolving nature of the Web requires this computation to be repeated many times. Due to the enormous size of the Web’s hyperlink structure, PageRank computations are usually carried out on parallel computers. Recently, a hypergraph-partitioning-based formulation for parallel sparse-matrix vector multiplication is proposed as a preprocessing step which will minimize the communication overhead of the parallel PageRank computations. Based on this work, we propose Website-based partitioning approaches in order to reduce the overhead of this preprocessing step. The conducted experiments show that the proposed approach produces comparable performance results for PageRank computation while achieving lower preprocessing overheads. 1

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The efficiency of parallel iterative methods for solving linear systems, arising from reallife applications, depends greatly on matrix characteristics and on the amount of parallel overhead. It is often viewed that a major part of this overhead can be caused by parallel matrix-vector multiplications. However, for difficult large linear systems, the preconditioning operations needed to accelerate convergence are to be performed in parallel and may also incur substantial overhead. To obtain an efficient preconditioning, it is desirable to consider certain matrix numerical properties in the matrix partitioning process. In general, graph partitioners consider the nonzero structure of a matrix to balance the number of unknowns and to decrease communication volume among parts. The present work builds upon hypergraph partitioning techniques because of their ability to handle nonsymmetric and irregular structured matrices and because they correctly minimize communication volume. First, several hyperedge weight schemes are proposed to account for the numerical matrix property called diagonal dominance of rows and columns. Then, an algorithm for the independent partitioning of certain submatrices followed by the matching of the obtained parts is presented in detail along with a proof that it correctly minimizes the total communication volume. For the proposed variants of hypergraph partitioning models, numerical experiments compare the iterations to converge, investigate the diagonal dominance of the obtained parts, and show the values of the partitioning cost functions. 1

We present the PaToH MATLAB Matrix Partitioning Interface. The interface provides support for hypergraph-based sparse matrix partitioning methods which are used for efficient parallelization of sparse matrix-vector multiplication operations. The interface also offers tools for visualizing and measuring the quality of a given matrix partition. We propose a novel, multilevel, 2D coarsening-based 2D matrix partitioning method and implement it using the interface. We have performed extensive comparison of the proposed method against our implementation of orthogonal recursive bisection and fine-grain methods on a large set of publicly available test matrices. The conclusion of the experiments is that the new method can compete with the fine-grain method while also suggesting new research directions.

The need to have efficient storage schemes for spatial networks is apparent when the volume of query processing in some road networks (e.g., the navigation systems) is considered. Specifically, under the assumption that the road network is stored in a central server, the adjacent data elements in the network must be clustered on the disk in such a way that the number of disk page accesses is kept minimal during the processing of network queries. In this work, we introduce the link-based storage scheme for clustered road networks and compare it with the previously proposed junction-based storage scheme. In order to investigate the performance of aggregate network queries in clustered spatial networks, we extend our recently proposed clustering hypergraph model from junctionbased storage to link-based storage. We propose techniques for additional storage savings in bidirectional networks that make the link-based storage scheme even more preferable in terms of the storage efficiency. We evaluate the performance of our link-based storage scheme against the junction-based storage scheme both theoretically and empirically. The results of the experiments conducted on a wide range of road network datasets show that the link-based storage scheme is preferable in terms of both storage and query processing efficiency.

Abstract. The modeling flexibility provided by hypergraphs has drawn a lot of interest from the combinatorial scientific community, leading to novel models and algorithms, their applications, and development of associated tools. Hypergraphs are now a standard tool in combinatorial scientific computing. The modeling flexibility of hypergraphs however, comes at a cost: algorithms on hypergraphs are inherently more complicated than those on graphs, which sometimes translates to nontrivial increases in processing times. Neither the modeling flexibility of hypergraphs, nor the runtime efficiency of graph algorithms can be overlooked. Therefore, the new research thrust should be how to cleverly trade-off between the two. This work addresses one method for this trade-off by solving the hypergraph partitioning problem by finding vertex separators on graphs. Specifically, we investigate how to solve the hypergraph partitioning problem by seeking a vertex separator on its net intersection graph (NIG), where each net of the hypergraph is represented by a vertex, and two vertices share an edge if their nets have a common vertex. We propose a vertex-weighting scheme to attain good node-balanced hypergraphs, since the NIG model cannot preserve node balancing information. Vertex-removal and vertex-splitting techniques are described to optimize cutnet and connectivity metrics, respectively, under the recursive bipartitioning paradigm. We also developed implementations of our proposed hypergraph partitioning formulations by adopting and modifying a state-of-the-art graph partitioning by vertex separator tool onmetis. Experiments conducted on a large collection of sparse matrices demonstrate the effectiveness of our proposed techniques. Key words. hypergraph partitioning; combinatorial scientific computing; graph partitioning by vertex separator; sparse matrices. AMS subject classifications.

Given a partitioning of a sparse matrix for parallel matrix–vector multiplication, which determines the total communication volume, we try to find a suitable vector partitioning that balances the communication load among the processors. We present a new lower bound for the maximum communication cost per processor, an optimal algorithm that attains this bound for the special case where each matrix column is owned by at most two processors, and a new heuristic algorithm for the general case that often attains the lower bound. This heuristic algorithm tries to avoid raising the current lower bound when assigning vector components to processors. Experimental results show that the new algorithm often improves upon the heuristic algorithm that is currently implemented in the sparse matrix partitioning package Mondriaan. Trying both heuristics combined with a greedy improvement procedure solves the problem optimally in most practical cases. The vector partitioning problem is proven to be NP-complete.

Abstract. We propose a directed hypergraph model and a refinement heuristic to distribute communicating tasks among the processing units in a distributed memory setting. The aim is to achieve load balance and minimize the maximum data sent by a processing unit. We also take two other communication metrics into account with a tie-breaking scheme. With this approach, task distributions causing an excessive use of network or a bottleneck processor which participates to almost all of the communication are avoided. We show on a large number of problem instances that our model improves the maximum data sent by a processor up to 34 % for parallel environments with 4, 16, 64 and 256 processing units compared to the state of the art which only minimizes the total communication volume.