Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc. of the 6th SIAM Conference on Parallel Processing for Scientific Computing, 1993, 445--452; Hendrickson and Leland, A Multilevel Algorithm for Partitioning Graphs, Tech. report SAND 93-1301, Sandia National Laboratories, Albuquerque, NM, 1993]. From the early work it was clear that multilevel techniques held great promise; however, it was not known if they can be made to consistently produce high quality partitions for graphs arising in a wide range of application domains. We investigate the effectiveness of many different choices for all three phases: coarsening, partition of the coarsest graph, and refinement. In particular, we present a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of the size of the final partition obtained after multilevel refinement. We also present a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening. We test our scheme on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that our scheme produces partitions that are consistently better than those produced by spectral partitioning schemes in substantially smaller time. Also, when our scheme is used to compute fill-reducing orderings for sparse matrices, it produces orderings that have substantially smaller fill than the widely used multiple minimum degree algorithm.

/pdf/a-fast-and-high-quality-multilevel-scheme-for-partitioning-2i240yved4.pdf

A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs

http://www.mathcs.emory.edu/%7Ebenzi/Web_papers/survey.pdf

Preconditioning techniques for large linear systems: a survey

http://glaros.dtc.umn.edu/gkhome/fetch/papers/pmlJPDC98.pdf

A Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering

In this paper we present a parallel formulation of a multilevel k-way graph partitioning algorithm. The multilevel k-way partitioning algorithm reduces the size of the graph by collapsing vertices and edges (coarsening phase), finds a k-way partition of the smaller graph, and then it constructs a k-way partition for the original graph by projecting and refining the partition to successively finer graphs (uncoarsening phase). A key innovative feature of our parallel formulation is that it utilizes graph coloring to effectively parallelize both the coarsening and the refinement during the uncoarsening phase. Our algorithm is able to achieve a high degree of concurrency, while maintaining the high quality partitions produced by the serial algorithm. We test our scheme on a large number of graphs from finite element methods, and transportation domains. Our parallel formulation on Cray T3D, produces high quality 128-way partitions on 128 processors in a little over two seconds, for graphs with a million vertices. Thus our parallel algorithm makes it possible to perform dynamic graph partition in adaptive computations without compromising quality.

/pdf/parallel-multilevel-k-way-partitioning-scheme-for-irregular-93yf46q02l.pdf

Parallel Multilevel k-way Partitioning Scheme for Irregular Graphs

Extracting knowledge by performing computations on graphs is becoming increasingly challenging as graphs grow in size. A standard approach distributes the graph over a cluster of nodes, but performing computations on a distributed graph is expensive if large amount of data have to be moved. Without partitioning the graph, communication quickly becomes a limiting factor in scaling the system up. Existing graph partitioning heuristics incur high computation and communication cost on large graphs, sometimes as high as the future computation itself. Observing that the graph has to be loaded into the cluster, we ask if the partitioning can be done at the same time with a lightweight streaming algorithm.We propose natural, simple heuristics and compare their performance to hashing and METIS, a fast, offline heuristic. We show on a large collection of graph datasets that our heuristics are a significant improvement, with the best obtaining an average gain of 76%. The heuristics are scalable in the size of the graphs and the number of partitions. Using our streaming partitioning methods, we are able to speed up PageRank computations on Spark, a distributed computation system, by 18% to 39% for large social networks.

/pdf/streaming-graph-partitioning-for-large-distributed-graphs-2816sufsls.pdf

Streaming graph partitioning for large distributed graphs

The design of a parallel implementation of multilevel recursive spectral bisection on the Cray T3D is described. The code is intended to be fast enough to enable dynamic repartitioning of adaptive meshes and to partition meshes that are too large for workstations. Two innovations in the implementation are recursive asynchronous task teams and a parallel version of the multilevel accelerator. A performance improvement of a factor of 140 over the best available serial implementation is demonstrated.

PMRSB: Parallel Multilevel Recursive Spectral Bisection

The design of a parallel implementation of multilevel recursive spectral bisection is described The goal is to implement a code that is fast enough to enable dynamic repartitioning of adaptive meshes

/pdf/a-parallel-implementation-of-multilevel-recursive-spectral-1vvhk2z0a4.pdf

A Parallel Implementation of Multilevel Recursive Spectral Bisection for Application to Adaptive Unstructured Meshes

A parallel MPI implementation of the Sparse Approximate Inverse (SPAI) preconditioner is described. SPAI has proven to be a highly effective preconditioner, and is inherently parallel because it computes columns (or rows) of the preconditioning matrix independently. However, there are several problems that must be addressed for an efficient MPI implementation: load balance, latency hiding, and the need for one-sided communication. The effectiveness, efficiency, and scaling behavior of our implementation will be shown for different platforms.

Stephen T. Barnard

Papers

PMRSB: Parallel Multilevel Recursive Spectral Bisection

A Parallel Implementation of Multilevel Recursive Spectral Bisection for Application to Adaptive Unstructured Meshes

A Portable MPI Implementation of the SPAI Preconditioner in ISIS