Multilevel Graph Partitioning Research Articles

A new hybrid GPU-CPU sparse LDLT factorization algorithm with GPU and CPU factorizing concurrently

This paper proposes a new task assignment scheme for sparse LDLT factorization on a hybrid GPU-CPU platform, and correspondingly an efficient supernodal algorithm, GPU Node and CPU Pipelining (GNCP). GNCP assigns a large number of weakly coupled tasks to GPU and CPU respectively so that GPU and CPU can execute different factorization tasks concurrently without explicit synchronization, which has not been achieved by previous researches. More precisely, based on the number of CPU threads, the global memory of GPU as well the size of the matrix, we introduce the concept of the truncation level in the partition tree built by the multi-level graph partitioning. The sum of FLOP associated with decoupled nodes in the truncation level accounts for about 65% of the whole computations in tested numerical examples. GNCP assigns the factorization tasks of each node in the truncation level to GPU. The data of each node in the truncation level are copied to the global memory of GPU and factorized entirely and independently by GPU one-by-one. Once factorization of any node in the truncation level has been completed, the cross-factorizations of this node to its ancestors and the following related operations are decoupled from other GPU tasks. These tasks are assigned to CPU threads with a reentrant pipelining strategy, accounting for about 35% of the whole calculations. Using this novel design, CPU and GPU are working concurrently and a large part of factorization is performed in overlapped time. Numerical tests for matrices from SuiteSparse matrix collection show higher performance than the well-known hybrid GPU-CPU solver CHOLMOD.

A massively parallel implicit 3D unstructured grid solver for computing turbulent flows on latest distributed memory computational architectures

An implicit unstructured grid density-based solver–PRAVAHA based on a parallel variant of the lower upper symmetric Gauss-Seidel (LUSGS) method is developed to compute large-scale engineering problems. A four-layered parallel algorithm is designed to efficiently compute three-dimensional turbulent flows on massively parallel modern multiple instruction multiple data-stream (MIMD) computational hardware. This data parallel approach achieves multiple layers of parallelism including continuity of flow solution, transfer of solution gradients, and calculation of drag/lift/solution residuals, right up to the innermost implicit LUSGS solver sub-routine, which is relatively less explored in the literature. Domain decomposition is performed using the METIS software based on multi-level graph partitioning algorithms. Non-blocking message-passing interface library functions are used to manage inter-processor communication through explicit message passing, efficiently. Super-linear scalability of the parallel solver is established on the current state-of-the-art supercomputing facility, the 838 teraflops PARAM seva on up to 6144 cores. Linear or even super-linear speedup on problems of significant size is observed even on ad-hoc parallel computing platforms like workstations and multi-node clusters, for turbulent flow simulations.

Load Balancing Based on Firefly and Ant Colony Optimization Algorithms for Parallel Computing.

With the wide application of computational fluid dynamics in various fields and the continuous growth of the complexity of the problem and the scale of the computational grid, large-scale parallel computing came into being and became an indispensable means to solve this problem. In the numerical simulation of multi-block grids, the mapping strategy from grid block to processor is an important factor affecting the efficiency of load balancing and communication overhead. The multi-level graph partitioning algorithm is an important algorithm that introduces graph network dynamic programming to solve the load-balancing problem. This paper proposed a firefly-ant compound optimization (FaCO) algorithm for the weighted fusion of two optimization rules of the firefly and ant colony algorithm. For the graph, results after multi-level graph partitioning are transformed into a traveling salesman problem (TSP). This algorithm is used to optimize the load distribution of the solution, and finally, the rough graph segmentation is projected to obtain the most original segmentation optimization results. Although firefly algorithm (FA) and ant colony optimization (ACO), as swarm intelligence algorithms, are widely used to solve TSP problems, for the problems for which swarm intelligence algorithms easily fall into local optimization and low search accuracy, the improvement of the FaCO algorithm adjusts the weight of iterative location selection and updates the location. Experimental results on publicly available datasets such as the Oliver30 dataset and the eil51 dataset demonstrated the effectiveness of the FaCO algorithm. It is also significantly better than the commonly used firefly algorithm and other algorithms in terms of the search results and efficiency and achieves better results in optimizing the load-balancing problem of parallel computing.

Correction to: Multilevel Graph Partitioning for Three-Dimensional Discrete Fracture Network Flow Simulations

The publication of this article unfortunately contained a mistake. The assignment to the affiliations of author Satish Karra was not correct

Multilevel Graph Partitioning for Three-Dimensional Discrete Fracture Network Flow Simulations

We present a topology-based method for mesh-partitioning in three-dimensional discrete fracture network (DFN) simulations that takes advantage of the intrinsic multi-level nature of a DFN. DFN models are used to simulate flow and transport through low-permeability fractured media in the subsurface by explicitly representing fractures as discrete entities. The governing equations for flow and transport are numerically integrated on computational meshes generated on the interconnected fracture networks. Modern high-fidelity DFN simulations require high-performance computing on multiple processors where performance and scalability depends partially on obtaining a high-quality partition of the mesh to balance work-loads and minimize communication across all processors. The discrete structure of a DFN naturally lends itself to various graph representations, which can be thought of as coarse-scale representations of the computational mesh. Using this concept, we develop two applications of the multilevel graph partitioning algorithm to partition the mesh of a DFN. In the first, we project a partition of the graph based on the DFN topology onto the mesh of the DFN and in the second, this DFN-based projection is used as the initial condition for further partitioning refinement of the mesh. We compare the performance of these methods with standard multi-level graph partitioning using graph-based metrics (cut, imbalance, partitioning time), computational-based metrics (FLOPS, iterations, solver time), and total run time. The DFN-based and the mesh-based partitioning methods are comparable in terms of the graph-based metrics, but the time required to obtain the partition is several orders of magnitude faster using the DFN-based partitions. The computation-based metrics show comparable performance between both methods so, in combination, the DFN-based partitions are several orders of magnitude faster than the mesh-based partition. Moreover, the method which uses the DFN-partition solution as the initial condition of the mesh partition provided cut and imbalance values that were close to the mesh-based partition but in a fraction of the time. In turn, this hybrid method outperformed both of the other methods in terms of the total run time.

Fast shared-memory streaming multilevel graph partitioning

A fast parallel graph partitioner can benefit many applications by reducing data transfers. The online methods for partitioning graphs have to be fast and they often rely on simple one-pass streaming algorithms, while the offline methods for partitioning graphs contain more involved algorithms and the most successful methods in this category belong to the multilevel approaches. In this work, we assess the feasibility of using streaming graph partitioning algorithms within the multilevel framework. Our end goal is to come up with a fast parallel offline multilevel partitioner that can produce competitive cutsize quality. We rely on a simple but fast and flexible streaming algorithm throughout the entire multilevel framework. This streaming algorithm serves multiple purposes in the partitioning process: a clustering algorithm in the coarsening, an effective algorithm for the initial partitioning, and a fast refinement algorithm in the uncoarsening. Its simple nature also lends itself easily for parallelization. The experiments on various graphs show that our approach is on the average up to 5.1x faster than the multi-threaded MeTiS, which comes at the expense of only 2x worse cutsize.

Evolutionary multi-level acyclic graph partitioning

Directed graphs are widely used to model data flow and execution dependencies in streaming applications. This enables the utilization of graph partitioning algorithms for the problem of parallelizing execution on multiprocessor architectures under hardware resource constraints. However due to program memory restrictions in embedded multiprocessor systems, applications need to be divided into parts without cyclic dependencies. We found that this can be done by a subsequent second graph partitioning step with an additional acyclicity constraint. We have four main contributions. First, we show that this more constrained version of the graph partitioning problem is NP-complete and present linear time heuristics. We then integrate them into an existing multi-level graph partitioning framework to better handle large graphs. This achieves a 9% reduction of the edge cut compared to the previous single-level algorithm. Based on this, we engineer an evolutionary algorithm to further reduce the cut, achieving a 30% reduction on average compared to the state of the art. Finally, we integrate the partitioning heuristics into a graph compiler for an embedded multiprocessor architecture and show that this can reduce the amount of communication for a real-world imaging application and thereby accelerate it by an average of 11%. It is shown that the compiler can emit optimized code for vastly different hardware platforms using the heuristics. In addition, we demonstrate how a custom fitness function for the evolutionary algorithm can be used to optimize other objectives like load balancing if the communication volume is not predominantly important on a given hardware platform.

Performance analysis of parallel stabilized mixed Galerkin method for three-dimensional transient Darcy flow using mesh reordering techniques

This study is related to parallel stabilized mixed Galerkin method for 3-D transient flow through porous media using OpenMP specifications for shared type memory architectures. The stabilized mixed Galerkin formulation employs standard finite elements and first-order backward difference for transient analysis. The computational complexity of the finite element method increases by increasing the number of elements requisite for large and complicated systems specifically for petroleum crude reservoirs, which leads to a large number of data elements for high performance. The shared memory architecture provides a platform for solving such difficult and big problems of new areas. In parallel finite element methods, the mesh reorganizing/partitioning is considered to be a significant preprocessing process to improve efficiency. The focus of the present case-study is to analyze the performances of octree-based and multilevel-based mesh reordering techniques. The analysis is carried out by running the parallel Darcy flow algorithms on AMD Opteron processors using tetrahedral and hexahedral meshes. The performance of the algorithm is estimated by computing the respective execution time, running time and algorithm efficiency. Major findings of this study are: (1) the octree-based mesh reordering outclasses multilevel graph-based method by at least 200% in terms of execution time, (2) the hexahedral meshes which are structured meshes show better scalability as compared to the tetrahedral meshes by a factor of 5, and (3) finally, octree mesh reordering demonstrates poor scalability in multithreaded environment as compared to multilevel graph partitioning method due to O(P2) overhead.

A reactive self-tuning scheme for multilevel graph partitioning

We propose a new multilevel graph bi-partitioning approach (M-RRTS) using greedy construction and reactive-randomized tabu search (RRTS). RRTS builds upon local search by adding prohibitions (to enforce diversification) and self-tuning mechanisms to adapt meta-parameters in an online manner to the instance being solved. The novel M-RRTS approach adds a multi-scale structure to the previous method. The original graph is summarized through a hierarchy of coarser graphs. At each step, more densely-interconnected nodes at a given level of the hierarchy are coalesced together. The coarsest graph is then partitioned, and uncoarsening phases followed by refinement steps build solutions at finer levels until the original graph is partitioned. A variation of RRTS is applied for the refinement of partitions after each uncoarsening phase. We investigate various building blocks of the proposed multilevel scheme, such as different initial greedy constructions, different tie-breaking options and various matching mechanisms to build the coarser levels. Detailed experimental results are presented on the benchmark graphs from Walshaw’s graph partitioning repository and potentially hard graphs. The proposed approach produces the record results for 14 of 34 graphs from the repository in lower CPU times with respect to competing approaches. These results confirm the value of the new self-tuning and multilevel strategy to rapidly adapt to new instances.

Parallel Graph Partitioning for Complex Networks

Processing large complex networks like social networks or web graphs has attracted considerable interest. To do this in parallel, we need to partition them into pieces of roughly equal size. Unfortunately, previous parallel graph partitioners originally developed for more regular mesh-like networks do not work well for complex networks. Here we address this problem by parallelizing and adapting the label propagation technique originally developed for graph clustering. By introducing size constraints, label propagation becomes applicable for both the coarsening and the refinement phase of multilevel graph partitioning. This way we exploit the hierarchical cluster structure present in many complex networks. We obtain very high quality by applying a highly parallel evolutionary algorithm to the coarsest graph. The resulting system is both more scalable and achieves higher quality than state-of-the-art systems like ParMetis or PT-Scotch. For large complex networks the performance differences are very big. As an example, our algorithm partitions a web graph with 3.3 G edges in 16 seconds using 512 cores of a high-performance cluster while producing a high quality partition—none of the competing systems can handle this graph on our system.

LBMIC: communication-aware load balancing in distributed ASMs with evolving social networks

Multi-agent-based simulation for artificial stock market (ASM) is an important method in behavioural finance. The social network in ASM will influence the coordination and decision making of the intelligent agents. To improve the performance of an ASM with evolving social networks in a distributed computing environment, the computational load balancing and inter-nodes communication should be considered jointly. This paper proposes a scheduling algorithm called LBMIC to partition the agents onto different computing nodes while keeping the degree of load imbalance lower than a given threshold with minimized inter-nodes communication between agents. LBMIC models the scheduling into a graph partitioning problem and uses the multi-level graph partitioning algorithm to achieve an efficient scheduling. When the network evolves, LBMIC refines the partitioning by migrating parts of the agents. The experiments indicate that LBMIC can efficiently improve the performance of communication-intensive ASMs by both initial partitioning and refining partitioning.

Partitioning (hierarchically clustered) complex networks via size-constrained graph clustering

The most commonly used method to tackle the graph partitioning problem in practice is the multilevel metaheuristic. In this paper we introduce size-constrained label propagation (SCLaP) and show how it can be used to instantiate both the coarsening phase and the refinement phase of multilevel graph partitioning. We mainly target networks with highly irregular and hierarchically clustered structure (but other network types can be partitioned as well). Additionally, we augment the basic algorithm with several extensions to further improve its speed and/or solution quality. Depending on the configuration of the resulting partitioner using SCLaP, we are able to compute high-quality partitions outperforming all competitors, or instead, to compute similarly good partitions as the best competitor in terms of quality, hMetis, while being an order of magnitude faster. Our fastest configuration partitions the largest real-world graph in our study (it has 3.3 billion edges) with sequential code in about ten minutes while cutting less than half of the edges than the fastest competitor, kMetis.

Balanced, non-contiguous partitioning of power systems considering operational constraints

This paper presents an integer programming based partitioning of dynamic graphs that arise in power systems. The proposed approach allows for explicitly expressing power systems operational constraints in the partitioning algorithm. Balanced, non-contiguous graphs appear in several power systems applications such as network partitioning for high performance computing based parallel transient simulators and in wide area control. Quality of partition obtained using the developed algorithm compares favorably with a well known multi-level graph partitioning approach – METIS. Several test systems, ranging from a 9 bus test system to the 2383 bus western polish test system is used to demonstrate the applicability of proposed approach for power system partitioning problems. In addition, applicability of the proposed approach for a non-stationary system such as power networks is demonstrated by partitioning in real-time. Real-time in this context is defined as the interval between two dispatches as the edge weights of the dynamic graph is expected to change at every dispatch.

A Parallel Maximal Matching Algorithm for Large Graphs Using Pregel

Graph matching is to find an independent edge set in a graph. It can be used for various purposes such as finding a cover in a graph, chemical structural computations, multi-level graph partitioning and so on. When a graph is too large to be handled by a single machine, we should use multiple machines. In this paper, we use Pregel, a cloud graph processing architecture which is able to process massive scale graph data in scalable and fault-tolerant ways. We propose a parallel maximal matching algorithm described in the Pregel's vertex-centric BSP model. We test our algorithm on an 8 node cluster and the results show that our algorithm can realize high quality matching for a large graph in a short time. Also, our algorithm is linearly scalable with the number of machines.

Multilevel Segmentation of Histopathological Images Using Cooccurrence of Tissue Objects

This paper presents a new approach for unsupervised segmentation of histopathological tissue images. This approach has two main contributions. First, it introduces a new set of high-level texture features to represent the prior knowledge of spatial organization of the tissue components. These texture features are defined on the tissue components, which are approximately represented by tissue objects, and quantify the frequency of two component types being cooccurred in a particular spatial relationship. As they are defined on components, rather than on image pixels, these object cooccurrence features are expected to be less vulnerable to noise and variations that are typically observed at the pixel level of tissue images. Second, it proposes to obtain multiple segmentations by multilevel partitioning of a graph constructed on the tissue objects and combine them by an ensemble function. This multilevel graph partitioning algorithm introduces randomization in graph construction and refinements in its multilevel scheme to increase diversity of individual segmentations, and thus, improve the final result. The experiments on 200 colon tissue images reveal that the proposed approach--the object cooccurrence features together with the multilevel segmentation algorithm--is effective to obtain high-quality results. The experiments also show that it improves the segmentation results compared to the previous approaches.

A New Coarsening Strategy for Multilevel Graph Partitioning

When applying multilevel scheme to solve the graph partitioning problem, shortcomings and limitations exist in the state-of-the-art coarsening schemes depend mainly on finding maximal matchings to obtain the coarse graphs, which can cause the multilevel algorithms to produce poor-quality solutions. This paper proposes an improved coarsening scheme by improving vertex combining strategy and edge ordering criteria. The new coarsening scheme is more effective in quality, which is proved by both theoretical analysis and experimental results.

Fast affinity propagation clustering: A multilevel approach

In this paper, we propose a novel Fast Affinity Propagation clustering approach (FAP). FAP simultaneously considers both local and global structure information contained in datasets, and is a high-quality multilevel graph partitioning method that can implement both vector-based and graph-based clustering. First, a new Fast Sampling algorithm (FS) is proposed to coarsen the input sparse graph and choose a small number of final representative exemplars. Then a density-weighted spectral clustering method is presented to partition those exemplars on the global underlying structure of data manifold. Finally, the cluster assignments of all data points can be achieved through their corresponding representative exemplars. Experimental results on two synthetic datasets and many real-world datasets show that our algorithm outperforms the state-of-the-art original affinity propagation and spectral clustering algorithms in terms of speed, memory usage, and quality on both vector-based and graph-based clustering.

Scalable clustering methods for the name disambiguation problem

When non-unique values are used as the identifier of entities, due to their homonym, confusion can occur. In particular, when (part of) “names” of entities are used as their identifier, the problem is often referred to as a name disambiguation problem, where goal is to sort out the erroneous entities due to name homonyms (e.g., If only last name is used as the identifier, one cannot distinguish “Masao Obama” from “Norio Obama”). In this paper, in particular, we study the scalability issue of the name disambiguation problem—when (1) a small number of entities with large contents or (2) a large number of entities get un-distinguishable due to homonyms. First, we carefully examine two of the state-of-the-art solutions to the name disambiguation problem and point out their limitations with respect to scalability. Then, we propose two scalable graph partitioning algorithms known as multi-level graph partitioning and multi-level graph partitioning and merging to solve the large-scale name disambiguation problem. Our claim is empirically validated via experimentation—our proposal shows orders of magnitude improvement in terms of performance while maintaining equivalent or reasonable accuracy compared to competing solutions.

Multi-level sequential circuit partitioning for test vector generation for low power test in VLSI

Sequential graph partitioning algorithms have been developed to fulfill the requirements of emerging multi-phase problems in circuit testing models. In this paper, we present a multi-level graph partitioning algorithm for circuit partitioning, which will minimize the number of test vectors during a low power test in VLSI circuits. By reducing the number of test vectors, we can reduce the energy consumption during the test. Our experimental results with ISCAS bench mark circuits have shown that the power can be reduced up to 55%.

Using graph partitioning to discover regions of correlated spatio-temporal change in evolving graphs

There is growing interest in studying dynamic graphs, or graphs that evolve with time. In this work, we investigate a new type of dynamic graph analysis - finding regions of a graph that are evolving in a similar manner and are topologically similar over a period of time. For example, these regions can be used to group a set of changes having a common cause in event detection and fault diagnosis. Prior work [6] has proposed a greedy framework called cSTAG to find these regions. It was accurate in datasets where the regions are temporally and spatially well separated. However, in cases where the regions are not well separated, cSTAG produces incorrect groupings. In this paper, we propose a new algorithm called regHunter. It treats the region discovery problem as a multi-objective optimisation problem, and it uses a multi-level graph partitioning algorithm to discover the regions of correlated change. In addition, we propose an external clustering validation technique, and use several existing internal measures to evaluate the accuracy of regHunter. Using synthetic datasets, we found regHunter is significantly more accurate than cSTAG in dynamic graphs that have regions with small separation. Using two real datasets - the access graph of the 1998 World Cup website, and the BGP connectivity graph during the landfall of Hurricane Katrina - we found regHunter obtained more accurate results than cSTAG. Furthermore, regHunter was able to discover two interesting regions for the World Cup access graph that CSTAG was not able to find.