Multilevel Partitioning Method Research Articles

Critical Path Awareness Techniques for Large-Scale Graph Partitioning

Graph partitioning is one of the fundamental problems in many graph-based applications and systems. It enables the division of a graph into smaller sub-graphs for subsequent parallel processing, reducing the processing latency of the graph. The critical path of a graph is the logical path with the longest delay from input to output. The processing time of the graph mainly depends on the delay incurred by the critical path, independent of other paths with small delays. Therefore, it can reduce the processing time of the graph by protecting the critical path of the graph from partition. However, existing approaches to graph partitioning only focus on metrics such as minimum cut and partition balance. As a result, the critical paths of graphs may be destroyed in the partitioning procedure. To address this problem, we present a critical path awareness approach, namely path-metis, to protect the critical paths and alleviate the processing latency after graph partitioning. In path-metis, two efficient strategies, including Slack and critical path fix strategies, are introduced. The Slack strategy, which incorporates critical path information into the weights of DAG, is used as pre-processing before traditional multi-level partitioning methods, like Metis. Then, for the generated partitioning scheme, the critical path fix strategy is proposed to further protect critical paths from being cut. We demonstrate the effectiveness of our approach on both real and synthetic datasets. From the experimental results, compared to Metis, our method improves critical path performance by 17.70%.

Multilevel Partitioning with Multiple Strategies for Complex Water Distribution Network

The operation and dispatching of large-scale water supply systems is becoming more and more complex, and effective partitioning is a good response measure to this increasing complexity. This paper presents a multilevel partitioning method using freely configurable strategies suitable for multiobjective situations such as allocation of water sources, pressure regulation, and flow measurement. The method mainly relies on two strategies. Using Strategy A, water supply features are constructed based on water sources and a partition clustering is implemented that is suitable for the overall partitioning of a water supply from multiple sources. Using Strategy B, important pipes are first extracted to simplify the search path and then various constraints (pressure boundary, flow shunt, cross-river management) are used to select control points and partition by graph theory searching. This strategy is suitable for refining partitioning of the structure of a specific source. A combination of the aforementioned strategies was applied to the first-level and second-level partitioning in a case study. By flexibly configuring strategies and constraints, good partitioning is achieved and multiple management goals are accomplished satisfactorily. This research enhances understanding of the operational issues in complex water systems. The overall management of water networks can be disassembled into tasks at all levels and with their own objectives performed in stages. The solution provided in this study has the advantages of flexibility and effectiveness for a variety of complex water distribution situations, contributing insights into the real challenges of partitioning large water distribution networks.

Relaxation-Based Coarsening for Multilevel Hypergraph Partitioning

Multilevel partitioning methods that are inspired by principles of multiscaling are the most powerful practical hypergraph partitioning solvers. Hypergraph partitioning has many applications in disciplines ranging from scientific computing to data science. In this paper we introduce the concept of algebraic distance on hypergraphs and demonstrate its use as an algorithmic component in the coarsening stage of multilevel hypergraph partitioning solvers. The algebraic distance is a vertex distance measure that extends hyperedge weights for capturing the local connectivity of vertices which is critical for hypergraph coarsening schemes. The practical effectiveness of the proposed measure and corresponding coarsening scheme is demonstrated through extensive computational experiments on a diverse set of problems. Finally, we propose a benchmark of hypergraph partitioning problems to compare the quality of other solvers.

Numerical Simulation Bistatic Scattering from Three-Dimensional Conducting Rough Surface with UV Multilevel Partitioning Method Using Pulse Vector Basic Function

Vector wave three-dimensional (3D) conducting rough surface scattering problem solved by a UV method with multilevel partitioning procession (UV-MLP) using pulse vector basic function is developed in this paper, in which pulse vector basic function is more appropriately to be used for truly describe the vector inducted currents’ distribution along 3D PEC rough surface. For a 3D conducting rough surface scattering problem, the scattering structure is partitioned into multilevel blocks. By investigating the rank in the static problem, the impedance matrix for given transmitting and receiving blocks is expressed into products of U and V matrices. The UV method is illustrated by applying to a 3D scattering problem of random conducting rough surface. Finally, numerical simulation results are carried and discussed.

Wave scattering with the UV multilevel partitioning method: Three‐dimensional problem of dielectric rough‐surface scattering

AbstractA scalar wave 3D dielectric rough‐surface scattering problem solved by a UV method with multilevel partitioning (UV‐MLP) is developed in this paper. For a 3D dielectric rough‐surface scattering problem, the scattering structure is partitioned into a multilevel block. By looking up the rank in the static problem, the impedance matrix for a given transmitting and receiving block is expressed into a product of U and V matrices. The UV method is illustrated by applying a 3D scattering problem to a random dielectric rough surface in this paper. The coupled scalar integral equations for dielectric rough‐surface boundary conditions are treated. Numerical simulation results are illustrated. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1313–1317, 2006;Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.21613

Correction to “Wave scattering with the UV multilevel partitioning method: 1. Two‐dimensional problem of perfect electric conductor surface scattering”

[1] In the paper “Wave scattering with the UV multilevel partitioning method: 1. Two-dimensional problem of perfect electric conductor surface scattering” (Radio Science, 39, RS5010, doi:10.1029/2003RS003010, 2004), the following correction is needed. [2] In paragraph [34], the last sentence should read: “Then .”

Correction to “Wave scattering with UV multilevel partitioning method: 2. Three‐dimensional problem of nonpenetrable surface scattering”

Correction to “Wave scattering with UV multilevel partitioning method: 2. Three‐dimensional problem of nonpenetrable surface scattering”

Wave scattering with the UV multilevel partitioning method: 1. Two‐dimensional problem of perfect electric conductor surface scattering

A UV method with multilevel partitioning (UVMLP) is developed to solve electromagnetic problem. In this paper we demonstrate the technique to treat electromagnetic problem for large surface in a two‐dimensional (2‐D) problem. Using the multilevel partitioning (MLP), the decomposition preprocessing, and the matrix vector multiplication require CPU of O(N logN) per iteration with a smaller constant factor for matrix column multiplication than decomposition. The memory requirement is of O(N logN). We demonstrate the technique for a rough surface scattering problem with surface length up to 13,000 wavelengths and RMS height up to ten wavelengths. Computations are based on using PC with a single Pentium 4–2.4 GHz processor and 1 G RAM. For the case of 65,536 unknowns, it requires about CPU from 3.3 to 5.33 s per iteration and a total CPU of 3.9–14.3 min with 49–146 conjugate gradient iterations.

Wave scattering with UV multilevel partitioning method: 2. Three‐dimensional problem of nonpenetrable surface scattering

A UV multilevel partitioning method (UV‐MLP) is developed to solve scalar wave three‐dimensional (3‐D) scattering problem. The method consists of setting up a table of transmitting and receiving block size and their separation using fast coarse‐coarse sampling. For a specific scattering problem with given geometry, the scattering structure is partitioned into multilevel blocks. By looking up the rank in the static problem, the impedance matrix for a given transmitting and receiving block is expressed into a product of U and V matrix. In this paper the method is illustrated by applying to a 3‐D scattering problem of random nonpenetrable rough surface. The cases of Dirichelt and Neumann boundary conditions are treated. Numerical simulation results are illustrated. For 65,536 boundary unknowns on a rough surface, and using a single processor of 2.66 GHz, it takes about 34 CPU min and 1.8 Gb of memory to compute the solution using conjugate gradient iterations and multilevel UV to accelerate the matrix‐column vector multiplication.

Wave scattering with UV multilevel partitioning method for volume scattering by discrete scatterers

AbstractA UV multilevel partitioning method (UV‐MLP) is developed to solve the volume‐scattering problem. The method involves creating a rank table of transmitting and receiving block sizes and their separation. The table can be set up speedily using coarse‐coarse sampling. For a specific scattering problem with given geometry, the scattering structure is partitioned into multilevel blocks. By looking up the rank in the predetermined table, the impedance matrix for a given transmitting and receiving block is expressed as a product of U and V matrices. We illustrate the method for 2D volume scattering by discrete scatterers. Multiple scattering is cast into the Foldy–Lax equations of partial waves. We show that UV decomposition can be applied directly to the impedance matrix of partial waves of higher order than the usual lowest‐order Green's function. Numerical results are illustrated for randomly distributed cylinders that are one wavelength in diameter. For scattering by 1024 cylinders on a single PC processor with a 2.6‐GHz CPU and 2‐GB memory, only 14 CPU min are required to obtain the numerical solution. If subsectional volumetric discretization with the method of moments (MoM) is applied to this problem, the equivalent number of volumetric unknowns is 80,425. Furthermore, for 4096 cylinders that have 321,700 equivalent numbers of volumetric unknowns, it takes only 7.34 sec for one matrix‐vector multiplication. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 41: 354–361, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20140

Efficient parallel algorithms for numerical simulation

COUPL+ is a programming environment for applications using unstructured and hybrid grids for numerical simulations. It automates parallelization by handling the partitioning of data and dependent data and maintaining halo interfaces and copy coherency. We explore some algorithms behind this package. A multi-level partitioning method is described which is effective in the presence of skewed data, solving the multi-set median-finding problem. Partitioning elements over a set of pre-partitioned nodes is explored and a novel method is suggested for reducing communication in the resulting distribution.

Minimization of memory access overhead for multidimensional DSP applications via multilevel partitioning and scheduling

Massive uniform nested loops are broadly used in multidimensional digital signal processing (DSP) applications. Due to the large amount of data handled by such applications, the optimization of data accesses by fully utilizing the local memory and minimizing communication overhead is important in order to improve the overall system performance. Most of the traditional partition strategies do not consider the effect of data access on the computational performance. In this paper, a multilevel partitioning method, based on a static data scheduling technique known as carrot-hole data scheduling, is proposed to control the data traffic between different levels of memory. Based on this data schedule, optimal partition vector, scheduling vector and the partition size are chosen in such a way to minimize communication overhead. Nonhomogeneous size partitions are the final result of the partition scheme which produces a significant performance improvement. Experiments show that by using this technique, local memory misses are significantly reduced as compared to results obtained from traditional methods. This method can be used in application specific DSP system design and compiler for DSP processors.