Parallel multilevel k way partitioning scheme for irregular graphs george karypisy vipin kumary abstract. A parallel algorithm for multilevel k way hypergraph partitioning aleksandar trifunovic william j. This problem arises in the air traffic control area. The multilevel graph partitioning schemes include three phases 10, 11,12. The basic structure of a multilevel bisection algorithm is very simple. The multilevel kway partitioning algorithm reduces the size of the graph by successively collapsing vertices and edges coarsening phase, finds a kway. Global optimization of multilevel electricity market. It instantiates the multilevel approach in its most extreme version, removing only a single vertex in every level of the hierarchy. Parallel multilevel algorithms for multiconstraint graph.
This k way partitioning refinement scheme is substantially simpler and faster than either the k way fm 3, or the k pwlr algorithm 17, but is equally effective in the multi level context. A key feature of this parallel formulation is that it is able to achieve a high. Mar 16, 2007 in this paper a new graph partitioning problem is introduced, the relaxed k way graph partitioning problem. A highquality partition is calculated for this reduced graph and this partition is mapped back to the original graph. Rrts builds upon local search by adding prohibitions to enforce diversification and selftuning mechanisms to adapt metaparameters in an online manner to the instance being solved. The multilevel kway partitioning algorithm reduces the size of the graph by collapsing vertices and edges coarsening phase, finds a kway partitioning of the smaller graph, and then it constructs a kway partitioning for the original graph by projecting and refining the partition to successively finer graphs uncoarsening phase. Multilevel algorithms for multiconstraint graph partitioning. A fast and high quality multilevel scheme for partitioning irregular graphs. The idea of the multilevel technique is to reduce the magnitude of a graph by merging vertices together, compute a partition on this reduced graph, and finally project this partition on the original graph. Knottenbelt department of computing, imperial college london south kensington campus, london sw7 2az, uk email. Gggpgreedy graph growing partitioning ggpgraph growing partitioning gpaglobal path algorithm graspgreedy randomized adaptive search procedure hemheavy edge matching hypalhybrid k way graph partitioning algorithm klkernighanlin lglockgain mmgminmax greedy prpathrelinking sbspectral bisection shemsorted heavy edge matching tfmtabu based. Parallel multilevel algorithms for multiconstraint graph partitioning kirk schloegel, george karypis, and vipin kumar. Just as graphs naturally represent many kinds of information in mathematical and computer science problems, hypergraphs also arise naturally in important practical problems, including circuit layout, boolean satisability, numerical linear algebra, etc. Parallel multilevel series kway partitioning scheme for.
The general idea behind multilevel partitioning involves producing a smaller graph which is an approximation of the original input graph. It performs graph partitioning quickly, taking advantage of geometry information. Balanced graph partitioning is an npcomplete problem with a wide range of applications. We conducted extensive experiments using a variety of large graphs and data sets, and obtained very promising results. A successful heuristic for partitioning large graphs is the multilevel graph partitioning mgp approach depicted in.
A coarsegrain parallel formulation of multilevel kway graph. The multilevel kway partitioning algorithm reduces the size of the graph by successively collapsing vertices and edges coarsening phase, finds a kway partitioning of the smaller graph, and. A parallel algorithm for multilevel kway hypergraph partitioning aleksandar trifunovic william j. The k way graph partitioning problem is to split v into k. Parallel multilevel kway partitioning scheme for irregular graphs. Engineering multilevel graph partitioning algorithms. Towards a parallel diskbased algorithm for multilevel kway. The various phases of the multilevel k way partitioning algorithm. In this paper we discuss problems encountered in parallelizing different phases of multilevel k way partitioning schemes, and present a parallel formulation for the multilevel k way partitioning algorithm 17. A key feature of this parallel formulation is that it is able to achieve a high degree of concurrency while maintaining the high quality of the partitions produced by the serial multilevel k way partitioning algorithm. Applications of this multiconstraint graph partitioning problem include parallel solution of multiphysics and multiphase computations, that underly many existing and emerging. Pdf a coarsegrain parallel formulation of multilevel kway.
This kway partitioning refinement scheme is substantially simpler and faster than either the kway fm 3, or the kpwlr algorithm 17, but is equally effective in the multi level context. Multilevelkway partitioning scheme for irregular graphs. A study of the kway graph partitioning problem bruno. Section 3 compares the results produced by our algorithm to those. Builds on hendrickson and leland 1995 work, uses the same overall scheme but proposes different algorithms in each of the. A key feature of this parallel formulation is that it is able to achieve a high degree of concurrency while maintaining the high quality of the partitions produced by the serial multilevel kway partitioning.
Pdf a modified multilevel kway partitioning algorithm for trip. In this paper we present a parallel formulation of a multilevel kway graph partitioning algorithm. Graph partitioning is a wellknown optimization problem of great interest in theoretical and applied studies. A new method, the fusion fission, for the relaxed kway graph. A new graph partitioning method is presented, the fusion fission, which can be used to resolve the relaxed kway graph partitioning problem. More recently, another class of algorithms, called multilevel k way mlkw, proposes the use of the multilevel paradigm in order to directly construct a k way partitioning of a graph, following the vcycle paradigm shown in fig. Multilevel kway partitioning techniques are generally faster and provide better quality solutions than multilevel recursive bisection schemes 18. It is close to the kway, also called multiway, graph partitioning problem, but with relaxed imbalance constraints. Department of computer science and engineering, army hpc research. Nearlylinear time spectral graph reduction for scalable. Pdf multilevel kway partitioning scheme for irregular. Engineering multilevel graph partitioning algorithms peter sanders, christian schulz karlsruhe institute of technology kit, 76128 karlsruhe, germany sanders,christian.
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. In this paper we present a parallel formulation of a multilevel k way graph partitioning algorithm, that is particu larly suited for messagepassing libraries that have high latency. In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. However, since partitioning is critical in several practical applications, heuristic algorithms were developed with nearlinear runtime. In this paper we present a parallel formulation of a multilevel k way graph partitioning algorithm. The main components of the proposed algorithm and their relationship with one another are given in sections 4, 5. We present a multilevel algorithm for graph partitioning in which the graph is approximated by a sequence of increasingly.
This leads us to believe that a kway graph partitioning may be more suitable for such graph. Applications of k way graph partitioning i scheduling work on k processors. These applications include many largescale distributed problems, including the optimal storage of large set. 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 brief discussion of multilevel graph partitioning appears in section 2. In mathematics, the multilevel technique is a technique used to solve the graph partitioning problem. A successful heuristic for partitioning large graphs is the multilevel graph partition ing mgp approach depicted in figure 1 where the graph is recursively contracted to achieve smaller graphs which should re. During the last 40 years, the literature has strongly increased and big improvements have been made. A astf and high quality multilevel scheme for partitioning irregular graphs. Kahypar is a multilevel hypergraph partitioning framework providing direct kway and recursive bisection based partitioning algorithms. Furthermore, the multilevel kway partitioning algorithms implemented bymetis are 40 to 160 times faster than multilevel spectral bisection and 8 to 16 times faster than chaco multilevel.
First, the given graph g is coarsened down to a few hundred vertices coarsening phase, then a partition is computed on this smaller graph by some algorithm partitioning phase, and finally, this partition is projected back to the original graph g uncoarsening phase. Comparison of initial partitioning methods for multilevel direct k way graph partitioning with fixed vertices i maria predaria, aur elien esnarda, jean romanb auniv. These algorithms consist of three phases, namely, coarsening phase, initialpartitioning phase, and uncoarsen ing and re. Introduction the graph partitioning problem is to partition the vertices of a graph in p roughly equal partitions such that the number of edges connecting vertices in different partitions is minimized. A parallel algorithm for multilevel kway hypergraph partitioning. Pdf parallel multilevel kway partitioning scheme for. Parallel multilevel kway partitioning scheme for irregular. It is close to the k way, also called multi way, graph partitioning problem, but with relaxed imbalance constraints. A fast kernelbased multilevel algorithm for graph clustering. Given a hypergraph h, kway partitioning of h assigns vertices of h to k disjoint. Towards a parallel diskbased algorithm for multilevel k. A fast kernelbased multilevel algorithm for graph clustering inderjit dhillon.
Graphs with over half a millionvertices can be partitionedin 256 parts, in under a minuteon scienti. The multilevel kway partitioning algorithm reduces the size of the graph by collapsing vertices and edges coarsening phase, finds a kway partition of the smaller graph, and then it constructs. We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in grimm et al. Comparison of coarsening schemes for multilevel graph. Unstructured graph partitioning and sparse matrix ordering. Recently 2, 12, 16, a multilevel recursive bisection mlrb algorithm has emerged as a highly effective method for computing a kway partitioning of a graph.
It is designed especially for large numerical simulation codes. Kumar 18 also developed a multilevel kway partitioning scheme in which a kway partitioning of the coarsened graph is computed and re ned using a variation of the kl re nement scheme. In this paper, we present and study a class of graph partitioning algorithms that reduces the size of the graph by collapsing vertices and edges, we find a k way partitioning of the smaller graph, and then we uncoarsen and refine it to construct a k way partitioning for the original graph. Graph partitioning for highperformance scientific simulations advanced topics spring 2008. In many schemes, the projected partition is further improved using the fm re. The graph partitioning problem is npcomplete 3, 4 and there is no approximation algorithm with a constant ratio factor for general graphs 5. The k way graph partitioning problem is to split v. A fast and high quality multilevel scheme for partitioning. Pdf multilevel kway hypergraph partitioning hirendra.
A fast and high quality multilevel scheme for partitioning irregular graphs george karypis yand vipin kumar siam j. Reorder matrix for better locality by graph partitioning x y p0 p1 p2 p3 p0 p1 p2 p3 parallel sparse matrixvector product. 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. In this paper, we present a new multilevel kway hypergraph partitioning algorithm that substantially outperforms the existing stateoftheart kpmlr algorithm. The basic structure of a multilevel kway partitioning algorithm is very simple. Multilevel kway hypergraph partitioning proceedings of the.
A coarsegrain parallel formulation of multilevel kway graph partitioning algorithm. A coarsegrain parallel formulation of multilevel kway. But the coarsest hypergraph is now directly partitioned into k parts, and this kway partitioning is successively re. We propose a new multilevel graph bi partitioning approach mrrts using greedy construction and reactiverandomized tabu search rrts.
A new method, the fusion fission, for the relaxed k way. A coarsegrain parallel formulation of multilevel k way graph partitioning algorithm. Such movebased heuristics for k way hypergraph partitioning appear in 46, 27, 14, with renements given by 47, 58, 32, 49, 24, 10, 20, 35, 41. A parallel algorithm for multilevel kway hypergraph. Then, we apply the proposed modified multilevel kway partitioning algorithm to obtain optimal number of partitions from the developed road graph. In this paper, we present a new multilevel k way hypergraph partitioning algorithm that substantially outperforms the existing stateoftheart kpmlr algorithm for multiway partitioning, both for optimizing local as well as global objectives. Comparison of initial partitioning methods for multilevel direct kway graph partitioning with fixed vertices. We evaluate the performance of our multilevel kway partition ing algorithm both in terms of the partitioning quality as well as computational requirements on the ispd98 benchmark 16. Multilevel direct kway hypergraph partitioning with multiple. A study of the kway graph partitioning problem thesis presented in partial ful. Multilevel kway partitioning scheme for irregular graphs. Graph partitioning for highperformance scientific simulations. Related and prior work in graph partitioning is discussed in sections 3.
After applying an initial partitioning algorithm to the smallest graph, the. 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 the. Multilevel kway partitioning \ n initial partittoning phase figure 1. Graph partitioning, ordering, and clustering for multicore architectures a dissertation submitted to the faculty of the graduate school of the university of minnesota by dominique lasalle in partial fulfillment of the requirements for the degree of doctor of philosophy dr. Parallel multilevel kway partitioning scheme for irregular graphs george karypist vipin kumart abstract. First, movement of a single node across partition boundaryin a coarse graph can lead to movement of a large number of related nodes in the original graph. Edges of the original graph that cross between the groups will produce edges in the partitioned graph. Introduction to graph partitioning stanford university. A key contribution of our work is a simple and yet powerful scheme for re. Multilevel k way partitioning \ n initial partittoning phase figure 1. Graph partitioning, ordering, and clustering for multicore. Pdf a coarsegrain parallel formulation of multilevel k. Since the 1990s, many multilevel schemes have been introduced as a practical tool to solve this problem.
Recently a new class of multilevel partitioning techniques was developed 3, 12, 11, 4. A distributed algorithm for largescale graph partitioning. Comparison of initial partitioning methods for multilevel direct kway graph partitioning with fixed vertices i maria predaria, aur elien esnarda, jean romanb auniv. A new graph partitioning method is presented, the fusion fission, which can be used to resolve the relaxed k way graph. The multilevel k way partitioning algorithm reduces the size of the graph by successively collapsing vertices and edges coarsening phase, finds a k way partitioning of the smaller graph, and then it constructs a k way partitioning for the original graph by projecting and refining the partition to successively finer graphs uncoarsening phase. Graph partitioning is a theoretical subject with applications in many areas, principally. The various phases of the multilevel kway partitioning algorithm.
The algorithms it implements are based on k way multilevel graph partitioning and adaptive repartitioning. The kway graph partitioning problem is to split v into k disjoint subsets s j. We present a multilevel graph partitioning algorithm using novel local improvement algorithms and global search strategies transferred from multi. In this paper, we present and study a class of graph partitioning algorithms that reduces the size of the graph by collapsing vertices and edges, we find ak way partitioning of the smaller graph, and then we uncoarsen and refine it to construct ak way partitioning for the original graph. Unedited notes 1 graph partition a graph partition problem is to cut a graph into 2 or more good pieces. Comparison of initial partitioning methods for multilevel. A new method, the fusion fission, for the relaxed kway. The coarsening phase is to reduce the size of the graph by collapsing vertex and edge until its size is smaller than a given threshold. The multilevel kway partitioning algorithm reduces the size of the graph by collapsing vertices and edges coarsening phase, finds a kway partition of the smaller graph, and then it constructs a kway partition for the original graph by projecting and refining the partition to successively finer graphs. A key feature of this parallel formulation is that it is able to achieve a high degree of concurrency while maintaining the high quality of. In this paper, we present and study a class of graph partitioning algorithms that reduces the size of the graph by collapsing vertices and edges, we find akway partitioning of the smaller graph, and then we uncoarsen and refine it to construct akway partitioning for the original graph.
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