Padma Raghavan. Parallel ordering using edge contraction. Technical Report CS-95-293, Department of Computer Science, University of Tennessee, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....3. Developing parallel formulations of multilevel graph partitioning schemes is quite challenging. Coarsening requires that nodes connected via edges be merged together. Since the graph is distributed randomly across the processors, parallel coarsening schemes can require a lot of communication [33, 1, 22]. The Kernighan Lin refinement heuristic and its variant, that are used during the uncoarsening phase, appear serial in nature [9] and previous attempts to parallelize them have had mixed success [9, 6, 22] In this paper we present a parallel formulation for the multilevel k way partitioning ....

....the improvements are not as dramatic (somewhere between 27 and 40 on 16 processors) This is because, during k way refinement only a few vertices get moved; hence, there is limited cache reuse. 6 Related Work Developing parallel graph partitioning algorithms has received a lot of attention [9, 14, 33, 6, 18, 2, 1, 22] due to its extensive applications in many areas. However, most of this work was concentrated on parallelizing algorithms that produce poor quality partitions, such as serial algorithms based on geometric graph partitioning [14, 6] or algorithms that have very high computational requirements, ....

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Padma Raghavan. Parallel ordering using edge contraction. Technical Report CS-95-293, Department of Computer Science, University of Tennessee, 1995.

....several minutes for these problems. In this paper we present a parallel formulation of the multilevel graph partitioning and sparse matrix ordering al gorithm. A key feature of our parallel formulation (that distinguishes it from other proposed parallel formulations of multilevel algorithms [2, 1, 24, 14]) is that it partitions the vertices of the graph into x parts while distributing the overall adjacency matrix of the graph among all p processors. This mapping results in substantially smaller commu nication than one dimensional distribution for graphs with relatively high degree, especially if ....

....of up to 56 on 128 processor Cray T3D. Due to the two dimensional mapping scheme used in the parallel formulation, its asymptotic speedup is limited to O (x ) because the matching operation is performed only on the diagonal processors. In contrast, for one dimensional mapping scheme used in [24, 1, 14], the asymptotic speedup can be O(p) for large enough graphs. However, this two dimensional mapping has the following advantages. First, the actual speedup on graphs with large average degrees is quite good as shown in Figure 8. The reason is that for these graphs, the formation of the next level ....

Padma Raghavan. Parallel ordering using edge contraction. Technical Report CS-95-293, Department of Computer Science, University of Tennessee, 1995.

.... we present a parallel formulation of a graph partitioning and sparse matrix ordering algorithm that is based on a multilevel algorithm we developed recently [14] A key feature of our parallel formulation (that distinguishes it from other proposed parallel formulations of multilevel algorithms [2, 1, 22]) is that it partitions the vertices of the graph into # p parts while distributing the overall adjacency matrix of the graph among all p processors. As 1 shown in [16] this mapping is usually much better than onedimensional distribution, when no partitioning information about the graph is ....

Padma Raghavan. Parallel ordering using edge contraction. Technical Report CS-95-293, Department of Computer Science, University of Tennessee, 1995.

.... better partitions than those provided by spectral partitioning techniques [22] and are generally at least an order of magnitude faster than even the state of the art implementation of spectral techniques [3] Developing parallel graph partitioning algorithms has received a lot of attention [11, 23, 6, 13, 2, 1, 17] due to its extensive applications in many areas. However, most of this work was concentrated on algorithms based on geometric graph partitioning [11, 6] or algorithms that have very high computational requirements, such as spectral bisection [2, 1, 13] Geometric graph partitioning algorithms ....

....processor. Development of formulations of multilevel graph partitioning schemes is quite challenging. Coarsening requires that nodes connected via edges be merged together. Since the graph is distributed randomly across the processors, parallel coarsening schemes can require a lot of communication [23, 1, 17]. The Kernighan Lin refinement heuristic and its variant, that are used during the uncoarsening phase, appear serial in nature [8] and previous attempts to parallelize them have had mixed success [8, 6, 17] Recently, we developed [18] a parallel formulation for the multilevel k way partitioning ....

Padma Raghavan. Parallel ordering using edge contraction. Technical Report CS-95-293, Department of Computer Science, University of Tennessee, 1995. 8

....to be repartitioned due to the dynamic nature of the underlying computation. In such cases, having to bring the graph to one processor for repartitioning can create a serious bottleneck that could adversely impact the scalability of the overall application. Work in parallel graph partitioning [3,23,30,49,51,77,94] has been focused on geometric [30,77] spectral [3] and multilevel partitioning schemes [49,51,94] Geometric graph partitioning algorithms tend to be quite easy to parallelize. Typically, these require a parallel sorting algorithm. Spectral and multilevel partitioners are more difficult to ....

....the underlying computation. In such cases, having to bring the graph to one processor for repartitioning can create a serious bottleneck that could adversely impact the scalability of the overall application. Work in parallel graph partitioning [3,23,30,49,51,77,94] has been focused on geometric [30,77], spectral [3] and multilevel partitioning schemes [49,51,94] Geometric graph partitioning algorithms tend to be quite easy to parallelize. Typically, these require a parallel sorting algorithm. Spectral and multilevel partitioners are more difficult to parallelize. Their parallel asymptotic run ....

P. Raghavan. Parallel ordering using edge contraction. Technical Report CS-95-293, Department of Computer Science, UniversityofTennessee, 1995.

....the solution are now becoming more critical on parallel computers. The package [61] executes all phases in parallel, and there has been much recent work in finding parallel methods for performing the reordering. This has been another reason for the growth in dissection approaches (for example, see [65, 77]) Parallelism in the triangular solve can be obtained either using the identical tree to the numerical factorization [12] or by generating a tree from the sparsity pattern of the triangular factor [15] However, in order to avoid the intrinsically sequential nature of a sparse triangular solve, ....

P. Raghavan. Parallel ordering using edge contraction. Parallel Computing, 23(8):1045--1067, 1997.

.... better partitions than those provided by spectral partitioning techniques [29] and are generally at least an order of magnitude faster than even the state of the art implementation of spectral techniques [3] Developing parallel graph partitioning algorithms has received a lot of attention [13, 31, 6, 15, 2, 1, 19] due to its extensive applications in many areas. However, most of this work was concentrated on algorithms based on geometric graph partitioning [13, 6] or algorithms that have very high computational requirements, such as spectral bisection [2, 1, 15] Geometric graph partitioning algorithms ....

....processor. Development of formulations of multilevel graph partitioning schemes is quite challenging. Coarsening requires that nodes connected via edges be merged together. Since the graph is distributed randomly across the processors, parallel coarsening schemes can require a lot of communication [31, 1, 19]. The Kernighan Lin refinement heuristic and its variant, that are used during the uncoarsening phase, appear serial in nature [8] and previous attempts to parallelize them have had mixed success [8, 6, 19] Recently, we developed [20] a parallel formulation for the multilevel k way partitioning ....

Padma Raghavan. Parallel ordering using edge contraction. Technical Report CS-95-293, Department of Computer Science, University of Tennessee, 1995.

....takes several minutes for these problems. In this paper we present a parallel formulation of the multilevel graph partitioning and sparse matrix ordering algorithm. A key feature of our parallel formulation (that distinguishes it from other proposed parallel formulations of multilevel algorithms [2, 1, 24, 14]) is that it partitions the vertices of the graph into p p parts while distributing the overall adjacency matrix of the graph among all p processors. This mapping results in substantially smaller communication than one dimensional distribution for graphs with relatively high degree, especially if ....

....of up to 56 on 128 processor Cray T3D. Due to the two dimensional mapping scheme used in the parallel formulation, its asymptotic speedup is limited to O( p p) because the matching operation is performed only on the diagonal processors. In contrast, for one dimensional mapping scheme used in [24, 1, 14], the asymptotic speedup can be O(p) for large enough graphs. However, this twodimensional mapping has the following advantages. First, the actual speedup on graphs with large average degrees is quite good as shown in Figure 8. The reason is that for these graphs, the formation of the next level ....

Padma Raghavan. Parallel ordering using edge contraction. Technical Report CS-95-293, Department of Computer Science, University of Tennessee, 1995.

....solution are now becoming more critical on parallel computers. The package [118] executes all phases in parallel, and there has been much recent work in finding parallel methods for performing the reordering. This has been another reason for the growth in dissection approaches (for example, see [130, 168]) Parallelism in the triangular solve can be obtained either using the identical tree to the numerical factorization [10] or by generating a tree from the sparsity pattern of the triangular factor [12] However, in order to avoid the intrinsically sequential nature of a sparse triangular solve, ....

P. Raghavan. Parallel ordering using edge contraction. Technical Report CS95 -293, Department of Computer Science, University of Tennessee, Knoxville, Tennessee, 1995. Submitted to Parallel Computing.

....uses the multisector nodes to compute the degrees of nodes in the domains and so usually generates a better ordering than the former on the domain subgraphs. There are many examples of incomplete nested dissection in the literature [3] 5] 6] 9] 16] 17] 20] 22] 25] 27] 33] [34], including two excellent state of the art software packages, chaco from Sandia National Laboratories [18] and metis from the University of Minnesota [21] The above methods are all members of the multisection family of ordering algorithms. In the following sections we will compare incomplete ....

P. Raghavan. Parallel ordering using edge contraction. Technical Report CS-95-293, Dept. of Computer Science, The University of Tennessee, Knoxville, Tennessee, 1995.

....on a graph, first find a separator on a coarse graph and project back to the original. Early implementations include [7] and [8] Multilevel 26 SPOOLES 2. 0 : June 26, 1998 algorithms are very popular in current software including CHACO [14] 15] METIS [18] 19] BEND [16] WGGP [12] and PCO [26]. SPOOLES also includes a hybrid ordering approach called multi section [4] 5] 6] and [27] For some types of graphs, nested dissection does much better than minimum degree, for others much worse. Multisection is an ordering that uses both nested dissection and minimum degree to create an ....

P. Raghavan. Parallel ordering using edge contraction. Technical Report CS-95-293, Dept. of Computer Science, The University of Tennessee, Knoxville, Tennessee, 1995.

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Raghavan, P. (1995c), Parallel ordering using edge contraction, Technical Report CS-95-293, Department of Computer Science, University of Tennessee, Knoxville, Tennessee. Submitted to Parallel Computing.

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