H. D. Simon. Partitioning of unstructured mesh problems for parallel processing. Computing Systems in Engineering, 2, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we describe an ordering algorithm based on spectral methods. Spectral algorithms are important tools for solving graph partitioning problems, and they have been used in a wide range of applications, such as solving linear systems [18] domain decomposition [6] scientific numerical algorithms [20], and clustering problems [16] Spectral algorithms have also been used for ordering vertices in a graph, and in particular for the linear arrangement problem, as discussed in [13] The spectral algorithm we discuss in this section is attractive because of its simplicity and its intuitive appeal. ....

H. D. Simon. Partitioning of unstructured mesh problems for parallel processing. Computing Systems in Engineering, 2, 1991.

....each data item. In comparison, graph partitioning algorithms can be applied automatically based on the loop data access pattern. Partitioning algorithms were first developed for load balancing in parallel codes [32] Spectral partitioning methods can be effective but are computationally intensive [31]. More recently, people have employed multi level graph partitioning algorithms encapsulated in library packages such as metis [19, 20] to achieve good partitions with low overhead. These algorithms compute a succession of coarsened graphs (with fewer nodes) which approximate the original graph, ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

....depict one of the meshes used in our experimentation (we do not show the 804K mesh due to printing and resolution limitations) For this case, the freestream Mach number is 0.768 and the incidence is 1.16 degrees. We employed the recursive spectral partitioning algorithm to carry out partitioning [11, 13]. Partitioning was performed on a sequential machine as a preprocessing operation. l 0 l 1 l 2 l 3 l 4 l 5 1 LIVE all 1111 1111 1111 11 ( 11 11 11 1 LIVE any 1111 1111 1111 11 ( 11 11 11 11 11 1 1 BUFFD 11 1111 1111 1 1 = 11 11 11 11 1 HOIST 1111 1111 1111 1 Gamma 11 11 11 11 ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Pergamon Press, 1991. 12

....and data nodes. In comparison, graph partitioning algorithms can be applied automatically based on the loop data access pattern. Partitioning algorithms were first developed for load balancing in parallel codes [45] Spectral partitioning methods can be effective but are computationally intensive [44]. More recently, people have employedmulti level graph partitioning algorithms encapsulated in library packages such as METIS [26, 27] achieve good partitions with low overhead. These algorithms compute a succession of coarsened graphs (with fewer nodes) which approximate the original graph, ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

....each data item. In comparison, graph partitioning algorithms can be applied automatically based on the loop data access pattern. Partitioning algorithms were first developed for load balancing in parallel codes [30] Spectral partitioning methods can be effective but are computationally intensive [29]. More recently, people have employedmulti level graph partitioning algorithms encapsulated in library packages such as METIS [17, 18] to achieve good partitions with low overhead. These algorithms compute a succession of coarsened graphs (with fewer nodes) which approximate the original graph, ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

....information for each data item. Instead of relying on coordinate information, partitions may be computed using the underlying graph structure that is constructed by connecting data elements accessed in the same loop iteration. Spectral methods can be effective but are computationally intensive [45]. More recently, people have employed multi level graph partitioning algorithms encapsulated in library packages such as METIS [28, 29] Multi level graph partitioning algorithms work by first computing a succession of coarsened graphs (with fewer nodes) which approximate the original graph. ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

....solutions to the graph partitioning problem. Important heuristics include recursive coordinate bisection, recursive graph bisection, recursive spectral bisection, mincut based methods, clustering techniques, geometry based mapping, block based spatial decomposition, and scattered decomposition [3, 11, 12, 15]. We present genetic algorithms for graph partitioning, using new crossover operators that utilize information available from the history of genetic search. Our work is characterized by the following features: 1. Use of prior information to improve solutions. 2. Efficient partitioning of graphs ....

....as a graph partitioning strategy. 4.1 Improving solutions obtained using other methods Fast heuristic algorithms can be used to obtain an initial candidate solution which is then improved by applying the genetic algorithm. Table 1 compares the results of Recursive Spectral Bisection (RSB) [11, 12, 13] with the GA initialized by a solution obtained by the Index Based Partitioning algorithm (IBP) 10] described in the Appendix. Number of Parts 2 4 8 167 Nodes Cut Using DKNUX 20 63 109 Cut Using RSB 20 59 120 144 Nodes Cut Using DKNUX 33 65 120 Cut Using RSB 36 78 119 Table 1: A Comparison of the ....

H. Simon, "Partitioning of unstructured mesh problems for parallel processing," Proc. Conf. Parallel Methods on Large Scale Structural Analysis and Physics Applications, Pergamon Press, 1991.

....tetrahedra. For this case, the freestream Mach number is 0.768 and the incidence is 1.16 degrees. The computed Mach contours are also shown in Figure 8, where good resolution of the shock on the wing is observed. We employed the recursive spectral partitioning algorithm to carry out partitioning [33, 39]. Williams [43] compared this algorithm with binary dissection [5] and simulated annealing methods for partitioning two dimensional unstructured mesh calculations. He found that recursive spectral partitioning produced better partitions than binary dissection. Simulated annealing in some cases ....

....produced better partitions than binary dissection. Simulated annealing in some cases produced better partitions but the overhead for simulated annealing proved to be prohibitive even for the relatively small meshes employed (the largest had 5772 elements) Venkatakrishnan [42] and Simon [39] also reported favorable results with the spectral partitioner. We carried out preliminary performance comparisons between binary dissection and the recursive spectral partitioning and found that recursive spectral partitioning gave superior results on an iPSC 860 hypercube on our three ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Pergamon Press, 1991.

....achieved were about 100 Mflops. For the multigrid run, engineering solutions (3 to 4 orders of convergence) for this case could thus be obtained in roughly 30 minutes of CRAY YMP single processor CPU time. We employed the recursive spectral partitioning algorithm to carry out partitioning [10] [13]. Williams [16] compared this algorithm with binary dissection [1] and simulated annealing methods for partitioning two dimensional unstructured mesh calculations. He found that recursive spectral partitioning produced better partitions than binary dissection. Simulated annealing in some cases ....

....produced better partitions than binary dissection. Simulated annealing in some cases produced better partitions but the overhead for simulated annealing proved to be prohibitive even for the relatively small meshes employed (the largest had 5772 elements) Venkatakrishnan [14] and Simon [13] also reported favorable results with this partitioner. We carried out preliminary performance comparisons between binary dissection and the recursive spectral partitioning and found that recursive spectral partitioning gave superior results on the iPSC 860 on our three dimensional meshes. The ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

....P0 P2 P6 P4 P1 P3 P5 P7 Figure 1: The partitioning of irregular mesh However, we emphasize that good suboptimal solutions are sufficient for effective parallelization of a large class of irregular problems. There are a large number of partitioning algorithms available in the literature [2] [6], 9] Depending on the requirement application, one may be more useful than the other. The following are some important features of a partitioning algorithm. 1. Cost of partitioning vs. quality: For a given application, a cheaper algorithm generating a solution of reasonable quality may be ....

....problems. Index based mapping has been used for sorting on a two dimensional mesh [7] parallelizing quadtrees and sparse images [4] 5] and for n body simulations on parallel machines [8] The quality of the mappings produced by our algorithms is comparable to co ordinate recursive bisection [6]. Although the algorithm does not perform as well as spectral bisection methods, it is easily parallelizable and should be useful for parallelizing problems that are adaptive in nature. 2 The Mapping Problem We are given a graph G = V; E) where V represents a set of vertices, and E represents ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

....been widely observed (e.g. 5] 13] that performance on distributed memory systems can be enhanced by distributing data using a non standard format. Researchers have developed a variety of methods to obtain data mappings that are designed to optimize irregular problem communication requirements [1, 16, 18]. The distribution produced by these methods needs to be represented by a table that associates a processor assignment with each array element. This kind of distribution is often called an irregular distribution. Figure 2 depicts three different distributions of data arrays over two processors. ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Pergamon Press, 1991.

.... locality, application programmers have been taking advantage of this kind of locality when hand coding in low level parallel languages (for example with message passing) They have developed a wealth a different mapping strategies, both value based [BB87, HS91, DHU 93] and connectivity based [Sim91] which can be use as mapping strategies for irregular distributions. LPAR is a programming model for implementing numerical algorithms with a local structure, such as Particle in Cell or Multigrid, on distributed memory MIMD multiprocessors [BK93] Given a data structure which already reflects ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Pergamon Press, 1991.

....topology, due to the NP completeness, only weak criteria can be found such as minimizing the number of edges cut and the number of adjacent partitions, and balancing the computational and communicational load. Much ongoing research is devoted to improving the partitioning of unstructured grids [6][7] For the further discussion, we assume that a good partitioning for each configuration can be found. That is, each algorithm for the partitioning problem able to calculate a configuration dependent partitioning is already adaptive in the sense defined above. By that, internal and boundary ....

H. Simon, Partitioning of Unstructured Mesh Problems for Parallel processing, in Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physic Applications, Pergamon Press, 1991

....and arrays are distributed among processors in a known uniform manner. Dybbuk primitives can be called to construct a distributed data structure, the Runtime Data Graph to represent the access patterns of the distributed arrays in the loops. The runtime data graph is passed to a partitioner [7, 11, 18]. The partitioner returns the new array distributions. Once the new array distribution is obtained, loops iterations that access the distributed arrays must also be partitioned among processors to balance computation and to reduce communication. To partition the loop iterations, the primitives ....

....in 100 iterations. In these experiments, the sum of the costs associated with generating distributed data and iteration graphs, remapping arrays and preprocessing loop, are less than the cost of two iterations of the Euler solver. We used a parallelized version of Simon s eigenvalue partitioner [18] [14] for the data partitioning. Williams [21] compared this algorithm with binary dissection [1] and simulated annealing methods for partitioning two dimensional unstructured mesh calculations. He found that recursive spectral partitioning produced better partitions than binary dissection. ....

[Article contains additional citation context not shown here]

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Pergamon Press, 1991.

....such partitioning. These methods include recursive coordinate bisection, inertial bisection, scattered decomposition, geometry based partitioners, simulated annealing, mean field annealing, recursive spectral bisection, recursive spectral multisection, mincut based methods, and genetic algorithms [1, 11, 12, 14, 15, 17, 19, 23, 24, 26, 28, 29, 36, 41]. The computational graphs derived from many applications are such that the vertices correspond to two or three dimensional coordinates, and the interaction between computations is limited to physically proximate vertices. Examples of such applications include molecular dynamics, static and ....

....coordinate value along the dimension to be bisected. The computational time requirement is proportional to O(N log N ) Recursive spectral bisection (RSB) is derived from a graph bisection strategy based on the Fiedler vector (the second eigenvector of the Laplacian matrix of the given graph) [2, 36]. The computational complexity for RSB has been empirically observed to be O(N p N ) dominated by the Lanczos iterative solver used to find the bisecting eigenvector at every recursive step. Several improvements have been proposed recently to improve on the time as well as quality of the simple ....

H. Simon. Partitioning of Unstructured Mesh Problems for Parallel Processing. Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

....flexibility in the allocation of iterations of a given wavefront to processors. There are in fact sophisticated algorithms that partition array elements (and thereby iterations of parallel loops operating on these elements) across processors so as to minimize inter processor communication [10, 1]. Unfortunately, the existence of loop carried dependences limits, if not precludes, the possibility of adapting them for use in our case. Conceivably, one can still apply some form of such algorithms to iterations within a given wavefront (which are independent) but because these iterations are ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications, 1991.

....in a round robin fashion. These two data distribution schemes are often called BLOCK and CYCLIC data distributions [13] respectively. Researchers have developed a variety of heuristic methods to obtain data mappings that are designed to optimize irregular problem communication requirements [39, 44, 27, 25, 3, 17]. The distribution produced by these methods typically results in a table that lists a processor assignment for each array element. This kind of distribution is often called an irregular distribution. Partitioners typically make use of one or more of the following types of information: 1. a ....

....mesh point. For instance, a user might choose a partitioner that is based on coordinates [3] to partition data. A coordinate bisection partitioner decomposes data using the spatial location of vertices in the mesh. If the user chooses a graph based partitioner, such as the spectral partitioner [39], the connectivity of the mesh could be used to decompose the data. The next step in parallelizing this application involves assigning equal amounts of work to processors. An unstructured Euler solver consists of a sequence of loops that sweep over a mesh. Computational work associated with each ....

[Article contains additional citation context not shown here]

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Pergamon Press, 1991.

.... application programmers have already been taking advantage of value based locality when hand coding in low level parallel languages (for example with message passing) They have developed a wealth of different mapping strategies, both value based [BB87, HS91, DHU 93] and connectivity based [Sim91] which can be used as mapping strategies for irregular distributions. Run time iteration graphs can assist in improving load balance and access locality when distributing loop iterations across processors [PSC93a] High level library routines, such as CHAOS [DMS 92, DHU 93] can assist ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Pergamon Press, 1991.

....could be extended for other languages. 5.1 Language Support for Data Distribution On distributed memory machines, large data arrays need to be partitioned over the local memories of processors. Since most irregular problems do not run efficiently with standard BLOCK and CYCLIC data distributions [25], S1 REAL 8 x(N) y(N) S2 INTEGER map(N) S3 DECOMPOSITION reg(N) irreg(N) S4 DISTRIBUTE reg(block) S5 ALIGN map with reg S6 . set map array using some partitioner S7 DISTRIBUTE irreg(map) S8 ALIGN x,y with irreg Figure 6: Irregular distribution using Fortran D L2: do i = 1, n step L2: forall i ....

.... 6: Irregular distribution using Fortran D L2: do i = 1, n step L2: forall i = 1, sizeof indirection arrays S1 REDUCE(SUM, x(ia(i) y(ib(i) end forall end do Figure 7: Reduction loop in Fortran D researchers have developed a variety of heuristic methods to obtain irregular data distributions [25, 1]. The distributions produced by these methods typically result in a data structure called a maparray or translation table, that lists the processor assignment for each array element. Fortran D provides an interface that allows users to define irregular distributions and align arrays with such ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications, 1991.

.... The important deterministic heristics include recursive coordinate bisection, recursive graph bisection, recursive spectral bisection, minicut based methods, clustering techniques, geometry based mapping, block based spatial decomposition and scattered decomposition [1] 5] 6] 9] 8] 15] 17] [18] [19] Simulated Annealing Mean Field Annealing are techniques for general combinatorial optimization problems [12] 2] The SA heuristic is inherently sequential and hence hard to parallelize. Depending on the application, one may be more useful than the other. The following are some of the ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

....of interprocessor communication, 2. the computation involved in sweeping over each mesh should exhibit good load balance and 3. interpolations and projections should only require modest amounts of data movement. We have partitioned the grids in our example using the partitioner described in [35] with good results, but there are many other possible partitioners. 3.3 Adaptive Irregular Computations An adaptive irregular computation consists of a loosely synchronous computation executed repeatedly in which the data access pattern changes between iterations. The changes may be gradual, ....

....partitioners are designed using programmers a priori knowledge about a problem s computational structure and its expected computational behavior. There has been significant progress in the development of robust partitioners for static single phase loosely synchronous calculations see e.g. [35, 22] but much work remains to be done in order to deal with other problem classes. Similarly, we have proposed a scheme for integrating data partitioners into compilers that appears to be appropriate for static single and perhaps for multiphase loops [29] Much work is needed to generalize these ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

....in a roundrobin fashion. These two data distribution schemes are often called BLOCK and CYCLIC data distributions [14] respectively. Researchers have developed a variety of heuristic methods to obtain data mappings that are designed to optimize irregular problem communication requirements [37, 41, 29, 27, 3, 20]. The distribution produced by these methods typically results in a table that lists a processor assignment for each array element. This kind of distribution is often called an irregular distribution. Partitioners typically make use of one or more of the following types of information: 1. a ....

....mesh point. For instance, a user might choose a partitioner that is based on coordinates [3] to partition data. A coordinate bisection partitioner decomposes data using the spatial location of vertices in the mesh. If the user chooses a graph based partitioner, such as the spectral partitioner [37], the connectivity of the mesh could be used to decompose the data. The next step in parallelizing this application involves assigning equal amount of work to processors. An Euler solver consists of a sequence of loops that sweep over a mesh. Computational work associated with each loop must be ....

[Article contains additional citation context not shown here]

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Pergamon Press, 1991.

....[5] A simple and fast heuristic for achieving partitioning is to cluster physically proximate points in two or three dimensions. Important heuristics include recursive coordinate bisection, inertial bisection, scattered decomposition, geometry based partitioners and index based partitioners [7, 8, 11, 29, 24, 31, 32]. There are a number of methods that use explicit edge information to achieve better partitioning. Important heuristics include simulated annealing, meanfield annealing, recursive spectral bisection, recursive spectral multisection, mincut based methods, and genetic algorithms [1, 9, 10, 12, 14, ....

H. Simon. Partitioning of Unstructured Mesh Problems for Parallel Processing. Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

....set 3: 2800 point, the minimum message size in each level is 2 Theta Msg unit bytes, and the maximum size is 36 Theta Msg unit bytes. There is a large amount of literature on how to partition the task graph so as to minimize the communication cost. Many of these methods are iterative in nature, [15, 22] are a few of them (The reader is referred to [15] for a complete list) After a particular threshold any improvement in partitioning is expensive. For problems which require runtime partitioning, it is critical that this partitioning be completed extremely fast. For such problems, the gains ....

H. Simon. Partitioning of unstructured mesh problems for parallel processing. In Proceedings of the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Permagon Press, 1991.

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H. Simon, `Partitioning of unstructured mesh problems for parallel processing', Proceedingsof the Conference on Parallel Methods on Large Scale Structural Analysis and Physics Applications. Pergamon Press, 1991.

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