## Parallel Decomposition of Unstructured FEM-Meshes (1995)

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Venue: | Concurrency: Practice & Experience |

Citations: | 42 - 15 self |

### BibTeX

@INPROCEEDINGS{Diekmann95paralleldecomposition,

author = {Ralf Diekmann and Derk Meyer and Burkhard Monien},

title = {Parallel Decomposition of Unstructured FEM-Meshes},

booktitle = {Concurrency: Practice & Experience},

year = {1995},

pages = {199--215},

publisher = {Springer}

}

### Years of Citing Articles

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### Abstract

. We present a massively parallel algorithm for static and dynamic partitioning of unstructured FEM-meshes. The method consists of two parts. First a fast but inaccurate sequential clustering is determined which is used, together with a simple mapping heuristic, to map the mesh initially onto the processors of a massively parallel system. The second part of the method uses a massively parallel algorithm to remap and optimize the mesh decomposition taking several cost functions into account. It first calculates the amount of nodes that have to be migrated between pairs of clusters in order to obtain an optimal load balancing. In a second step, nodes to be migrated are chosen according to cost functions optimizing the amount and necessary communication and other measures which are important for the numerical solution method (like for example the aspect ratio of the resulting domains). The parallel parts of the method are implemented in C under Parix to run on the Parsytec GCel systems. R...

### Citations

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Citation Context ...to other Heuristics We compare the performance of Par 2 to a number of other efficient and often used partitioning heuristics. The HS-heuristic [12] is our own implementation [36], the versions of KL =-=[25]-=-, the Inertial method (In) [38], the Spectral method (SP) [35] and the Multilevel method (ML) [19] are implemented in the Chaco 1 library of partitioning algorithms [20]. The results of Farhat's algor... |

794 | A fast and high quality multilevel scheme for partitioning irregular graphs
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Citation Context ...ad balancing. Related work: A large number of efficient graph partitioning heuristics have been designed in the past, most of them for recursive bisection but also some for direct k-partitioning. See =-=[5, 7, 12, 14, 16]-=- for overviews of different techniques. Many of the most efficient methods have been collected into the Chaco library by Hendrickson and Leland [13]. Farhat describes in [8] a simple and efficient seq... |

486 |
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Citation Context ...nates of all nodes on each processor. Both parallel versions can be used in an adaptive environment. Walshaw and Berzins apply Recursive Spectral Bisection (RSB) which was introduced by Simon et. al. =-=[21, 22]-=- to adaptive refining meshes by combining the method with a contraction scheme [24]. The sequential algorithm performs a remapping taking the existing clustering into account. The partitioning algorit... |

443 |
A multilevel algorithm for partitioning graphs
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Citation Context ...sed partitioning heuristics. The HS-heuristic [12] is our own implementation [36], the versions of KL [25], the Inertial method (In) [38], the Spectral method (SP) [35] and the Multilevel method (ML) =-=[19]-=- are implemented in the Chaco 1 library of partitioning algorithms [20]. The results of Farhat's algorithm [14] are obtained by an own implementation which is also used to determine one of the initial... |

289 |
Partitioning of unstructured problems for parallel processing
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Citation Context ...nates of all nodes on each processor. Both parallel versions can be used in an adaptive environment. Walshaw and Berzins apply Recursive Spectral Bisection (RSB) which was introduced by Simon et. al. =-=[21, 22]-=- to adaptive refining meshes by combining the method with a contraction scheme [24]. The sequential algorithm performs a remapping taking the existing clustering into account. The partitioning algorit... |

277 | A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems. Concurrency: Practice and Experience
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Citation Context ...andle adaptively refining meshes in a parallel environment. Its results are comparable to those of recursive coordinate bisection and the method is faster than recursive multilevel spectral bisection =-=[1, 20]-=-, especially if the number of processors is large [34]. IBP can easily be applied to heterogeneous computing environments like for example workstation clusters which may, additionally, change their si... |

179 |
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Citation Context ...ad balancing. Related work: A large number of efficient graph partitioning heuristics have been designed in the past, most of them for recursive bisection but also some for direct k-partitioning. See =-=[5, 7, 12, 14, 16]-=- for overviews of different techniques. Many of the most efficient methods have been collected into the Chaco library by Hendrickson and Leland [13]. Farhat describes in [8] a simple and efficient seq... |

168 |
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Citation Context ...ents one subdomain. The choice of the P nodes uses two strategies: Feder's and Green's method to determine a set of points with largest geometrical distance between each other (farthest point method) =-=[10]-=- and a pure random choice. Experiments showed for this application a slightly better cut size if random choice is used. The method produces a non-balanced clustering with good cut size, well shaped su... |

157 |
19810n the mapping problem
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Citation Context ...nd the resulting communication structure. 0 1 2 3 4 5 6 7 0 1 2 3 4 6 7 5 Fig. 1. The GEOM MAP algorithm. Bokhari's Algorithm: Bokhari introduces an iterative improvement heuristic for graph mappings =-=[3]-=-. The heuristic repeatedly exchanges pairs of nodes increasing the number of dilation-1-edges of the embedding until no further improvement is possible. It then permutes p P randomly chosen nodes and ... |

157 | Performance of dynamic load balancing algorithms for unstructured mesh calculations,” Concurrency: Practice and Experience
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Citation Context ...e results of Par 2 to other partitioning heuristics. Note that Par 2 optimizes the cost function shown in Sec. 3.2 and not directly those shown here. Function Description Ref. \Gamma 1 Total cut size =-=[5, 7, 11, 15, 20, 21, 22, 26]-=- \Gamma 2 Maximal cut size [11] \Gamma 3 Degree of cluster graph [11] \Gamma 4 Aspect ratio \Gamma 5 Shape The total cut size (\Gamma 1 ) is the sum of all mesh edges connecting different subdomains. ... |

86 | How good is recursive bisection
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Citation Context ...y recursive bisection. There are several disadvantages attached to recursive bisection. Compared to direct kpartitioning the results can be worse, especially if the number of partitions is very large =-=[39]-=-. The time (and space) needed to compute a single bisection grows linear with the number of nodes (in the best case). Thus for very large graphs that have to be mapped onto large parallel systems ther... |

82 |
The chaco user’s guide
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Citation Context ...o some for direct k-partitioning. See [5, 7, 12, 14, 16] for overviews of different techniques. Many of the most efficient methods have been collected into the Chaco library by Hendrickson and Leland =-=[13]-=-. Farhat describes in [8] a simple and efficient sequential algorithm for the partitioning of FEM-meshes. This front-technique is a breath-first-search based method which is widely used by engineers a... |

76 |
A simple and efficient automatic FEM domain decomposer
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- 1988
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Citation Context ...f the parallel FEM-simulation. But there are several other measures that are often much more important than cut size and that depend very much on the numerical solution method used for the simulation =-=[8]-=-. Examples are the aspect ratio of subdomains (i.e. the ratio between the length of the longest and the shortest boundary-segment, where a segment is a part of the boundary leading to a single neighbo... |

75 |
Automatic Mesh Partitioning
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Citation Context ...medes environment [3] is based on a geometric approach where a d-dimensional mesh is projected onto a (d+1)-dimensional sphere and partitioned by searching for a center point and cutting hyper-planes =-=[29]-=-. This sequential method uses global information on the node coordinates as well as the graph structure to obtain good partitions. Currently there are no parallel versions available and the use in an ... |

62 | Dynamic load balancing for PDE solvers an adaptive unstructured meshes
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- 1992
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Citation Context ... environment. Walshaw and Berzins apply Recursive Spectral Bisection (RSB) which was introduced by Simon et. al. [35, 38] to adaptive refining meshes by combining the method with a contraction scheme =-=[41]-=-. The sequential algorithm performs a remapping taking the existing clustering into account. Van Driessche and Roose generalize RSB and apply it to refining meshes by introducing virtual vertices and ... |

60 |
TOP/DOMDEC—A Software Tool for Mesh Partitioning and Parallel Processing
- Farhat, Simon
- 1993
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Citation Context ...for the partitioning of FEM-meshes. This front-technique is a breath-first-search based method which is widely used by engineers and influenced the development of a number of other partitioning tools =-=[9, 17]-=-. We describe it in more detail in Section 2.1, as it is, among others, used within our mesh-generation simulation. Recursive Orthogonal Bisection (ROB) and Unbalanced Recursive Bisection (URB) use no... |

46 | Mapping unstructured grid computations to massively parallel computers
- Hammond
- 1992
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Citation Context ...ad balancing. Related work: A large number of efficient graph partitioning heuristics have been designed in the past, most of them for recursive bisection but also some for direct k-partitioning. See =-=[5, 7, 12, 14, 16]-=- for overviews of different techniques. Many of the most efficient methods have been collected into the Chaco library by Hendrickson and Leland [13]. Farhat describes in [8] a simple and efficient seq... |

44 | The generalized dimension exchange method for load balancing in k-ary n-cubes and variants
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- 1995
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Citation Context ...hase again splits into a number of steps. 3.1 Flow Calculation The balancing flow calculation uses the Generalized Dimension Exchange (GDE) method which was introduced by Xu and Lau in 1991 (see e.g. =-=[27, 28]-=-). The method is completely parallel and local. It requires no global information and converges on certain networks like e.g. grids very fast. P i P j Link Exchange Load x Load y Load Load + l * y (1-... |

40 |
An improved spectral bisection algorithm and its application to dynamic load balancing
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- 1995
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Citation Context ...al algorithm performs a remapping taking the existing clustering into account. Van Driessche and Roose generalize RSB and apply it to refining meshes by introducing virtual vertices and virtual edges =-=[13]-=-. This formulation allows the minimization of node transfers in order to obtain balanced load after a refining step. Incremental Graph Partitioning (IGP) by Ou and Ranka uses a linear-programming form... |

38 | Using helpful sets to improve graph bisections
- Diekmann, Monien, et al.
- 1995
(Show Context)
Citation Context |

28 | Plassmann: Parallel algorithms for the adaptive refinement and partitioning of unstructured meshes
- Jones, Paul
- 1994
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Citation Context ...ers, used within our mesh-generation simulation. Recursive Orthogonal Bisection (ROB) and Unbalanced Recursive Bisection (URB) use node-coordinates to partition a mesh and neglect the graph structure =-=[15]-=-. Both methods are fast and easy to implement whereas URB offers larger flexibility. There are ways to parallelize both methods but the parallelization requires to hold the coordinates of all nodes on... |

26 |
Parallelism in graph–partitioning
- Savage, Wloka
- 1991
(Show Context)
Citation Context ...umbers of processors and to minimize the number of edges crossing the partition boundaries [30]. The general partitioning problem is still very hard to solve and the existing heuristics like those in =-=[10, 18, 37]-=- are often very timeand space-consuming, even if they are parallel. Therefore, the partitioning is often handled by recursive bisection. There are several disadvantages attached to recursive bisection... |

23 | Partitioning of unstructured meshes for load balancing. Concurrency: Practice and Experience - Martin, Otto - 1995 |

22 |
The PARTY partitioning—library user guide—version 1.1
- Preis, Diekmann
- 1996
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Citation Context ...24, 27, 33, 38] for overviews of different techniques. Many of the most efficient methods have been collected into libraries like Chaco by Hendrickson/Leland [20], Metis by Kumar [24], Party by Preis =-=[36]-=-, and others [15, 40, 42]. We will not discuss these algorithms and tools here but concentrate on methods explicitly designed to partition and map numerical grids. Farhat describes in [14] a simple an... |

19 | Nearest-neighbor algorithms for loadbalancing in parallel computers. Concurrency: Pract
- Xu, Lau, et al.
- 1995
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Citation Context ...hase again splits into a number of steps. 3.1 Flow Calculation The balancing flow calculation uses the Generalized Dimension Exchange (GDE) method which was introduced by Xu and Lau in 1991 (see e.g. =-=[44, 45]-=-). This iterative method is P i P j Link Exchange Load x Load y Load Load + l * y (1-l) * x x (1-l) * y + l * P i P j Link Sweep Figure 4: The basic operation of the GDE algorithm. completely parallel... |

16 | Fast and parallel mapping algorithms for irregular problems
- Ou, Ranka, et al.
- 1996
(Show Context)
Citation Context ...ed mesh calculations [40]. The partitioning with PUL-DM is done sequentially and there is currently no support for adaptive refining meshes. Ou, Ranka and Fox introduce Index Based Partitioning (IBP) =-=[32, 34]-=- where they reduce the partitioning problem to index sorting which is solved by a parallel version of SampleSort. Like ROB and URB this method ignores structural information but can be used to handle ... |

14 | Combining Helpful Sets and Parallel Simulated Annealing for the Graph-Partitioning Problem.” Parallel Algorithms and
- Diekmann, Luling, et al.
- 1996
(Show Context)
Citation Context ...umbers of processors and to minimize the number of edges crossing the partition boundaries [30]. The general partitioning problem is still very hard to solve and the existing heuristics like those in =-=[10, 18, 37]-=- are often very timeand space-consuming, even if they are parallel. Therefore, the partitioning is often handled by recursive bisection. There are several disadvantages attached to recursive bisection... |

13 |
On robust and adaptive multigrid methods
- Bastian, Wittum
- 1994
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Citation Context ...eir convexity. Some of the most efficient numerical methods use adaptively refining meshes which change their structure during runtime in order to adapt to the characteristics of the physical problem =-=[1]-=-. Parallel adaptive environments have to include the ability to cope with these changing meshes. The new method: The heuristic described in this paper tries to optimize the shape of subdomains in addi... |

13 | An analytical comparison of nearest neighbor algorithms for load balancing in parallel computers
- Xu, Monien, et al.
- 1995
(Show Context)
Citation Context ...hase again splits into a number of steps. 3.1 Flow Calculation The balancing flow calculation uses the Generalized Dimension Exchange (GDE) method which was introduced by Xu and Lau in 1991 (see e.g. =-=[27, 28]-=-). The method is completely parallel and local. It requires no global information and converges on certain networks like e.g. grids very fast. P i P j Link Exchange Load x Load y Load Load + l * y (1-... |

13 |
Kropf: A fast distributed Mapping Algorithm
- Boillat, G
- 1990
(Show Context)
Citation Context ...for example, a value ofs= 1 1+sin(��=k 1 ) leads to an optimal number of O(k 1 ) sweeps. It can be shown [45] that the GDE method converges asymptotically faster than for example the Diffusion met=-=hod [4]-=-. Unfortunately, the optimal -values are not known for arbitrary graphs and wrong values lead to decreased convergence rates. But if the algorithm is run on the processor network (grid) it can happen ... |

12 | Automated parallel solution of unstructured PDE problems, to appear. Available from http://www.cs.cmu.edu/afs/cs.cmu.edu/project/qu~e/public/ papers/pde.color.ps
- Ghattas, Miller, et al.
(Show Context)
Citation Context ...the method with a contraction scheme [24]. The sequential algorithm performs a remapping taking the existing clustering into account. The partitioning algorithm included in the Archimedes environment =-=[2]-=- is based on a geometric approach where a d-dimensional mesh is projected onto a (d + 1)-dimensional sphere and partitioned by searching for a center point and cutting hyper-planes. This sequential me... |

12 | A parallelisable algorithm for optimising unstructured mesh partitions. Mathematics research report
- Walshaw, Cross, et al.
- 1995
(Show Context)
Citation Context ...mbedding by a pairwise exchange of nodes until a local minimum of a cost function counting the amount of resulting communication is achieved. Finally Walshaw et. al. designed the software tool JOSTLE =-=[25]-=- which is able to support adaptively changing meshes. The method uses Farhat's algorithm for a sequential clustering, tries to optimize the subdomain shapes, restores an optimal load balancing and fin... |

12 |
and Sanjay Ranka. Parallel incremental graph partitioning using linear programming
- Ou
- 1994
(Show Context)
Citation Context ...balancing. 1.2 Related work A large number of efficient graph partitioning heuristics have been designed in the past, most of them for recursive bisection but also some for direct k-partitioning. See =-=[10, 12, 18, 21, 24, 27, 33, 38]-=- for overviews of different techniques. Many of the most efficient methods have been collected into libraries like Chaco by Hendrickson/Leland [20], Metis by Kumar [24], Party by Preis [36], and other... |

11 |
Automatic load balanced partitioning strategies for PDE computations
- Chrisochoides, Houstis, et al.
- 1989
(Show Context)
Citation Context ...e the shape is based on a migration of nodes lying far away from the center of gravity. We normalize the distance of a node from the center of its cluster by the radius of the corresponding subdomain =-=[4]-=- and set the change in shape to the changing value of this measure for the originating and the destinating cluster of a node: f(v; D j ) = d(ffi i ; v) r i \Gamma 1 + d(ffi j ; v) r j \Gamma 1 Then th... |

10 | Distributed combinatorial optimization
- Diekmann, Luling, et al.
- 1993
(Show Context)
Citation Context ...apped onto a smaller one (the processor network) [19]. Unfortunately, this mapping problem is NP-complete and there exist almost no efficient sequential or parallel heuristics solving it sufficiently =-=[6]-=-. With growing performance of interconnection networks and especially with the establishment of independent routing networks, it is appropriate to reduce the mapping problem to the task of partitionin... |

10 |
PUL-TUF Prototype User Guide
- Trewin
- 1993
(Show Context)
Citation Context ...ode-exchange strategy. The method is neither parallel nor used in an adaptive environment. PUL-SM and PUL-DM are software tools designed at the EPCC to support parallel unstructured mesh calculations =-=[23]-=-. The partitioning with PUL-DM is done sequentially and there is currently no support for adaptive refining meshes. Ou and Ranka introduce Index Based Partitioning (IBP) [20] where they reduce the par... |

10 | Padfem: A portable parallel FEM-tool
- Diekmann, Dralle, et al.
- 1996
(Show Context)
Citation Context ..., or their convexity. Especially preconditioned conjugate gradient solvers (PCG) which use the subdomain-structure to determine preconditioning matrices usually take benefit from well shaped clusters =-=[9]-=-. Their convergence behavior is determined by the form of subdomains and the smoothness of inner boundaries. Some of the most efficient numerical methods use adaptively refining meshes which change th... |

10 | Communication throughput of interconnection networks
- Diekmann, uling, et al.
- 1994
(Show Context)
Citation Context ...d if the communication structure of the application produces no "hot-spots", then a combination of \Gamma 2 and \Gamma 3 can give a measure for the time needed to complete a single communica=-=tion step [31]-=-. Certain load balancing heuristics minimize \Gamma 2 explicitly [33]. \Gamma 3 is often minimized indirectly as some partitioning heuristics optimize the dilation if the cluster graph is mapped onto ... |

7 |
and Sanjay Ranka. Parallel remapping algorithms for adaptive problems
- Ou
- 1994
(Show Context)
Citation Context ...uctured mesh calculations [23]. The partitioning with PUL-DM is done sequentially and there is currently no support for adaptive refining meshes. Ou and Ranka introduce Index Based Partitioning (IBP) =-=[20]-=- where they reduce the partitioning problem to index sorting which is solved by a parallel version of Sample-Sort. Like ROB and URB this method ignores structural information but can be used to handle... |

6 | A parallel localsearch algorithm for the k-partitioning problem
- Diekmann, Luling, et al.
- 1995
(Show Context)
Citation Context |

6 | Efficient Use of Parallel & Distributed Systems: From Theory to Practice
- Monien, Diekmann, et al.
- 1995
(Show Context)
Citation Context ...The mesh decomposition and assignments of subdomains to processors can be modeled as graph embedding problem where a large graph (the mesh) has to be mapped onto a smaller one (the processor network) =-=[19]-=-. Unfortunately, this mapping problem is NP-complete and there exist almost no efficient sequential or parallel heuristics solving it sufficiently [6]. With growing performance of interconnection netw... |

5 | Lonsdale: Comparative Efficiencies of Domain Decompositions
- Floros, Reeve, et al.
- 1994
(Show Context)
Citation Context ...e results of Par 2 to other partitioning heuristics. Note that Par 2 optimizes the cost function shown in Sec. 3.2 and not directly those shown here. Function Description Ref. \Gamma 1 Total cut size =-=[5, 7, 11, 15, 20, 21, 22, 26]-=- \Gamma 2 Maximal cut size [11] \Gamma 3 Degree of cluster graph [11] \Gamma 4 Aspect ratio \Gamma 5 Shape The total cut size (\Gamma 1 ) is the sum of all mesh edges connecting different subdomains. ... |

4 |
Run-Time Load Balancing Techniques for a Parallel Unstructured Multi-Grid Euler Solver with Adaptive Grid Refinement
- Keyser, Roose
- 1995
(Show Context)
Citation Context ...er boundaries. Some of the most efficient numerical methods use adaptively refining meshes which change their structure during runtime in order to adapt to the characteristics of the physical problem =-=[2, 8]-=-. Parallel adaptive environments have to include the ability to cope with these changing meshes, too. 1.1 The new method The heuristic described in this paper tries to optimize the shape of subdomains... |

3 | Ranka: Mapping unstructured Computational Graphs for Adaptive and Nonumiform Computational Environments
- Kaddoura, Ou, et al.
- 1995
(Show Context)
Citation Context ...cessors is large [34]. IBP can easily be applied to heterogeneous computing environments like for example workstation clusters which may, additionally, change their size and structure during run-time =-=[23]-=-. Hammond's Cyclic Pairwise Exchange (CPE) heuristic [18] maps the mesh randomly on a hypercube and optimizes the embedding by a pairwise exchange of nodes until a local minimum of a cost function cou... |

3 |
Reinefeld: Distributed Combinatorial Optimization
- Diekmann, L��uling, et al.
- 1993
(Show Context)
Citation Context ...mapped onto a smaller one (the processor network) [30]. Unfortunately, this mapping problem is NPcomplete and there exist almost no efficient sequential or parallel heuristics solving it sufficiently =-=[11]-=-. With growing performance of interconnection networks and especially with the establishment of independent routing networks, it is appropriate to reduce the mapping problem to the task of partitionin... |

2 |
Automatische Netzeinteilungsalgorithmen zum effektiven Einsatz der parallelen Substrukturtechnik
- Lengen, Krome
- 1994
(Show Context)
Citation Context ...for the partitioning of FEM-meshes. This front-technique is a breath-first-search based method which is widely used by engineers and influenced the development of a number of other partitioning tools =-=[9, 17]-=-. We describe it in more detail in Section 2.1, as it is, among others, used within our mesh-generation simulation. Recursive Orthogonal Bisection (ROB) and Unbalanced Recursive Bisection (URB) use no... |

2 |
Dynamic Load-balancing for PDE solvers on apative unstructured meshes. Concurrency: Practice and Experience 7(1
- Walshaw, Berzins
- 1995
(Show Context)
Citation Context ... environment. Walshaw and Berzins apply Recursive Spectral Bisection (RSB) which was introduced by Simon et. al. [21, 22] to adaptive refining meshes by combining the method with a contraction scheme =-=[24]-=-. The sequential algorithm performs a remapping taking the existing clustering into account. The partitioning algorithm included in the Archimedes environment [2] is based on a geometric approach wher... |

1 |
Parallel Grid Generation on
- Chrisochoides, Fox, et al.
- 1993
(Show Context)
Citation Context ...code the mesh generation should be done in parallel, too. But as mesh generation itself is a very complex problem and an area of active research, only a small number of parallel mesh generators exist =-=[7]-=-. If the mesh generation is performed sequentially or if existing meshes are supposed to be used, the elements have to be distributed over the processors of the massively parallel system (which is sup... |

1 |
Datenparallele k-Partitionierung unstrukturierter Finite Elemente Netze
- Meyer
- 1995
(Show Context)
Citation Context ...ent information the structure of the cluster graph may change and sometimes a cluster may split into disconnected components. These problems can be avoided by a careful choice of nodes to be migrated =-=[28]. Eli-=-minating Node-Chains: Near domain boundaries, especially near inner boundaries but sometimes also between subdomain boundaries, the described node migration strategies may produce small node "off... |